diff --git a/packs/erdos-open-problems/data/erdos_open_problems.md b/packs/erdos-open-problems/data/erdos_open_problems.md index 46e175f..4e80101 100644 --- a/packs/erdos-open-problems/data/erdos_open_problems.md +++ b/packs/erdos-open-problems/data/erdos_open_problems.md @@ -1,62 +1,57 @@ # Erdos Open Problems (Active Snapshot) -- generated_at_utc: `2026-03-05T15:52:31Z` +- generated_at_utc: `2026-05-27T07:44:18Z` - source_url: `https://erdosproblems.com/range/1-end` -- total_open: `691` +- total_open: `671` -- [#1](https://erdosproblems.com/1) — OPEN | $500 | number theory, additive combinatorics — edited: 23 January 2026 — If $A\subseteq \{1,\ldots,N\}$ with $\lvert A\rvert=n$ is such that the subset sums $\sum_{a\in S}a$ are distinct for all $S\subseteq A$... -- [#3](https://erdosproblems.com/3) — OPEN | $5000 | number theory, additive combinatorics, arithmetic progressions — edited: 23 January 2026 — If $A\subseteq \mathbb{N}$ has $\sum_{n\in A}\frac{1}{n}=\infty$ then must $A$ contain arbitrarily long arithmetic progressions? +- [#1](https://erdosproblems.com/1) — OPEN | $500 | number theory, additive combinatorics — edited: 06 April 2026 — If $A\subseteq \{1,\ldots,N\}$ with $\lvert A\rvert=n$ is such that the subset sums $\sum_{a\in S}a$ are distinct for all $S\subseteq A$... +- [#3](https://erdosproblems.com/3) — OPEN | $5000 | number theory, additive combinatorics, arithmetic progressions — edited: 04 April 2026 — If $A\subseteq \mathbb{N}$ has $\sum_{n\in A}\frac{1}{n}=\infty$ then must $A$ contain arbitrarily long arithmetic progressions? - [#5](https://erdosproblems.com/5) — OPEN | number theory, primes — Let $C\geq 0$. Is there an infinite sequence of $n_i$ such that\[\lim_{i\to \infty}\frac{p_{n_i+1}-p_{n_i}}{\log n_i}=C?\] - [#7](https://erdosproblems.com/7) — VERIFIABLE | $25 | number theory, covering systems — edited: 22 January 2026 — Is there a distinct covering system all of whose moduli are odd? -- [#9](https://erdosproblems.com/9) — OPEN | number theory, additive basis, primes — edited: 20 January 2026 — Let $A$ be the set of all odd integers $\geq 1$ not of the form $p+2^{k}+2^l$ (where $k,l\geq 0$ and $p$ is prime). Is the upper density... -- [#10](https://erdosproblems.com/10) — OPEN | number theory, additive basis, primes — edited: 24 January 2026 — Is there some $k$ such that every large integer is the sum of a prime and at most $k$ powers of 2? -- [#11](https://erdosproblems.com/11) — FALSIFIABLE | number theory, additive basis — edited: 20 January 2026 — Is every large odd integer $n$ the sum of a squarefree number and a power of 2? -- [#12](https://erdosproblems.com/12) — OPEN | number theory — Let $A$ be an infinite set such that there are no distinct $a,b,c\in A$ such that $a\mid (b+c)$ and $b,c>a$. Is there such an $A$ with\[\... +- [#9](https://erdosproblems.com/9) — OPEN | number theory, additive basis, primes — edited: 07 April 2026 — Let $A$ be the set of all odd integers $\geq 1$ not of the form $p+2^{k}+2^l$ (where $k,l\geq 0$ and $p$ is prime). Is the upper density... +- [#10](https://erdosproblems.com/10) — OPEN | number theory, additive basis, primes — edited: 11 April 2026 — Is there some $k$ such that every large integer is the sum of a prime and at most $k$ powers of 2? +- [#11](https://erdosproblems.com/11) — OPEN | number theory, additive basis — edited: 05 April 2026 — Is every large odd integer $n$ the sum of a squarefree number and a power of 2? +- [#12](https://erdosproblems.com/12) — OPEN | number theory — edited: 08 April 2026 — Let $A$ be an infinite set such that there are no distinct $a,b,c\in A$ such that $a\mid (b+c)$ and $b,c>a$. Is there such an $A$ with\[\... - [#14](https://erdosproblems.com/14) — OPEN | number theory, sidon sets, additive combinatorics — edited: 14 September 2025 — Let $A\subseteq \mathbb{N}$. Let $B\subseteq \mathbb{N}$ be the set of integers which are representable in exactly one way as the sum of... - [#15](https://erdosproblems.com/15) — OPEN | number theory, primes — Is it true that\[\sum_{n=1}^\infty(-1)^n\frac{n}{p_n}\]converges, where $p_n$ is the sequence of primes? - [#17](https://erdosproblems.com/17) — OPEN | number theory, primes — edited: 28 December 2025 — Are there infinitely many primes $p$ such that every even number $n\leq p-3$ can be written as a difference of primes $n=q_1-q_2$ where $... -- [#18](https://erdosproblems.com/18) — OPEN | $250 | number theory, divisors, factorials — edited: 20 January 2026 — We call $m$ practical if every integer $1\leq n0$,\[h(N) = N^{1/2}+O_\epsilon(N^... +- [#28](https://erdosproblems.com/28) — OPEN | $500 | number theory, additive basis — edited: 06 April 2026 — If $A\subseteq \mathbb{N}$ is such that $A+A$ contains all but finitely many integers then $\limsup 1_A\ast 1_A(n)=\infty$. +- [#30](https://erdosproblems.com/30) — OPEN | $1000 | number theory, sidon sets, additive combinatorics — edited: 06 April 2026 — Let $h(N)$ be the maximum size of a Sidon set in $\{1,\ldots,N\}$. Is it true that, for every $\epsilon>0$,\[h(N) = N^{1/2}+O_\epsilon(N^... - [#32](https://erdosproblems.com/32) — OPEN | $50 | number theory, additive basis — edited: 23 January 2026 — Is there a set $A\subset\mathbb{N}$ such that\[\lvert A\cap\{1,\ldots,N\}\rvert = o((\log N)^2)\]and such that every large integer can be... - [#33](https://erdosproblems.com/33) — OPEN | number theory, additive basis — edited: 27 December 2025 — Let $A\subset\mathbb{N}$ be such that every large integer can be written as $n^2+a$ for some $a\in A$ and $n\geq 0$. What is the smallest... - [#36](https://erdosproblems.com/36) — OPEN | number theory, additive combinatorics — edited: 23 January 2026 — Find the optimal constant $c>0$ such that the following holds. For all sufficiently large $N$, if $A\sqcup B=\{1,\ldots,2N\}$ is a partit... -- [#38](https://erdosproblems.com/38) — OPEN | number theory — edited: 16 September 2025 — Does there exist $B\subset\mathbb{N}$ which is not an additive basis, but is such that for every set $A\subseteq\mathbb{N}$ of Schnirelma... -- [#39](https://erdosproblems.com/39) — OPEN | $500 | number theory, sidon sets, additive combinatorics — edited: 23 January 2026 — Is there an infinite Sidon set $A\subset \mathbb{N}$ such that\[\lvert A\cap \{1\ldots,N\}\rvert \gg_\epsilon N^{1/2-\epsilon}\]for all $... +- [#39](https://erdosproblems.com/39) — OPEN | $500 | number theory, sidon sets, additive combinatorics — edited: 06 April 2026 — Is there an infinite Sidon set $A\subset \mathbb{N}$ such that\[\lvert A\cap \{1\ldots,N\}\rvert \gg_\epsilon N^{1/2-\epsilon}\]for all $... - [#40](https://erdosproblems.com/40) — OPEN | $500 | number theory, additive basis — For what functions $g(N)\to \infty$ is it true that\[\lvert A\cap \{1,\ldots,N\}\rvert \gg \frac{N^{1/2}}{g(N)}\]implies $\limsup 1_A\ast... -- [#41](https://erdosproblems.com/41) — OPEN | $500 | number theory, sidon sets, additive combinatorics — edited: 23 January 2026 — Let $A\subset\mathbb{N}$ be an infinite set such that the triple sums $a+b+c$ are all distinct for $a,b,c\in A$ (aside from the trivial c... -- [#42](https://erdosproblems.com/42) — OPEN | number theory, sidon sets, additive combinatorics — edited: 23 January 2026 — Let $M\geq 1$ and $N$ be sufficiently large in terms of $M$. Is it true that for every Sidon set $A\subset \{1,\ldots,N\}$ there is anoth... -- [#43](https://erdosproblems.com/43) — OPEN | $100 | number theory, sidon sets, additive combinatorics — edited: 20 December 2025 — If $A,B\subset \{1,\ldots,N\}$ are two Sidon sets such that $(A-A)\cap(B-B)=\{0\}$ then is it true that\[ \binom{\lvert A\rvert}{2}+\bino... +- [#41](https://erdosproblems.com/41) — OPEN | $500 | number theory, sidon sets, additive combinatorics — edited: 06 April 2026 — Let $A\subset\mathbb{N}$ be an infinite set such that the triple sums $a+b+c$ are all distinct for $a,b,c\in A$ (aside from the trivial c... - [#44](https://erdosproblems.com/44) — OPEN | number theory, sidon sets, additive combinatorics — edited: 09 January 2026 — Let $N\geq 1$ and $A\subset \{1,\ldots,N\}$ be a Sidon set. Is it true that, for any $\epsilon>0$, there exist $M$ and $B\subset \{N+1,\l... - [#50](https://erdosproblems.com/50) — OPEN | $250 | number theory — Schoenberg proved that for every $c\in [0,1]$ the density of\[\{ n\in \mathbb{N} : \phi(n)0$\[\max( \lvert A+A\rvert,\lvert AA\rvert)\gg_\epsilon \lvert A... +- [#52](https://erdosproblems.com/52) — OPEN | $250 | number theory, additive combinatorics — edited: 08 April 2026 — Let $A$ be a finite set of integers. Is it true that for every $\epsilon>0$\[\max( \lvert A+A\rvert,\lvert AA\rvert)\gg_\epsilon \lvert A... - [#60](https://erdosproblems.com/60) — OPEN | graph theory, cycles — edited: 18 November 2025 — Does every graph on $n$ vertices with $>\mathrm{ex}(n;C_4)$ edges contain $\gg n^{1/2}$ many copies of $C_4$? -- [#61](https://erdosproblems.com/61) — OPEN | graph theory — edited: 23 January 2026 — For any graph $H$ is there some $c=c(H)>0$ such that every graph $G$ on $n$ vertices that does not contain $H$ as an induced subgraph con... +- [#61](https://erdosproblems.com/61) — OPEN | graph theory — edited: 10 April 2026 — For any graph $H$ is there some $c=c(H)>0$ such that every graph $G$ on $n$ vertices that does not contain $H$ as an induced subgraph con... - [#62](https://erdosproblems.com/62) — OPEN | graph theory — edited: 23 January 2026 — If $G_1,G_2$ are two graphs with chromatic number $\aleph_1$ then must there exist a graph $G$ whose chromatic number is $4$ (or even $\a... -- [#64](https://erdosproblems.com/64) — FALSIFIABLE | $1000 | graph theory, cycles — edited: 18 January 2026 — Does every finite graph with minimum degree at least 3 contain a cycle of length $2^k$ for some $k\geq 2$? +- [#64](https://erdosproblems.com/64) — FALSIFIABLE | $1000 | graph theory, cycles — edited: 10 April 2026 — Does every finite graph with minimum degree at least 3 contain a cycle of length $2^k$ for some $k\geq 2$? - [#65](https://erdosproblems.com/65) — OPEN | graph theory, cycles — edited: 08 February 2026 — Let $G$ be a graph with $n$ vertices and $kn$ edges, and $a_10$ if $n$ is sufficiently large and... - [#77](https://erdosproblems.com/77) — OPEN | $250 | graph theory, ramsey theory — edited: 08 February 2026 — If $R(k)$ is the Ramsey number for $K_k$, the minimal $n$ such that every $2$-colouring of the edges of $K_n$ contains a monochromatic co... - [#78](https://erdosproblems.com/78) — OPEN | $100 | graph theory, ramsey theory — edited: 23 January 2026 — Let $R(k)$ be the Ramsey number for $K_k$, the minimal $n$ such that every $2$-colouring of the edges of $K_n$ contains a monochromatic c... -- [#80](https://erdosproblems.com/80) — OPEN | graph theory, ramsey theory — Let $c>0$ and let $f_c(n)$ be the maximal $m$ such that every graph $G$ with $n$ vertices and at least $cn^2$ edges, where each edge is c... +- [#80](https://erdosproblems.com/80) — OPEN | graph theory, ramsey theory — edited: 07 April 2026 — Let $c>0$ and let $f_c(n)$ be the maximal $m$ such that every graph $G$ with $n$ vertices and at least $cn^2$ edges, where each edge is c... - [#81](https://erdosproblems.com/81) — OPEN | graph theory — edited: 28 December 2025 — Let $G$ be a chordal graph on $n$ vertices - that is, $G$ has no induced cycles of length greater than $3$. Can the edges of $G$ be parti... -- [#82](https://erdosproblems.com/82) — OPEN | graph theory — edited: 06 October 2025 — Let $F(n)$ be maximal such that every graph on $n$ vertices contains a regular induced subgraph on at least $F(n)$ vertices. Prove that $... +- [#82](https://erdosproblems.com/82) — OPEN | graph theory — edited: 10 April 2026 — Let $F(n)$ be maximal such that every graph on $n$ vertices contains a regular induced subgraph on at least $F(n)$ vertices. Prove that $... - [#84](https://erdosproblems.com/84) — OPEN | graph theory, cycles — The cycle set of a graph $G$ on $n$ vertices is a set $A\subseteq \{3,\ldots,n\}$ such that there is a cycle in $G$ of length $\ell$ if a... -- [#85](https://erdosproblems.com/85) — FALSIFIABLE | graph theory, ramsey theory — edited: 06 December 2025 — Let $n\geq 4$ and $f(n)$ be minimal such that every graph on $n$ vertices with minimal degree $\geq f(n)$ contains a $C_4$. Is it true th... +- [#85](https://erdosproblems.com/85) — OPEN | graph theory, ramsey theory — edited: 06 December 2025 — Let $n\geq 4$ and $f(n)$ be minimal such that every graph on $n$ vertices with minimal degree $\geq f(n)$ contains a $C_4$. Is it true th... - [#86](https://erdosproblems.com/86) — OPEN | $100 | graph theory — edited: 27 December 2025 — Let $Q_n$ be the $n$-dimensional hypercube graph (so that $Q_n$ has $2^n$ vertices and $n2^{n-1}$ edges). Is it true that every subgraph... - [#87](https://erdosproblems.com/87) — OPEN | graph theory, ramsey theory — edited: 17 January 2026 — Let $\epsilon >0$. Is it true that, if $k$ is sufficiently large, then\[R(G)>(1-\epsilon)^kR(k)\]for every graph $G$ with chromatic numbe... - [#89](https://erdosproblems.com/89) — OPEN | $500 | geometry, distances — edited: 23 January 2026 — Does every set of $n$ distinct points in $\mathbb{R}^2$ determine $\gg n/\sqrt{\log n}$ many distinct distances? -- [#90](https://erdosproblems.com/90) — OPEN | $500 | geometry, distances — edited: 23 January 2026 — Does every set of $n$ distinct points in $\mathbb{R}^2$ contain at most $n^{1+O(1/\log\log n)}$ many pairs which are distance 1 apart? -- [#91](https://erdosproblems.com/91) — OPEN | geometry, distances — edited: 16 January 2026 — Let $n$ be a sufficently large integer. Suppose $A\subset \mathbb{R}^2$ has $\lvert A\rvert=n$ and minimises the number of distinct dista... -- [#92](https://erdosproblems.com/92) — OPEN | $500 | geometry, distances — edited: 28 December 2025 — Let $f(n)$ be maximal such that there exists a set $A$ of $n$ points in $\mathbb{R}^2$ in which every $x\in A$ has at least $f(n)$ points... +- [#91](https://erdosproblems.com/91) — OPEN | geometry, distances — edited: 13 April 2026 — Let $n$ be a sufficiently large integer. Suppose $A\subset \mathbb{R}^2$ has $\lvert A\rvert=n$ and minimises the number of distinct dist... - [#96](https://erdosproblems.com/96) — OPEN | geometry, distances, convex — edited: 23 January 2026 — If $n$ points in $\mathbb{R}^2$ form a convex polygon then there are $O(n)$ many pairs which are distance $1$ apart. - [#97](https://erdosproblems.com/97) — FALSIFIABLE | $100 | geometry, distances, convex — edited: 27 October 2025 — Does every convex polygon have a vertex with no other $4$ vertices equidistant from it? - [#98](https://erdosproblems.com/98) — OPEN | geometry, distances — edited: 15 October 2025 — Let $h(n)$ be such that any $n$ points in $\mathbb{R}^2$, with no three on a line and no four on a circle, determine at least $h(n)$ dist... @@ -66,19 +61,18 @@ - [#102](https://erdosproblems.com/102) — OPEN | geometry — Let $c>0$ and $h_c(n)$ be such that for any $n$ points in $\mathbb{R}^2$ such that there are $\geq cn^2$ lines each containing more than... - [#103](https://erdosproblems.com/103) — OPEN | geometry, distances — Let $h(n)$ count the number of incongruent sets of $n$ points in $\mathbb{R}^2$ which minimise the diameter subject to the constraint tha... - [#104](https://erdosproblems.com/104) — OPEN | $100 | geometry — Given $n$ points in $\mathbb{R}^2$ the number of distinct unit circles containing at least three points is $o(n^2)$. -- [#106](https://erdosproblems.com/106) — FALSIFIABLE | geometry — Draw $n$ squares inside the unit square with no common interior point. Let $f(n)$ be the maximum possible sum of the side-lengths of the... -- [#107](https://erdosproblems.com/107) — FALSIFIABLE | $500 | geometry, convex — edited: 23 January 2026 — Let $f(n)$ be minimal such that any $f(n)$ points in $\mathbb{R}^2$, no three on a line, contain $n$ points which form the vertices of a... +- [#106](https://erdosproblems.com/106) — FALSIFIABLE | geometry — edited: 06 March 2026 — Draw $n$ squares inside the unit square with no common interior point. Let $f(n)$ be the maximum possible sum of the side-lengths of the... +- [#107](https://erdosproblems.com/107) — FALSIFIABLE | $500 | geometry, convex — edited: 11 April 2026 — Let $f(n)$ be minimal such that any $f(n)$ points in $\mathbb{R}^2$, no three on a line, contain $n$ points which form the vertices of a... - [#108](https://erdosproblems.com/108) — OPEN | graph theory, chromatic number, cycles — edited: 23 January 2026 — For every $r\geq 4$ and $k\geq 2$ is there some finite $f(k,r)$ such that every graph of chromatic number $\geq f(k,r)$ contains a subgra... - [#111](https://erdosproblems.com/111) — OPEN | graph theory, chromatic number, set theory — If $G$ is a graph let $h_G(n)$ be defined such that any subgraph of $G$ on $n$ vertices can be made bipartite after deleting at most $h_G... - [#112](https://erdosproblems.com/112) — OPEN | graph theory, ramsey theory — Let $k=k(n,m)$ be minimal such that any directed graph on $k$ vertices must contain either an independent set of size $n$ or a transitive... -- [#114](https://erdosproblems.com/114) — OPEN | $250 | polynomials, analysis — edited: 23 January 2026 — If $p(z)\in\mathbb{C}[z]$ is a monic polynomial of degree $n$ then is the length of the curve $\{ z\in \mathbb{C} : \lvert p(z)\rvert=1\}... +- [#114](https://erdosproblems.com/114) — FALSIFIABLE | $250 | polynomials, analysis — edited: 23 January 2026 — If $p(z)\in\mathbb{C}[z]$ is a monic polynomial of degree $n$ then is the length of the curve $\{ z\in \mathbb{C} : \lvert p(z)\rvert=1\}... - [#117](https://erdosproblems.com/117) — OPEN | group theory — edited: 23 January 2026 — Let $h(n)$ be minimal such that any group $G$ with the property that any subset of $>n$ elements contains some $x\neq y$ such that $xy=yx... - [#119](https://erdosproblems.com/119) — OPEN | $100 | analysis, polynomials — edited: 23 January 2026 — Let $z_i$ be an infinite sequence of complex numbers such that $\lvert z_i\rvert=1$ for all $i\geq 1$, and for $n\geq 1$ let\[p_n(z)=\pro... - [#120](https://erdosproblems.com/120) — OPEN | $100 | combinatorics — edited: 23 January 2026 — Let $A\subseteq\mathbb{R}$ be an infinite set. Must there be a set $E\subset \mathbb{R}$ of positive measure which does not contain any s... -- [#122](https://erdosproblems.com/122) — OPEN | number theory — edited: 01 February 2026 — For which number theoretic functions $f$ is it true that, for any $F(n)$ such that $f(n)/F(n)\to 0$ for almost all $n$, there are infinit... +- [#122](https://erdosproblems.com/122) — OPEN | number theory — edited: 01 April 2026 — For which number theoretic functions $f$ is it true that, for any $F(n)$ such that $F(n)/f(n)\to 0$ for almost all $n$, there are infinit... - [#123](https://erdosproblems.com/123) — OPEN | $250 | number theory — edited: 20 January 2026 — Let $a,b,c\geq 1$ be three integers which are pairwise coprime. Is every large integer the sum of distinct integers of the form $a^kb^lc^... - [#124](https://erdosproblems.com/124) — OPEN | number theory, base representations, complete sequences — edited: 01 December 2025 — For any $d\geq 1$ and $k\geq 0$ let $P(d,k)$ be the set of integers which are the sum of distinct powers $d^i$ with $i\geq k$. Let $3\leq... -- [#125](https://erdosproblems.com/125) — OPEN | number theory, base representations — Let $A = \{ \sum\epsilon_k3^k : \epsilon_k\in \{0,1\}\}$ be the set of integers which have only the digits $0,1$ when written base $3$, a... - [#126](https://erdosproblems.com/126) — OPEN | $250 | number theory — Let $f(n)$ be maximal such that if $A\subseteq\mathbb{N}$ has $\lvert A\rvert=n$ then $\prod_{a\neq b\in A}(a+b)$ has at least $f(n)$ dis... - [#128](https://erdosproblems.com/128) — FALSIFIABLE | $250 | graph theory — edited: 31 October 2025 — Let $G$ be a graph with $n$ vertices such that every induced subgraph on $\geq \lfloor n/2\rfloor$ vertices has more than $n^2/50$ edges.... - [#129](https://erdosproblems.com/129) — OPEN | graph theory, ramsey theory — Let $R(n;k,r)$ be the smallest $N$ such that if the edges of $K_N$ are $r$-coloured then there is a set of $n$ vertices which does not co... @@ -86,39 +80,38 @@ - [#131](https://erdosproblems.com/131) — OPEN | number theory — edited: 30 September 2025 — Let $F(N)$ be the maximal size of $A\subseteq\{1,\ldots,N\}$ such that no $a\in A$ divides the sum of any distinct elements of $A\backsla... - [#132](https://erdosproblems.com/132) — OPEN | $100 | distances — Let $A\subset \mathbb{R}^2$ be a set of $n$ points. Must there be two distances which occur at least once but between at most $n$ pairs o... - [#137](https://erdosproblems.com/137) — OPEN | number theory, powerful — edited: 20 January 2026 — We say that $N$ is powerful if whenever $p\mid N$ we also have $p^2\mid N$. Let $k\geq 3$. Can the product of any $k$ consecutive positiv... -- [#138](https://erdosproblems.com/138) — OPEN | $500 | additive combinatorics — edited: 28 December 2025 — Let the van der Waerden number $W(k)$ be such that whenever $N\geq W(k)$ and $\{1,\ldots,N\}$ is $2$-coloured there must exist a monochro... +- [#138](https://erdosproblems.com/138) — OPEN | $500 | additive combinatorics — edited: 10 April 2026 — Let the van der Waerden number $W(k)$ be such that whenever $N\geq W(k)$ and $\{1,\ldots,N\}$ is $2$-coloured there must exist a monochro... - [#141](https://erdosproblems.com/141) — OPEN | additive combinatorics, primes, arithmetic progressions — edited: 28 September 2025 — Let $k\geq 3$. Are there $k$ consecutive primes in arithmetic progression? -- [#142](https://erdosproblems.com/142) — OPEN | $10000 | additive combinatorics, arithmetic progressions — edited: 23 January 2026 — Let $r_k(N)$ be the largest possible size of a subset of $\{1,\ldots,N\}$ that does not contain any non-trivial $k$-term arithmetic progr... -- [#143](https://erdosproblems.com/143) — OPEN | $500 | primitive sets — Let $A\subset (1,\infty)$ be a countably infinite set such that for all $x\neq y\in A$ and integers $k\geq 1$ we have\[ \lvert kx -y\rver... +- [#142](https://erdosproblems.com/142) — OPEN | $10000 | additive combinatorics, arithmetic progressions — edited: 04 April 2026 — Let $r_k(N)$ be the largest possible size of a subset of $\{1,\ldots,N\}$ that does not contain any non-trivial $k$-term arithmetic progr... +- [#143](https://erdosproblems.com/143) — OPEN | $500 | primitive sets — edited: 24 April 2026 — Let $A\subset (1,\infty)$ be a countably infinite set such that for all $x\neq y\in A$ and integers $k\geq 1$ we have\[ \lvert kx -y\rver... - [#145](https://erdosproblems.com/145) — OPEN | number theory — edited: 19 October 2025 — Let $s_10$ such that $$R(C_4,K_n) \ll n^{2-c}.$$ +- [#159](https://erdosproblems.com/159) — OPEN | $100 | graph theory, ramsey theory — edited: 07 March 2026 — There exists some constant $c>0$ such that $$R(C_4,K_n) \ll n^{2-c}.$$ - [#160](https://erdosproblems.com/160) — OPEN | additive combinatorics, arithmetic progressions — edited: 02 December 2025 — Let $h(N)$ be the smallest $k$ such that $\{1,\ldots,N\}$ can be coloured with $k$ colours so that every four-term arithmetic progression... - [#161](https://erdosproblems.com/161) — OPEN | $500 | combinatorics, hypergraphs, ramsey theory, discrepancy — edited: 16 January 2026 — Let $\alpha\in[0,1/2)$ and $n,t\geq 1$. Let $F^{(t)}(n,\alpha)$ be the smallest $m$ such that we can $2$-colour the edges of the complete... - [#162](https://erdosproblems.com/162) — OPEN | combinatorics, ramsey theory, discrepancy — edited: 30 December 2025 — Let $\alpha>0$ and $n\geq 1$. Let $F(n,\alpha)$ be the largest $k$ such that there exists some 2-colouring of the edges of $K_n$ in which... -- [#165](https://erdosproblems.com/165) — OPEN | $250 | graph theory, ramsey theory — edited: 28 December 2025 — Give an asymptotic formula for $R(3,k)$. +- [#165](https://erdosproblems.com/165) — OPEN | $250 | graph theory, ramsey theory — edited: 07 March 2026 — Give an asymptotic formula for $R(3,k)$. - [#167](https://erdosproblems.com/167) — FALSIFIABLE | graph theory — edited: 13 October 2025 — If $G$ is a graph with at most $k$ edge disjoint triangles then can $G$ be made triangle-free after removing at most $2k$ edges? -- [#168](https://erdosproblems.com/168) — OPEN | additive combinatorics — edited: 24 October 2025 — Let $F(N)$ be the size of the largest subset of $\{1,\ldots,N\}$ which does not contain any set of the form $\{n,2n,3n\}$. What is\[ \lim... -- [#169](https://erdosproblems.com/169) — OPEN | additive combinatorics, arithmetic progressions — Let $k\geq 3$ and $f(k)$ be the supremum of $\sum_{n\in A}\frac{1}{n}$ as $A$ ranges over all sets of positive integers which do not cont... +- [#168](https://erdosproblems.com/168) — OPEN | additive combinatorics — edited: 23 March 2026 — Let $F(N)$ be the size of the largest subset of $\{1,\ldots,N\}$ which does not contain any set of the form $\{n,2n,3n\}$. What is\[ \lim... +- [#169](https://erdosproblems.com/169) — OPEN | additive combinatorics, arithmetic progressions — edited: 04 April 2026 — Let $k\geq 3$ and $f(k)$ be the supremum of $\sum_{n\in A}\frac{1}{n}$ as $A$ ranges over all sets of positive integers which do not cont... - [#170](https://erdosproblems.com/170) — OPEN | additive combinatorics — Let $F(N)$ be the smallest possible size of $A\subset \{0,1,\ldots,N\}$ such that $\{0,1,\ldots,N\}\subset A-A$. Find the value of\[\lim_... -- [#172](https://erdosproblems.com/172) — OPEN | additive combinatorics, ramsey theory — edited: 01 February 2026 — Is it true that in any finite colouring of $\mathbb{N}$ there exist arbitrarily large finite $A$ such that all sums and products of disti... +- [#172](https://erdosproblems.com/172) — OPEN | additive combinatorics, ramsey theory — edited: 06 April 2026 — Is it true that in any finite colouring of $\mathbb{N}$ there exist arbitrarily large finite $A$ such that all sums and products of disti... - [#173](https://erdosproblems.com/173) — OPEN | geometry, ramsey theory — edited: 16 October 2025 — In any $2$-colouring of $\mathbb{R}^2$, for all but at most one triangle $T$, there is a monochromatic congruent copy of $T$. - [#174](https://erdosproblems.com/174) — OPEN | geometry, ramsey theory — edited: 16 October 2025 — A finite set $A\subset \mathbb{R}^n$ is called Ramsey if, for any $k\geq 1$, there exists some $d=d(A,k)$ such that in any $k$-colouring... -- [#176](https://erdosproblems.com/176) — OPEN | additive combinatorics, arithmetic progressions, discrepancy — edited: 28 December 2025 — Let $N(k,\ell)$ be the minimal $N$ such that for any $f:\{1,\ldots,N\}\to\{-1,1\}$ there must exist a $k$-term arithmetic progression $P$... +- [#176](https://erdosproblems.com/176) — OPEN | additive combinatorics, arithmetic progressions, discrepancy — edited: 04 April 2026 — Let $N(k,\ell)$ be the minimal $N$ such that for any $f:\{1,\ldots,N\}\to\{-1,1\}$ there must exist a $k$-term arithmetic progression $P$... - [#177](https://erdosproblems.com/177) — OPEN | discrepancy, arithmetic progressions — Find the smallest $h(d)$ such that the following holds. There exists a function $f:\mathbb{N}\to\{-1,1\}$ such that, for every $d\geq 1$,... - [#180](https://erdosproblems.com/180) — OPEN | graph theory, turan number — edited: 18 January 2026 — If $\mathcal{F}$ is a finite set of finite graphs then $\mathrm{ex}(n;\mathcal{F})$ is the maximum number of edges a graph on $n$ vertice... - [#181](https://erdosproblems.com/181) — OPEN | graph theory, ramsey theory — Let $Q_n$ be the $n$-dimensional hypercube graph (so that $Q_n$ has $2^n$ vertices and $n2^{n-1}$ edges). Prove that\[R(Q_n) \ll 2^n.\] -- [#183](https://erdosproblems.com/183) — OPEN | $250 | graph theory, ramsey theory — Let $R(3;k)$ be the minimal $n$ such that if the edges of $K_n$ are coloured with $k$ colours then there must exist a monochromatic trian... -- [#184](https://erdosproblems.com/184) — OPEN | graph theory, cycles — Any graph on $n$ vertices can be decomposed into $O(n)$ many edge-disjoint cycles and edges. -- [#187](https://erdosproblems.com/187) — OPEN | additive combinatorics, ramsey theory, arithmetic progressions — edited: 01 January 2026 — Find the best function $f(d)$ such that, in any 2-colouring of the integers, at least one colour class contains an arithmetic progression... +- [#183](https://erdosproblems.com/183) — OPEN | $250 | graph theory, ramsey theory — edited: 10 April 2026 — Let $R(3;k)$ be the minimal $n$ such that if the edges of $K_n$ are coloured with $k$ colours then there must exist a monochromatic trian... +- [#184](https://erdosproblems.com/184) — OPEN | graph theory, cycles — edited: 01 April 2026 — Any graph on $n$ vertices can be decomposed into $O(n)$ many edge-disjoint cycles and edges. +- [#187](https://erdosproblems.com/187) — OPEN | additive combinatorics, ramsey theory, arithmetic progressions — edited: 04 April 2026 — Find the best function $f(d)$ such that, in any 2-colouring of the integers, at least one colour class contains an arithmetic progression... - [#188](https://erdosproblems.com/188) — OPEN | geometry, ramsey theory — edited: 14 October 2025 — What is the smallest $k$ such that $\mathbb{R}^2$ can be red/blue coloured with no pair of red points unit distance apart, and no $k$-ter... - [#190](https://erdosproblems.com/190) — OPEN | additive combinatorics, arithmetic progressions — edited: 27 October 2025 — Let $H(k)$ be the smallest $N$ such that in any finite colouring of $\{1,\ldots,N\}$ (into any number of colours) there is always either... - [#193](https://erdosproblems.com/193) — OPEN | geometry — Let $S\subseteq \mathbb{Z}^3$ be a finite set and let $A=\{a_1,a_2,\ldots,\}\subset \mathbb{Z}^3$ be an infinite $S$-walk, so that $a_{i+... @@ -126,21 +119,21 @@ - [#196](https://erdosproblems.com/196) — OPEN | arithmetic progressions — Must every permutation of $\mathbb{N}$ contain a monotone 4-term arithmetic progression? In other words, given a permutation $x$ of $\mat... - [#197](https://erdosproblems.com/197) — OPEN | arithmetic progressions — Can $\mathbb{N}$ be partitioned into two sets, each of which can be permuted to avoid monotone 3-term arithmetic progressions? - [#200](https://erdosproblems.com/200) — OPEN | primes, arithmetic progressions — Does the longest arithmetic progression of primes in $\{1,\ldots,N\}$ have length $o(\log N)$? -- [#201](https://erdosproblems.com/201) — OPEN | additive combinatorics, arithmetic progressions — Let $G_k(N)$ be such that any set of $N$ integers contains a subset of size at least $G_k(N)$ which does not contain a $k$-term arithmeti... -- [#202](https://erdosproblems.com/202) — OPEN | covering systems — edited: 23 January 2026 — Let $n_1<\cdots < n_r\leq N$ with associated $a_i\pmod{n_i}$ such that the congruence classes are disjoint (that is, every integer is $\e... +- [#201](https://erdosproblems.com/201) — OPEN | additive combinatorics, arithmetic progressions — edited: 08 April 2026 — Let $G_k(N)$ be such that any set of $N$ integers contains a subset of size at least $G_k(N)$ which does not contain a $k$-term arithmeti... +- [#202](https://erdosproblems.com/202) — OPEN | covering systems — edited: 06 April 2026 — Let $n_1<\cdots < n_r\leq N$ with associated $a_i\pmod{n_i}$ such that the congruence classes are disjoint (that is, every integer is $\e... - [#203](https://erdosproblems.com/203) — OPEN | primes, covering systems — edited: 20 January 2026 — Is there an integer $m\geq 1$ with $(m,6)=1$ such that none of $2^k3^\ell m+1$ are prime, for any $k,\ell\geq 0$? - [#208](https://erdosproblems.com/208) — OPEN | number theory — edited: 19 October 2025 — Let $s_10$ and large $n$,\[s_{n+1}-s_n \ll_\epsilo... - [#212](https://erdosproblems.com/212) — OPEN | geometry, distances — edited: 16 October 2025 — Is there a dense subset of $\mathbb{R}^2$ such that all pairwise distances are rational? - [#213](https://erdosproblems.com/213) — OPEN | geometry, distances — edited: 16 October 2025 — Let $n\geq 4$. Are there $n$ points in $\mathbb{R}^2$, no three on a line and no four on a circle, such that all pairwise distances are i... -- [#217](https://erdosproblems.com/217) — OPEN | geometry, distances — edited: 02 October 2025 — For which $n$ are there $n$ points in $\mathbb{R}^2$, no three on a line and no four on a circle, which determine $n-1$ distinct distance... +- [#217](https://erdosproblems.com/217) — OPEN | geometry, distances — edited: 13 April 2026 — For which $n$ are there $n$ points in $\mathbb{R}^2$, no three on a line and no four on a circle, which determine $n-1$ distinct distance... - [#218](https://erdosproblems.com/218) — OPEN | number theory, primes — edited: 29 January 2026 — Let $d_n=p_{n+1}-p_n$. The set of $n$ such that $d_{n+1}\geq d_n$ has density $1/2$, and similarly for $d_{n+1}\leq d_n$. Furthermore, th... - [#222](https://erdosproblems.com/222) — OPEN | number theory, squares — Let $n_10$. Is it true that, for any sufficiently large $x$, there exist more than $c_1\log x$ many consecutive primes $\leq x$ such... -- [#241](https://erdosproblems.com/241) — OPEN | $100 | additive combinatorics, sidon sets — edited: 30 September 2025 — Let $f(N)$ be the maximum size of $A\subseteq \{1,\ldots,N\}$ such that the sums $a+b+c$ with $a,b,c\in A$ are all distinct (aside from t... -- [#242](https://erdosproblems.com/242) — FALSIFIABLE | number theory, unit fractions — edited: 28 January 2026 — For every $n>2$ there exist distinct integers $1\leq x2$ there exist distinct integers $1\leq x1$. Does the set of integers of the form $p+\lfloor C^k\rfloor$, for some prime $p$ and $k\geq 0$, have density $>0$? - [#247](https://erdosproblems.com/247) — OPEN | number theory, irrationality — edited: 20 January 2026 — Let $1\leq a_10$ such that, for any $K>1$, whenever $A$ is a sufficiently large finite multiset of positive integers with $\su... - [#313](https://erdosproblems.com/313) — OPEN | number theory, unit fractions — Are there infinitely many solutions to\[\frac{1}{p_1}+\cdots+\frac{1}{p_k}=1-\frac{1}{m},\]where $m\geq 2$ is an integer and $p_1<\cdots<... - [#317](https://erdosproblems.com/317) — OPEN | number theory, unit fractions — edited: 06 January 2026 — Is there some constant $c>0$ such that for every $n\geq 1$ there exists some $\delta_k\in \{-1,0,1\}$ for $1\leq k\leq n$ with\[0< \left\... -- [#318](https://erdosproblems.com/318) — OPEN | number theory, unit fractions — edited: 20 December 2025 — Let $A\subseteq \mathbb{N}$ be an infinite arithmetic progression and $f:A\to \{-1,1\}$ be a non-constant function. Must there exist a fi... - [#319](https://erdosproblems.com/319) — OPEN | number theory, unit fractions — What is the size of the largest $A\subseteq \{1,\ldots,N\}$ such that there is a function $\delta:A\to \{-1,1\}$ such that\[\sum_{n\in A}... - [#320](https://erdosproblems.com/320) — OPEN | number theory, unit fractions — edited: 28 September 2025 — Let $S(N)$ count the number of distinct sums of the form $\sum_{n\in A}\frac{1}{n}$ for $A\subseteq \{1,\ldots,N\}$. Estimate $S(N)$. - [#321](https://erdosproblems.com/321) — OPEN | number theory, unit fractions — edited: 28 September 2025 — What is the size of the largest $A\subseteq \{1,\ldots,N\}$ such that all sums $\sum_{n\in S}\frac{1}{n}$ are distinct for $S\subseteq A$? - [#322](https://erdosproblems.com/322) — OPEN | number theory, powers — edited: 30 September 2025 — Let $k\geq 3$ and $A\subset \mathbb{N}$ be the set of $k$th powers. What is the order of growth of $1_A^{(k)}(n)$, i.e. the number of rep... - [#323](https://erdosproblems.com/323) — OPEN | number theory, powers — Let $1\leq m\leq k$ and $f_{k,m}(x)$ denote the number of integers $\leq x$ which are the sum of $m$ many nonnegative $k$th powers. Is it... -- [#324](https://erdosproblems.com/324) — OPEN | number theory, powers — Does there exist a polynomial $f(x)\in\mathbb{Z}[x]$ such that all the sums $f(a)+f(b)$ with $a0$ such that every measurable $A\subseteq \mathbb{R}^2$ of measure $\geq c$ contains the vertices of a triangle of area 1? - [#354](https://erdosproblems.com/354) — OPEN | number theory, complete sequences — edited: 01 December 2025 — Let $\alpha,\beta\in \mathbb{R}_{>0}$ such that $\alpha/\beta$ is irrational. Is the multiset\[\{ \lfloor \alpha\rfloor,\lfloor 2\alpha\r... - [#357](https://erdosproblems.com/357) — OPEN | number theory — edited: 12 January 2026 — Let $1\leq a_1<\cdots 0$ and $n$ be some large integer. What is the size of the largest $A\subseteq \{1,\ldots,\lfloor cn\rfloor\}$ such that $n$ is not... -- [#364](https://erdosproblems.com/364) — VERIFIABLE | number theory, powerful — edited: 20 December 2025 — Are there any triples of consecutive positive integers all of which are powerful (i.e. if $p\mid n$ then $p^2\mid n$)? +- [#364](https://erdosproblems.com/364) — VERIFIABLE | number theory, powerful — edited: 13 April 2026 — Are there any triples of consecutive positive integers all of which are powerful (i.e. if $p\mid n$ then $p^2\mid n$)? - [#365](https://erdosproblems.com/365) — OPEN | number theory, powerful — edited: 31 October 2025 — Do all pairs of consecutive powerful numbers $n$ and $n+1$ come from solutions to Pell equations ? In other words, must either $n$ or $n+... -- [#366](https://erdosproblems.com/366) — VERIFIABLE | number theory, powerful — edited: 20 December 2025 — Are there any 2-full $n$ such that $n+1$ is 3-full? That is, if $p\mid n$ then $p^2\mid n$ and if $p\mid n+1$ then $p^3\mid n+1$. -- [#367](https://erdosproblems.com/367) — OPEN | number theory, powerful — edited: 08 February 2026 — Let $B_2(n)$ be the 2-full part of $n$ (that is, $B_2(n)=n/n'$ where $n'$ is the product of all primes that divide $n$ exactly once). Is... +- [#366](https://erdosproblems.com/366) — VERIFIABLE | number theory, powerful — edited: 15 April 2026 — Are there any $2$-full $n$ such that $n+1$ is $3$-full? That is, if $p\mid n$ then $p^2\mid n$ and if $p\mid n+1$ then $p^3\mid n+1$. +- [#367](https://erdosproblems.com/367) — OPEN | number theory, powerful — edited: 23 March 2026 — Let $B_2(n)$ be the $2$-full part of $n$ (that is, $B_2(n)=n/n'$ where $n'$ is the product of all primes that divide $n$ exactly once). I... - [#368](https://erdosproblems.com/368) — OPEN | number theory — How large is the largest prime factor of $n(n+1)$? -- [#369](https://erdosproblems.com/369) — OPEN | number theory — Let $\epsilon>0$ and $k\geq 2$. Is it true that, for all sufficiently large $n$, there is a sequence of $k$ consecutive integers in $\{1,... - [#371](https://erdosproblems.com/371) — OPEN | number theory — edited: 23 January 2026 — Let $P(n)$ denote the largest prime factor of $n$. Show that the set of $n$ with $P(n)a_1\geq a_2\geq \cdots \geq a_k\geq 2$, has only finitely many solutions. - [#374](https://erdosproblems.com/374) — OPEN | number theory — For any $m\in \mathbb{N}$, let $F(m)$ be the minimal $k\geq 2$ (if it exists) such that there are $a_1<\cdots 0$ such that\[\sum_{p\leq n}1_{p\nmid \binom{2n}{n}}\frac{1}{p}\leq C\]for all $n$ (where the summatio... -- [#380](https://erdosproblems.com/380) — OPEN | number theory — We call an interval $[u,v]$ 'bad' if the greatest prime factor of $\prod_{u\leq m\leq v}m$ occurs with an exponent greater than $1$. Let... - [#382](https://erdosproblems.com/382) — OPEN | number theory — Let $u\leq v$ be such that the largest prime dividing $\prod_{u\leq m\leq v}m$ appears with exponent at least $2$. Is it true that $v-u=v... - [#383](https://erdosproblems.com/383) — OPEN | number theory — Is it true that for every $k$ there are infinitely many primes $p$ such that the largest prime divisor of\[\prod_{0\leq i\leq k}(p^2+i)\]... - [#385](https://erdosproblems.com/385) — OPEN | number theory — Let\[F(n) = \max_{\substack{mn... - [#386](https://erdosproblems.com/386) — OPEN | number theory, binomial coefficients — Let $2\leq k\leq n-2$. Can $\binom{n}{k}$ be the product of consecutive primes infinitely often? For example\[\binom{21}{2}=2\cdot 3\cdot... - [#387](https://erdosproblems.com/387) — OPEN | number theory, binomial coefficients — edited: 23 January 2026 — Is there an absolute constant $c>0$ such that, for all $1\leq k< n$, the binomial coefficient $\binom{n}{k}$ has a divisor in $(cn,n]$? -- [#388](https://erdosproblems.com/388) — OPEN | number theory — Can one classify all solutions of\[\prod_{1\leq i\leq k_1}(m_1+i)=\prod_{1\leq j\leq k_2}(m_2+j)\]where $k_1,k_2>3$ and $m_1+k_1\leq m_2$... +- [#388](https://erdosproblems.com/388) — OPEN | number theory — edited: 11 April 2026 — Can one classify all solutions of\[\prod_{1\leq i\leq k_1}(m_1+i)=\prod_{1\leq j\leq k_2}(m_2+j)\]where $k_1,k_2>3$ and $m_1+k_1\leq m_2$... - [#389](https://erdosproblems.com/389) — OPEN | number theory — Is it true that for every $n\geq 1$ there is a $k$ such that\[n(n+1)\cdots(n+k-1)\mid (n+k)\cdots (n+2k-1)?\] - [#390](https://erdosproblems.com/390) — OPEN | number theory, factorials — Let $f(n)$ be the minimal $m$ such that\[n! = a_1\cdots a_k\]with $n< a_1<\cdots 1$? - [#420](https://erdosproblems.com/420) — OPEN | number theory — edited: 03 December 2025 — If $\tau(n)$ counts the number of divisors of $n$ then let\[F(f,n)=\frac{\tau((n+\lfloor f(n)\rfloor)!)}{\tau(n!)}.\]Is it true that\[\li... - [#421](https://erdosproblems.com/421) — OPEN | number theory — Is there a sequence $1\leq d_12$\[f(n) = f(n-f(n-1))+f(n-f(n-2)).\]Does $f(n)$ miss infinitely many integers? What is its behaviour? -- [#423](https://erdosproblems.com/423) — OPEN | number theory — edited: 16 January 2026 — Let $a_1=1$ and $a_2=2$ and for $k\geq 3$ choose $a_k$ to be the least integer $>a_{k-1}$ which is the sum of at least two consecutive te... -- [#424](https://erdosproblems.com/424) — OPEN | number theory — Let $a_1=2$ and $a_2=3$ and continue the sequence by appending to $a_1,\ldots,a_n$ all possible values of $a_ia_j-1$ with $i\neq j$. Is i... -- [#425](https://erdosproblems.com/425) — OPEN | number theory, sidon sets — edited: 28 December 2025 — Let $F(n)$ be the maximum possible size of a subset $A\subseteq\{1,\ldots,N\}$ such that the products $ab$ are distinct for all $aa_{k-1}$ which is the sum of at least two consecutive te... +- [#424](https://erdosproblems.com/424) — OPEN | number theory — edited: 31 March 2026 — Let $a_1=2$ and $a_2=3$ and continue the sequence by appending to $a_1,\ldots,a_n$ all possible values of $a_ia_j-1$ with $i\neq j$. Is i... +- [#425](https://erdosproblems.com/425) — OPEN | number theory, sidon sets — edited: 06 April 2026 — Let $F(n)$ be the maximum possible size of a subset $A\subseteq\{1,\ldots,N\}$ such that the products $ab$ are distinct for all $a1/2$, if $p$ is a sufficiently large prime then, for any $n\geq 0$, there exist $a,b\in(n,n+p^c)$ such that $... @@ -269,7 +254,6 @@ - [#454](https://erdosproblems.com/454) — OPEN | number theory, primes — edited: 07 October 2025 — Let\[f(n) = \min_{i0$ such that there are infinitely many $n$ where all primes $p\leq (2+\epsilon)\log n$ divide\[\prod_{1\leq i\leq... - [#458](https://erdosproblems.com/458) — FALSIFIABLE | number theory, primes — edited: 07 October 2025 — Let $[1,\ldots,n]$ denote the least common multiple of $\{1,\ldots,n\}$. Is it true that, for all $k\geq 1$,\[[1,\ldots,p_{k+1}-1]< p_k[1... - [#460](https://erdosproblems.com/460) — OPEN | number theory — edited: 14 January 2026 — Let $a_0=0$ and $a_1=1$, and in general define $a_k$ to be the least integer $>a_{k-1}$ for which $(n-a_k,n-a_i)=1$ for all $0\leq in\geq \max(A)$,\[\frac{\l... +- [#483](https://erdosproblems.com/483) — OPEN | number theory, additive combinatorics, ramsey theory — edited: 10 April 2026 — Let $f(k)$ be the minimal $N$ such that if $\{1,\ldots,N\}$ is $k$-coloured then there is a monochromatic solution to $a+b=c$. Estimate $... +- [#486](https://erdosproblems.com/486) — OPEN | number theory, primitive sets — edited: 08 April 2026 — Let $A\subseteq \mathbb{N}$, and for each $n\in A$ choose some $X_n\subseteq \mathbb{Z}/n\mathbb{Z}$. Let\[B = \{ m\in \mathbb{N} : m\not... +- [#488](https://erdosproblems.com/488) — FALSIFIABLE | number theory — edited: 08 April 2026 — Let $A$ be a finite set and\[B=\{ n \geq 1 : a\mid n\textrm{ for some }a\in A\}.\]Is it true that, for every $m>n\geq \max(A)$,\[\frac{\l... - [#489](https://erdosproblems.com/489) — OPEN | number theory — Let $A\subseteq \mathbb{N}$ be a set such that $\lvert A\cap [1,x]\rvert=o(x^{1/2})$. Let\[B=\{ n\geq 1 : a\nmid n\textrm{ for all }a\in... - [#495](https://erdosproblems.com/495) — OPEN | diophantine approximation, number theory — Let $\alpha,\beta \in \mathbb{R}$. Is it true that\[\liminf_{n\to \infty} n \| n\alpha \| \| n\beta\| =0\]where $\|x\|$ is the distance f... - [#500](https://erdosproblems.com/500) — OPEN | $500 | graph theory, hypergraphs, turan number — edited: 05 October 2025 — What is $\mathrm{ex}_3(n,K_4^3)$? That is, the largest number of $3$-edges which can placed on $n$ vertices so that there exists no $K_4^... @@ -297,7 +281,7 @@ - [#508](https://erdosproblems.com/508) — OPEN | geometry, ramsey theory — edited: 22 January 2026 — What is the chromatic number of the plane? That is, what is the smallest number of colours required to colour $\mathbb{R}^2$ such that no... - [#509](https://erdosproblems.com/509) — OPEN | analysis, polynomials — edited: 29 December 2025 — Let $f(z)\in\mathbb{C}[z]$ be a monic non-constant polynomial. Can the set\[\{ z\in \mathbb{C} : \lvert f(z)\rvert \leq 1\}\]be covered b... - [#510](https://erdosproblems.com/510) — OPEN | analysis — edited: 28 September 2025 — If $A\subset \mathbb{Z}$ is a finite set of size $N$ then is there some absolute constant $c>0$ and $\theta$ such that\[\sum_{n\in A}\cos... -- [#513](https://erdosproblems.com/513) — OPEN | analysis — edited: 28 December 2025 — Let $f=\sum_{n=0}^\infty a_nz^n$ be a transcendental entire function. What is the greatest possible value of\[\liminf_{r\to \infty} \frac... +- [#513](https://erdosproblems.com/513) — OPEN | analysis — edited: 02 April 2026 — Let $f=\sum_{n=0}^\infty a_nz^n$ be a transcendental entire function. What is the greatest possible value of\[\liminf_{r\to \infty} \frac... - [#514](https://erdosproblems.com/514) — OPEN | analysis — edited: 18 January 2026 — Let $f(z)$ be an entire transcendental function. Does there exist a path $L$ so that, for every $n$,\[\lvert f(z)/z^n\rvert \to \infty\]a... - [#517](https://erdosproblems.com/517) — OPEN | analysis — edited: 29 December 2025 — Let $f(z)=\sum_{k=1}^\infty a_kz^{n_k}$ be an entire function (with $a_k\neq 0$ for all $k\geq 1$). Is it true that if $n_k/k\to \infty$... - [#520](https://erdosproblems.com/520) — OPEN | number theory, probability — Let $f$ be a Rademacher multiplicative function: a random $\{-1,0,1\}$-valued multiplicative function, where for each prime $p$ we indepe... @@ -306,15 +290,15 @@ - [#524](https://erdosproblems.com/524) — OPEN | analysis, probability, polynomials — edited: 27 December 2025 — For any $t\in (0,1)$ let $t=\sum_{k=1}^\infty \epsilon_k(t)2^{-k}$ (where $\epsilon_k(t)\in \{0,1\}$). What is the correct order of magni... - [#528](https://erdosproblems.com/528) — OPEN | geometry — Let $f(n,k)$ count the number of self-avoiding walks of $n$ steps (beginning at the origin) in $\mathbb{Z}^k$ (i.e. those walks which do... - [#529](https://erdosproblems.com/529) — OPEN | geometry, probability — edited: 27 December 2025 — Let $d_k(n)$ be the expected distance from the origin after taking $n$ random steps from the origin in $\mathbb{Z}^k$ (conditional on no... -- [#530](https://erdosproblems.com/530) — OPEN | number theory, sidon sets — edited: 16 October 2025 — Let $\ell(N)$ be maximal such that in any finite set $A\subset \mathbb{R}$ of size $N$ there exists a Sidon subset $S$ of size $\ell(N)$... +- [#530](https://erdosproblems.com/530) — OPEN | number theory, sidon sets — edited: 08 April 2026 — Let $\ell(N)$ be maximal such that in any finite set $A\subset \mathbb{R}$ of size $N$ there exists a Sidon subset $S$ of size $\ell(N)$... - [#531](https://erdosproblems.com/531) — OPEN | number theory, ramsey theory — Let $F(k)$ be the minimal $N$ such that if we two-colour $\{1,\ldots,N\}$ there is a set $A$ of size $k$ such that all subset sums $\sum_... -- [#535](https://erdosproblems.com/535) — OPEN | number theory — Let $r\geq 3$, and let $f_r(N)$ denote the size of the largest subset of $\{1,\ldots,N\}$ such that no subset of size $r$ has the same pa... -- [#536](https://erdosproblems.com/536) — OPEN | number theory — edited: 12 January 2026 — Let $\epsilon>0$ and $N$ be sufficiently large. Is it true that if $A\subseteq \{1,\ldots,N\}$ has size at least $\epsilon N$ then there... +- [#535](https://erdosproblems.com/535) — OPEN | number theory — edited: 29 April 2026 — Let $r\geq 3$, and let $f_r(N)$ denote the size of the largest subset of $\{1,\ldots,N\}$ such that no subset of size $r$ has the same pa... +- [#536](https://erdosproblems.com/536) — OPEN | number theory — edited: 29 April 2026 — Let $f(N)$ be the largest size of $A\subseteq \{1,\ldots,N\}$ with the property that there are no distinct $a,b,c\in A$ such that\[[a,b]=... - [#538](https://erdosproblems.com/538) — OPEN | number theory — Let $r\geq 2$ and suppose that $A\subseteq\{1,\ldots,N\}$ is such that, for any $m$, there are at most $r$ solutions to $m=pa$ where $p$... - [#539](https://erdosproblems.com/539) — OPEN | number theory — edited: 22 January 2026 — Let $h(n)$ be such that, for any set $A\subseteq \mathbb{N}$ of size $n$, the set\[\left\{ \frac{a}{(a,b)}: a,b\in A\right\}\]has size at... -- [#544](https://erdosproblems.com/544) — OPEN | graph theory, ramsey theory — Show that\[R(3,k+1)-R(3,k)\to\infty\]as $k\to \infty$. Similarly, prove or disprove that\[R(3,k+1)-R(3,k)=o(k).\] +- [#544](https://erdosproblems.com/544) — OPEN | graph theory, ramsey theory — edited: 24 April 2026 — Show that\[R(3,k+1)-R(3,k)\to\infty\]as $k\to \infty$. Similarly, prove or disprove that\[R(3,k+1)-R(3,k)=o(k).\] - [#545](https://erdosproblems.com/545) — OPEN | graph theory, ramsey theory — edited: 02 December 2025 — Let $G$ be a graph with $m$ edges and no isolated vertices. Is the Ramsey number $R(G)$ maximised when $G$ is 'as complete as possible'?... -- [#548](https://erdosproblems.com/548) — FALSIFIABLE | graph theory — edited: 23 January 2026 — Let $n\geq k+1$. Every graph on $n$ vertices with at least $\frac{k-1}{2}n+1$ edges contains every tree on $k+1$ vertices. +- [#548](https://erdosproblems.com/548) — FALSIFIABLE | $100 | graph theory — edited: 07 March 2026 — Let $n\geq k+1$. Every graph on $n$ vertices with at least $\frac{k-1}{2}n+1$ edges contains every tree on $k+1$ vertices. - [#550](https://erdosproblems.com/550) — OPEN | graph theory, ramsey theory — Let $m_1\leq\cdots\leq m_k$ and $n$ be sufficiently large. If $T$ is a tree on $n$ vertices and $G$ is the complete multipartite graph wi... - [#552](https://erdosproblems.com/552) — OPEN | $100 | graph theory, ramsey theory — edited: 01 February 2026 — Determine the Ramsey number\[R(C_4,S_n),\]where $S_n=K_{1,n}$ is the star on $n+1$ vertices. In particular, is it true that, for any $c>0... - [#554](https://erdosproblems.com/554) — OPEN | graph theory, ramsey theory — edited: 08 February 2026 — Let $R_k(G)$ denote the minimal $m$ such that if the edges of $K_m$ are $k$-coloured then there is a monochromatic copy of $G$. Show that... @@ -330,15 +314,14 @@ - [#567](https://erdosproblems.com/567) — OPEN | graph theory, ramsey theory — edited: 18 January 2026 — Let $G$ be either $Q_3$ or $K_{3,3}$ or $H_5$ (the last formed by adding two vertex-disjoint chords to $C_5$). Is it true that, if $H$ ha... - [#568](https://erdosproblems.com/568) — OPEN | graph theory, ramsey theory — edited: 18 January 2026 — Let $G$ be a graph such that $R(G,T_n)\ll n$ for any tree $T_n$ on $n$ vertices and $R(G,K_n)\ll n^2$. Is it true that, for any $H$ with... - [#569](https://erdosproblems.com/569) — OPEN | graph theory, ramsey theory — edited: 18 January 2026 — Let $k\geq 1$. What is the best possible $c_k$ such that\[R(C_{2k+1},H)\leq c_k m\]for any graph $H$ on $m$ edges without isolated vertices? -- [#571](https://erdosproblems.com/571) — OPEN | graph theory, turan number — edited: 18 January 2026 — Show that for any rational $\alpha \in [1,2)$ there exists a bipartite graph $G$ such that\[\mathrm{ex}(n;G)\asymp n^{\alpha}.\] +- [#571](https://erdosproblems.com/571) — OPEN | graph theory, turan number — edited: 07 March 2026 — Show that for any rational $\alpha \in [1,2)$ there exists a bipartite graph $G$ such that\[\mathrm{ex}(n;G)\asymp n^{\alpha}.\] - [#572](https://erdosproblems.com/572) — OPEN | graph theory, turan number, cycles — edited: 18 January 2026 — Show that for $k\geq 3$\[\mathrm{ex}(n;C_{2k})\gg n^{1+\frac{1}{k}}.\] - [#573](https://erdosproblems.com/573) — OPEN | graph theory, turan number — edited: 18 January 2026 — Is it true that\[\mathrm{ex}(n;\{C_3,C_4\})\sim (n/2)^{3/2}?\] -- [#574](https://erdosproblems.com/574) — OPEN | graph theory, turan number — edited: 18 January 2026 — Is it true that, for $k\geq 2$,\[\mathrm{ex}(n;\{C_{2k-1},C_{2k}\})=(1+o(1))(n/2)^{1+\frac{1}{k}}.\] - [#575](https://erdosproblems.com/575) — OPEN | graph theory, turan number — edited: 18 January 2026 — If $\mathcal{F}$ is a finite set of finite graphs then $\mathrm{ex}(n;\mathcal{F})$ is the maximum number of edges a graph on $n$ vertice... - [#576](https://erdosproblems.com/576) — OPEN | graph theory, turan number — edited: 18 January 2026 — Let $Q_k$ be the $k$-dimensional hypercube graph (so that $Q_k$ has $2^k$ vertices and $k2^{k-1}$ edges). Determine the behaviour of\[\ma... - [#579](https://erdosproblems.com/579) — OPEN | graph theory, turan number — Let $\delta>0$. If $n$ is sufficiently large and $G$ is a graph on $n$ vertices with no $K_{2,2,2}$ and at least $\delta n^2$ edges then... - [#580](https://erdosproblems.com/580) — DECIDABLE | graph theory — edited: 24 October 2025 — Let $G$ be a graph on $n$ vertices such that at least $n/2$ vertices have degree at least $n/2$. Must $G$ contain every tree on at most $... -- [#583](https://erdosproblems.com/583) — FALSIFIABLE | graph theory — edited: 05 March 2026 — Every connected graph on $n$ vertices can be partitioned into at most $\lceil n/2\rceil$ edge-disjoint paths. +- [#583](https://erdosproblems.com/583) — FALSIFIABLE | graph theory — edited: 01 April 2026 — Every connected graph on $n$ vertices can be partitioned into at most $\lceil n/2\rceil$ edge-disjoint paths. - [#584](https://erdosproblems.com/584) — OPEN | graph theory, cycles — edited: 22 January 2026 — Let $G$ be a graph with $n$ vertices and $\delta n^{2}$ edges. Are there subgraphs $H_1,H_2\subseteq G$ such that $H_1$ has $\gg \delta^3... - [#585](https://erdosproblems.com/585) — OPEN | graph theory, cycles — What is the maximum number of edges that a graph on $n$ vertices can have if it does not contain two edge-disjoint cycles with the same v... - [#588](https://erdosproblems.com/588) — OPEN | $100 | geometry — Let $f_k(n)$ be minimal such that if $n$ points in $\mathbb{R}^2$ have no $k+1$ points on a line then there must be at most $f_k(n)$ many... @@ -353,14 +336,14 @@ - [#601](https://erdosproblems.com/601) — OPEN | $500 | graph theory, set theory — For which limit ordinals $\alpha$ is it true that if $G$ is a graph with vertex set $\alpha$ then $G$ must have either an infinite path o... - [#602](https://erdosproblems.com/602) — OPEN | combinatorics, set theory — Let $(A_i)$ be a family of sets with $\lvert A_i\rvert=\aleph_0$ for all $i$, such that for any $i\neq j$ we have $\lvert A_i\cap A_j\rve... - [#603](https://erdosproblems.com/603) — OPEN | combinatorics, set theory — edited: 28 December 2025 — Let $(A_i)$ be a family of countably infinite sets such that $\lvert A_i\cap A_j\rvert \neq 2$ for all $i\neq j$. Find the smallest cardi... -- [#604](https://erdosproblems.com/604) — OPEN | $500 | geometry, distances — edited: 15 October 2025 — Given $n$ distinct points $A\subset\mathbb{R}^2$ must there be a point $x\in A$ such that\[\#\{ d(x,y) : y \in A\} \gg n^{1-o(1)}?\]Or ev... +- [#604](https://erdosproblems.com/604) — OPEN | $500 | geometry, distances — edited: 23 March 2026 — Given $n$ distinct points $A\subset\mathbb{R}^2$ must there be a point $x\in A$ such that\[\#\{ d(x,y) : y \in A\} \gg n^{1-o(1)}?\]Or ev... - [#609](https://erdosproblems.com/609) — OPEN | graph theory, ramsey theory — Let $f(n)$ be the minimal $m$ such that if the edges of $K_{2^n+1}$ are coloured with $n$ colours then there must be a monochromatic odd... - [#610](https://erdosproblems.com/610) — OPEN | graph theory — edited: 02 December 2025 — For a graph $G$ let $\tau(G)$ denote the minimal number of vertices that include at least one from each maximal clique of $G$ (sometimes... - [#611](https://erdosproblems.com/611) — OPEN | graph theory — For a graph $G$ let $\tau(G)$ denote the minimal number of vertices that include at least one from each maximal clique of $G$ (sometimes... - [#612](https://erdosproblems.com/612) — OPEN | graph theory — Let $G$ be a connected graph with $n$ vertices, minimum degree $d$, and diameter $D$. Show if that $G$ contains no $K_{2r}$ and $(r-1)(3r... - [#614](https://erdosproblems.com/614) — OPEN | graph theory — Let $f(n,k)$ be minimal such that there is a graph with $n$ vertices and $f(n,k)$ edges where every set of $k+2$ vertices induces a subgr... - [#616](https://erdosproblems.com/616) — OPEN | graph theory — Let $r\geq 3$. For an $r$-uniform hypergraph $G$ let $\tau(G)$ denote the covering number (or transversal number), the minimum size of a... -- [#617](https://erdosproblems.com/617) — FALSIFIABLE | graph theory — Let $r\geq 3$. If the edges of $K_{r^2+1}$ are $r$-coloured then there exist $r+1$ vertices with at least one colour missing on the edges... +- [#617](https://erdosproblems.com/617) — FALSIFIABLE | graph theory — edited: 01 April 2026 — Let $r\geq 3$. If the edges of $K_{r^2+1}$ are $r$-coloured then there exist $r+1$ vertices with at least one colour missing on the edges... - [#619](https://erdosproblems.com/619) — OPEN | graph theory — For a triangle-free graph $G$ let $h_r(G)$ be the smallest number of edges that need to be added to $G$ so that it has diameter $r$ (whil... - [#620](https://erdosproblems.com/620) — OPEN | graph theory — If $G$ is a graph on $n$ vertices without a $K_4$ then how large a triangle-free induced subgraph must $G$ contain? - [#623](https://erdosproblems.com/623) — OPEN | set theory — Let $X$ be a set of cardinality $\aleph_\omega$ and $f$ be a function from the finite subsets of $X$ to $X$ such that $f(A)\not\in A$ for... @@ -370,16 +353,14 @@ - [#627](https://erdosproblems.com/627) — OPEN | graph theory, chromatic number — edited: 08 February 2026 — Let $\omega(G)$ denote the clique number of $G$ and $\chi(G)$ the chromatic number. If $f(n)$ is the maximum value of $\chi(G)/\omega(G)$... - [#628](https://erdosproblems.com/628) — FALSIFIABLE | graph theory, chromatic number — edited: 06 December 2025 — Let $G$ be a graph with chromatic number $k$ containing no $K_k$. If $a,b\geq 2$ and $a+b=k+1$ then must there exist two disjoint subgrap... - [#629](https://erdosproblems.com/629) — OPEN | graph theory, chromatic number — edited: 28 October 2025 — The list chromatic number $\chi_L(G)$ is defined to be the minimal $k$ such that for any assignment of a list of $k$ colours to each vert... -- [#633](https://erdosproblems.com/633) — OPEN | $25 | geometry — Classify those triangles which can only be cut into a square number of congruent triangles. - [#634](https://erdosproblems.com/634) — OPEN | $25 | geometry — edited: 30 December 2025 — Find all $n$ such that there is at least one triangle which can be cut into $n$ congruent triangles. - [#635](https://erdosproblems.com/635) — OPEN | number theory — edited: 01 February 2026 — Let $t\geq 1$ and $A\subseteq \{1,\ldots,N\}$ be such that whenever $a,b\in A$ with $b-a\geq t$ we have $b-a\nmid b$. How large can $\lve... -- [#638](https://erdosproblems.com/638) — OPEN | graph theory, ramsey theory — Let $S$ be a family of finite graphs such that for every $n$ there is some $G_n\in S$ such that if the edges of $G_n$ are coloured with $... +- [#638](https://erdosproblems.com/638) — OPEN | graph theory, ramsey theory — edited: 10 April 2026 — Let $S$ be a family of finite graphs such that for every $n$ there is some $G_n\in S$ such that if the edges of $G_n$ are coloured with $... - [#640](https://erdosproblems.com/640) — OPEN | graph theory, chromatic number — edited: 22 January 2026 — Let $k\geq 3$. Does there exist some $f(k)$ such that if a graph $G$ has chromatic number $\geq f(k)$ then $G$ must contain some odd cycl... - [#642](https://erdosproblems.com/642) — OPEN | graph theory, cycles — edited: 28 January 2026 — Let $f(n)$ be the maximal number of edges in a graph on $n$ vertices such that all cycles have more vertices than chords. Is it true that... - [#643](https://erdosproblems.com/643) — OPEN | graph theory, hypergraphs — edited: 26 October 2025 — Let $f(n;t)$ be minimal such that if a $t$-uniform hypergraph on $n$ vertices contains at least $f(n;t)$ edges then there must be four ed... - [#644](https://erdosproblems.com/644) — OPEN | combinatorics — Let $f(k,r)$ be minimal such that if $A_1,A_2,\ldots$ is a family of sets, all of size $k$, such that for every collection of $r$ of the... -- [#647](https://erdosproblems.com/647) — VERIFIABLE | $44 | number theory — edited: 05 October 2025 — Let $\tau(n)$ count the number of divisors of $n$. Is there some $n>24$ such that\[\max_{m24$ such that\[\max_{m0$ and $\omega(n)$ count the number of distinct prime factors of $n$. Are there infinitely many values of $n$ such that\[\o... +- [#679](https://erdosproblems.com/679) — OPEN | number theory — edited: 17 April 2026 — Let $\epsilon>0$ and $\omega(n)$ count the number of distinct prime factors of $n$. Are there infinitely many values of $n$ such that\[\o... - [#680](https://erdosproblems.com/680) — OPEN | number theory, primes — Is it true that, for all sufficiently large $n$, there exists some $k$ such that\[p(n+k)>k^2+1,\]where $p(m)$ denotes the least prime fac... - [#681](https://erdosproblems.com/681) — OPEN | number theory, primes — edited: 01 February 2026 — Is it true that for all large $n$ there exists $k$ such that $n+k$ is composite and\[p(n+k)>k^2,\]where $p(m)$ is the least prime factor... - [#683](https://erdosproblems.com/683) — OPEN | number theory, primes, binomial coefficients — edited: 31 December 2025 — Is it true that for every $1\leq k\leq n$ the largest prime divisor of $\binom{n}{k}$, say $P(\binom{n}{k})$, satisfies\[P\left(\binom{n}... -- [#684](https://erdosproblems.com/684) — OPEN | number theory, primes, binomial coefficients — edited: 23 January 2026 — For $0\leq k\leq n$ write\[\binom{n}{k} = uv\]where the only primes dividing $u$ are in $[2,k]$ and the only primes dividing $v$ are in $... +- [#684](https://erdosproblems.com/684) — OPEN | number theory, primes, binomial coefficients — edited: 01 April 2026 — For $0\leq k\leq n$ write\[\binom{n}{k} = uv\]where the only primes dividing $u$ are in $[2,k]$ and the only primes dividing $v$ are in $... - [#685](https://erdosproblems.com/685) — OPEN | number theory, primes, binomial coefficients — Let $\epsilon>0$ and $n$ be large depending on $\epsilon$. Is it true that for all $n^\epsilonr>2$, the value of\[\frac{\mathrm{ex}_r(n,K_k^r)}{\binom{n}{r}},\]where $\mathrm{ex}_r(n,K_k^r)$ is the largest num... -- [#713](https://erdosproblems.com/713) — OPEN | $500 | graph theory, turan number — edited: 06 October 2025 — Is it true that, for every bipartite graph $G$, there exists some $\alpha\in [1,2)$ and $c>0$ such that\[\mathrm{ex}(n;G)\sim cn^\alpha?\... +- [#713](https://erdosproblems.com/713) — OPEN | $500 | graph theory, turan number — edited: 07 March 2026 — Is it true that, for every bipartite graph $G$, there exists some $\alpha\in [1,2)$ and $c>0$ such that\[\mathrm{ex}(n;G)\sim cn^\alpha?\... - [#714](https://erdosproblems.com/714) — OPEN | graph theory, turan number — edited: 23 January 2026 — Is it true that\[\mathrm{ex}(n; K_{r,r}) \gg n^{2-1/r}?\] - [#719](https://erdosproblems.com/719) — OPEN | graph theory, hypergraphs — Let $\mathrm{ex}_r(n;K_{r+1}^r)$ be the maximum number of $r$-edges that can be placed on $n$ vertices without forming a $K_{r+1}^r$ (the... - [#723](https://erdosproblems.com/723) — FALSIFIABLE | combinatorics — If there is a finite projective plane of order $n$ then must $n$ be a prime power? A finite projective plane of order $n$ is a collection... @@ -439,24 +417,22 @@ - [#738](https://erdosproblems.com/738) — OPEN | graph theory, chromatic number — If $G$ has infinite chromatic number and is triangle-free (contains no $K_3$) then must $G$ contain every tree as an induced subgraph? - [#739](https://erdosproblems.com/739) — NOT PROVABLE | graph theory, chromatic number — edited: 01 October 2025 — Let $\mathfrak{m}$ be an infinite cardinal and $G$ be a graph with chromatic number $\mathfrak{m}$. Is it true that, for every infinite c... - [#740](https://erdosproblems.com/740) — OPEN | graph theory, chromatic number — Let $\mathfrak{m}$ be an infinite cardinal and $G$ be a graph with chromatic number $\mathfrak{m}$. Let $r\geq 1$. Must $G$ contain a sub... -- [#741](https://erdosproblems.com/741) — OPEN | additive combinatorics — Let $A\subseteq \mathbb{N}$ be such that $A+A$ has positive density. Can one always decompose $A=A_1\sqcup A_2$ such that $A_1+A_1$ and $... - [#743](https://erdosproblems.com/743) — FALSIFIABLE | graph theory — Let $T_2,\ldots,T_n$ be a collection of trees such that $T_k$ has $k$ vertices. Can we always write $K_n$ as the edge disjoint union of t... -- [#749](https://erdosproblems.com/749) — OPEN | additive combinatorics — Let $\epsilon>0$. Does there exist $A\subseteq \mathbb{N}$ such that the lower density of $A+A$ is at least $1-\epsilon$ and yet $1_A\ast... +- [#749](https://erdosproblems.com/749) — OPEN | additive combinatorics — edited: 06 April 2026 — Let $\epsilon>0$. Does there exist $A\subseteq \mathbb{N}$ such that the lower density of $A+A$ is at least $1-\epsilon$ and yet $1_A\ast... - [#750](https://erdosproblems.com/750) — OPEN | graph theory, chromatic number — edited: 14 October 2025 — Let $f(m)$ be some function such that $f(m)\to \infty$ as $m\to \infty$. Does there exist a graph $G$ of infinite chromatic number such t... -- [#757](https://erdosproblems.com/757) — OPEN | geometry, distances, sidon sets — edited: 08 January 2026 — Let $A\subset \mathbb{R}$ be a set of size $n$ such that every subset $B\subseteq A$ with $\lvert B\rvert =4$ has $\lvert B-B\rvert\geq 1... +- [#757](https://erdosproblems.com/757) — OPEN | geometry, distances, sidon sets — edited: 10 April 2026 — Let $A\subset \mathbb{R}$ be a set of size $n$ such that every subset $B\subseteq A$ with $\lvert B\rvert =4$ has $\lvert B-B\rvert\geq 1... - [#761](https://erdosproblems.com/761) — OPEN | graph theory, chromatic number — The cochromatic number of $G$, denoted by $\zeta(G)$, is the minimum number of colours needed to colour the vertices of $G$ such that eac... - [#766](https://erdosproblems.com/766) — OPEN | graph theory, turan number — Let $f(n;k,l)=\min \mathrm{ex}(n;G)$, where $G$ ranges over all graphs with $k$ vertices and $l$ edges. Give good estimates for $f(n;k,l)... - [#768](https://erdosproblems.com/768) — OPEN | number theory — edited: 14 September 2025 — Let $A\subset\mathbb{N}$ be the set of $n$ such that for every prime $p\mid n$ there exists some $d\mid n$ with $d>1$ such that $d\equiv... - [#769](https://erdosproblems.com/769) — OPEN | number theory, geometry — edited: 01 October 2025 — Let $c(n)$ be minimal such that if $k\geq c(n)$ then the $n$-dimensional unit cube can be decomposed into $k$ homothetic $n$-dimensional... - [#770](https://erdosproblems.com/770) — OPEN | number theory — edited: 24 September 2025 — Let $h(n)$ be minimal such that $2^n-1,3^n-1,\ldots,h(n)^n-1$ are mutually coprime. Does, for every prime $p$, the density $\delta_p$ of... -- [#773](https://erdosproblems.com/773) — OPEN | number theory, sidon sets, squares — What is the size of the largest Sidon subset $A\subseteq\{1,2^2,\ldots,N^2\}$? Is it $N^{1-o(1)}$? +- [#773](https://erdosproblems.com/773) — OPEN | number theory, sidon sets, squares — edited: 24 April 2026 — What is the size of the largest Sidon subset $A\subseteq\{1,2^2,\ldots,N^2\}$? Is it $N^{1-o(1)}$? - [#774](https://erdosproblems.com/774) — OPEN | number theory — edited: 28 December 2025 — We call $A\subset \mathbb{N}$ dissociated if $\sum_{n\in X}n\neq \sum_{m\in Y}m$ for all finite $X,Y\subset A$ with $X\neq Y$. Let $A\sub... -- [#776](https://erdosproblems.com/776) — OPEN | combinatorics — Let $r\geq 2$ and $A_1,\ldots,A_m\subseteq \{1,\ldots,n\}$ be such that $A_i\not\subseteq A_j$ for all $i\neq j$ and for any $t$ if there... +- [#776](https://erdosproblems.com/776) — OPEN | combinatorics — edited: 10 April 2026 — Let $r\geq 2$ and $A_1,\ldots,A_m\subseteq \{1,\ldots,n\}$ be such that $A_i\not\subseteq A_j$ for all $i\neq j$ and for any $t$ if there... - [#778](https://erdosproblems.com/778) — OPEN | graph theory — Alice and Bob play a game on the edges of $K_n$, alternating colouring edges by red (Alice) and blue (Bob). Alice goes first, and wins if... - [#779](https://erdosproblems.com/779) — FALSIFIABLE | number theory, primes — Let $n> 1$ and $p_1<\cdots0$ such that, for any $k$, the squ... -- [#783](https://erdosproblems.com/783) — OPEN | number theory — edited: 08 February 2026 — Fix some constant $C>0$ and let $N$ be large. Let $A\subseteq \{2,\ldots,N\}$ be such that $(a,b)=1$ for all $a\neq b\in A$ and $\sum_{n\... -- [#786](https://erdosproblems.com/786) — OPEN | number theory — edited: 02 February 2026 — Let $\epsilon>0$. Is there some set $A\subset \mathbb{N}$ of density $>1-\epsilon$ such that $a_1\cdots a_r=b_1\cdots b_s$ with $a_i,b_j\... +- [#786](https://erdosproblems.com/786) — OPEN | number theory — edited: 11 April 2026 — Let $\epsilon>0$. Is there some set $A\subset \mathbb{N}$ of density $>1-\epsilon$ such that $a_1\cdots a_r=b_1\cdots b_s$ with $a_i,b_j\... - [#787](https://erdosproblems.com/787) — OPEN | additive combinatorics — edited: 23 January 2026 — Let $g(n)$ be maximal such that given any set $A\subset \mathbb{R}$ with $\lvert A\rvert=n$ there exists some $B\subseteq A$ of size $\lv... - [#788](https://erdosproblems.com/788) — OPEN | additive combinatorics — edited: 26 January 2026 — Let $f(n)$ be maximal such that if $B\subset (2n,4n)\cap \mathbb{N}$ there exists some $C\subset (n,2n)\cap \mathbb{N}$ such that $c_1+c_... - [#789](https://erdosproblems.com/789) — OPEN | additive combinatorics — Let $h(n)$ be maximal such that if $A\subseteq \mathbb{Z}$ with $\lvert A\rvert=n$ then there is $B\subseteq A$ with $\lvert B\rvert \geq... @@ -467,8 +443,8 @@ - [#796](https://erdosproblems.com/796) — OPEN | number theory — edited: 16 January 2026 — Let $k\geq 2$ and let $g_k(n)$ be the largest possible size of $A\subseteq \{1,\ldots,n\}$ such that every $m$ has $g(n)\geq (\log n)^2$ is there a graph on $n$ vertices in which every induced subgraph on $g(n)$ vertic... -- [#809](https://erdosproblems.com/809) — OPEN | graph theory, ramsey theory — Let $k\geq 3$ and define $F_k(n)$ to be the minimal $r$ such that there is a graph $G$ on $n$ vertices with $\lfloor n^2/4\rfloor+1$ many... -- [#810](https://erdosproblems.com/810) — OPEN | graph theory, ramsey theory — Does there exist some $\epsilon>0$ such that, for all sufficiently large $n$, there exists a graph $G$ on $n$ vertices with at least $\ep... +- [#809](https://erdosproblems.com/809) — OPEN | graph theory, ramsey theory — edited: 01 April 2026 — Define the anti-Ramsey number $\chi_S(n,e,G)$ as the smallest $r$ such that there is a graph with $n$ vertices and $e$ edges with an $r$-... +- [#810](https://erdosproblems.com/810) — OPEN | graph theory, ramsey theory — edited: 01 April 2026 — Does there exist some $\epsilon>0$ such that, for all sufficiently large $n$, there exists a graph $G$ on $n$ vertices with at least $\ep... - [#811](https://erdosproblems.com/811) — OPEN | graph theory, ramsey theory — edited: 14 October 2025 — Suppose $n\equiv 1\pmod{m}$. We say that an edge-colouring of $K_n$ using $m$ colours is balanced if every vertex sees exactly $\lfloor n... - [#812](https://erdosproblems.com/812) — OPEN | graph theory, ramsey theory — Is it true that\[\frac{R(n+1)}{R(n)}\geq 1+c\]for some constant $c>0$, for all large $n$? Is it true that\[R(n+1)-R(n) \gg n^2?\] - [#813](https://erdosproblems.com/813) — OPEN | graph theory — Let $h(n)$ be minimal such that every graph on $n$ vertices where every set of $7$ vertices contains a triangle (a copy of $K_3$) must co... @@ -477,8 +453,8 @@ - [#820](https://erdosproblems.com/820) — OPEN | number theory — edited: 02 December 2025 — Let $H(n)$ be the smallest integer $l$ such that there exist $k0$, there exist infinitely many $n$ such t... - [#824](https://erdosproblems.com/824) — OPEN | number theory — edited: 28 September 2025 — Let $h(x)$ count the number of integers $1\leq a0$ such that in any subset of $A$ of size $n$ there... - [#849](https://erdosproblems.com/849) — OPEN | number theory, binomial coefficients — Is it true that, for every integer $t\geq 1$, there is some integer $a$ such that\[\binom{n}{k}=a\](with $1\leq k\leq n/2$) has exactly $... - [#850](https://erdosproblems.com/850) — OPEN | number theory, primes — edited: 28 September 2025 — Can there exist two distinct integers $x$ and $y$ such that $x,y$ have the same prime factors, $x+1,y+1$ have the same prime factors, and... -- [#851](https://erdosproblems.com/851) — PROVED | number theory — Let $\epsilon>0$. Is there some $r\ll_\epsilon 1$ such that the density of integers of the form $2^k+n$, where $k\geq 0$ and $n$ has at m... - [#852](https://erdosproblems.com/852) — OPEN | number theory, primes — Let $d_n=p_{n+1}-p_n$, where $p_n$ is the $n$th prime. Let $h(x)$ be maximal such that for some $na$.... - [#859](https://erdosproblems.com/859) — OPEN | number theory, divisors — Let $t\geq 1$ and let $d_t$ be the density of the set of integers $n\in\mathbb{N}$ for which $t$ can be represented as the sum of distinc... - [#860](https://erdosproblems.com/860) — OPEN | number theory, primes — edited: 30 September 2025 — Let $h(n)$ be such that, for any $m\geq 1$, in the interval $(m,m+h(n))$ there exist distinct integers $a_i$ for $1\leq i\leq \pi(n)$ suc... -- [#863](https://erdosproblems.com/863) — OPEN | number theory, sidon sets, additive combinatorics — Let $r\geq 2$ and let $A\subseteq \{1,\ldots,N\}$ be a set of maximal size such that there are at most $r$ solutions to $n=a+b$ with $a\l... - [#864](https://erdosproblems.com/864) — OPEN | number theory, sidon sets, additive combinatorics — Let $A\subseteq \{1,\ldots N\}$ be a set such that there exists at most one $n$ with more than one solution to $n=a+b$ (with $a\leq b\in... - [#865](https://erdosproblems.com/865) — OPEN | number theory, additive combinatorics — There exists a constant $C>0$ such that, for all large $N$, if $A\subseteq \{1,\ldots,N\}$ has size at least $\frac{5}{8}N+C$ then there... - [#866](https://erdosproblems.com/866) — OPEN | number theory, additive combinatorics — edited: 01 December 2025 — Let $k\geq 3$ and $g_k(N)$ be minimal such that if $A\subseteq \{1,\ldots,2N\}$ has $\lvert A\rvert \geq N+g_k(N)$ then there exist integ... -- [#869](https://erdosproblems.com/869) — OPEN | number theory, additive basis — If $A_1,A_2$ are disjoint additive bases of order $2$ (i.e. $A_i+A_i$ contains all large integers) then must $A=A_1\cup A_2$ contain a mi... - [#870](https://erdosproblems.com/870) — OPEN | number theory, additive basis — Let $k\geq 3$ and $A$ be an additive basis of order $k$. Does there exist a constant $c=c(k)>0$ such that if $r(n)\geq c\log n$ for all l... -- [#872](https://erdosproblems.com/872) — OPEN | number theory, primitive sets — Consider the two-player game in which players alternately choose integers from $\{2,3,\ldots,n\}$ to be included in some set $A$ (the sam... +- [#872](https://erdosproblems.com/872) — OPEN | number theory, primitive sets — edited: 24 April 2026 — Consider the two-player game in which players alternately choose integers from $\{2,3,\ldots,n\}$ to be included in some set $A$ (the sam... - [#873](https://erdosproblems.com/873) — OPEN | number theory — Let $A=\{a_10$. Is it true that, for all large $n$, the number of divisors of $n$ in $(n^{1/2},n^{1/2}+n^{1/2-\epsilon})$ is $O_\epsilo... -- [#887](https://erdosproblems.com/887) — OPEN | number theory, divisors — Is there an absolute constant $K$ such that, for every $C>0$, if $n$ is sufficiently large then $n$ has at most $K$ divisors in $(n^{1/2}... +- [#887](https://erdosproblems.com/887) — OPEN | number theory, divisors — edited: 10 April 2026 — Is there an absolute constant $K$ such that, for every $C>0$, if $n$ is sufficiently large then $n$ has at most $K$ divisors in $(n^{1/2}... - [#888](https://erdosproblems.com/888) — OPEN | number theory, squares — edited: 22 January 2026 — What is the size of the largest $A\subseteq \{1,\ldots,n\}$ such that if $a\leq b\leq c\leq d\in A$ are such that $abcd$ is a square then... - [#889](https://erdosproblems.com/889) — OPEN | number theory — edited: 02 January 2026 — For $k\geq 0$ and $n\geq 1$ let $v(n,k)$ count the prime factors of $n+k$ which do not divide $n+i$ for $0\leq ik$, then is it true that, for every $k\geq 1$,\[\liminf_{... - [#891](https://erdosproblems.com/891) — OPEN | number theory — Let $2=p_1k$ contains an integer divisible by a prime $>k$. Estimat... -- [#962](https://erdosproblems.com/962) — OPEN | number theory — edited: 30 December 2025 — Let $k(n)$ be the maximal $k$ such that there exists $m\leq n$ such that each of the integers\[m+1,\ldots,m+k\]are divisible by at least... +- [#961](https://erdosproblems.com/961) — OPEN | number theory — edited: 03 April 2026 — Let $f(k)$ be the minimal $n$ such that every set of $n$ consecutive integers $>k$ contains an integer divisible by a prime $>k$. Estimat... +- [#962](https://erdosproblems.com/962) — OPEN | number theory — edited: 03 April 2026 — Let $k(n)$ be the maximal $k$ such that there exists $m\leq n$ such that each of the integers\[m+1,\ldots,m+k\]are divisible by at least... - [#963](https://erdosproblems.com/963) — OPEN | number theory — edited: 23 January 2026 — Let $f(n)$ be the maximal $k$ such that in any set $A\subset \mathbb{R}$ of size $n$ there is a subset $B\subseteq A$ of size $\lvert B\r... -- [#968](https://erdosproblems.com/968) — OPEN | number theory — Let $u_n=p_n/n$, where $p_n$ is the $n$th prime. Does the set of $n$ such that $u_n0$ such that, for all large $d$,\[p(a,d) > (1+c)... @@ -575,38 +543,34 @@ - [#973](https://erdosproblems.com/973) — OPEN | analysis — edited: 23 January 2026 — Does there exist a constant $C>1$ such that, for every $n\geq 2$, there exists a sequence $z_i\in \mathbb{C}$ with $z_1=1$ and $\lvert z_... - [#975](https://erdosproblems.com/975) — OPEN | number theory, divisors, polynomials — edited: 27 December 2025 — Let $f\in \mathbb{Z}[x]$ be an irreducible non-constant polynomial such that $f(n)\geq 1$ for all large $n\in\mathbb{N}$. Does there exis... - [#976](https://erdosproblems.com/976) — OPEN | number theory — edited: 01 February 2026 — Let $f\in \mathbb{Z}[x]$ be an irreducible polynomial of degree $d\geq 2$. Let $F_f(n)$ be maximal such that there exists $1\leq m\leq n$... -- [#978](https://erdosproblems.com/978) — OPEN | number theory — edited: 05 March 2026 — Let $f\in \mathbb{Z}[x]$ be an irreducible polynomial of degree $k>2$ (and suppose that $k\neq 2^l$ for any $l\geq 1$) such that the lead... +- [#978](https://erdosproblems.com/978) — OPEN | number theory — edited: 31 March 2026 — Let $f\in \mathbb{Z}[x]$ be an irreducible polynomial of degree $k>2$ (and suppose that $k\neq 2^l$ for any $l\geq 1$) such that the lead... - [#979](https://erdosproblems.com/979) — OPEN | number theory — edited: 19 September 2025 — Let $k\geq 2$, and let $f_k(n)$ count the number of solutions to\[n=p_1^k+\cdots+p_k^k,\]where the $p_i$ are prime numbers. Is it true th... - [#982](https://erdosproblems.com/982) — FALSIFIABLE | geometry, convex, distances — edited: 19 October 2025 — If $n$ distinct points in $\mathbb{R}^2$ form a convex polygon then some vertex has at least $\lfloor \frac{n}{2}\rfloor$ different dista... - [#983](https://erdosproblems.com/983) — OPEN | number theory — edited: 18 January 2026 — Let $n\geq 2$ and $\pi(n)0$. -- [#987](https://erdosproblems.com/987) — OPEN | analysis, discrepancy — edited: 29 December 2025 — Let $x_1,x_2,\ldots \in (0,1)$ be an infinite sequence and let\[A_k=\limsup_{n\to \infty}\left\lvert \sum_{j\leq n} e(kx_j)\right\rvert,\... -- [#990](https://erdosproblems.com/990) — OPEN | analysis — Let $f=a_0+\cdots+a_dx^d\in \mathbb{C}[x]$ be a polynomial. Is it true that, if $f$ has roots $z_1,\ldots,z_d$ with corresponding argumen... - [#993](https://erdosproblems.com/993) — FALSIFIABLE | graph theory — edited: 01 February 2026 — The independent set sequence of any tree or forest is unimodal. In other words, if $i_k(G)$ counts the number of independent sets of vert... - [#995](https://erdosproblems.com/995) — OPEN | analysis, discrepancy — Let $n_10$, if $k$ is sufficiently large then, for all $n>0$ and interva... - [#1002](https://erdosproblems.com/1002) — OPEN | analysis, diophantine approximation — For any $0<\alpha<1$, let\[f(\alpha,n)=\frac{1}{\log n}\sum_{1\leq k\leq n}(\tfrac{1}{2}-\{ \alpha k\}).\]Does $f(\alpha,n)$ have an asym... -- [#1003](https://erdosproblems.com/1003) — OPEN | number theory — edited: 31 October 2025 — Are there infinitely many solutions to $\phi(n)=\phi(n+1)$, where $\phi$ is the Euler totient function? -- [#1004](https://erdosproblems.com/1004) — OPEN | number theory — Let $c>0$. If $x$ is sufficiently large then does there exist $n\leq x$ such that the values of $\phi(n+k)$ are all distinct for $1\leq k... +- [#1003](https://erdosproblems.com/1003) — OPEN | number theory — edited: 19 April 2026 — Are there infinitely many solutions to $\phi(n)=\phi(n+1)$, where $\phi$ is the Euler totient function? +- [#1004](https://erdosproblems.com/1004) — OPEN | number theory — edited: 12 April 2026 — Let $c>0$. If $x$ is sufficiently large then does there exist $n\leq x$ such that the values of $\phi(n+k)$ are all distinct for $1\leq k... - [#1005](https://erdosproblems.com/1005) — OPEN | number theory — Let $\frac{a_1}{b_1},\frac{a_2}{b_2},\ldots$ be the Farey fractions of order $n\geq 4$. Let $f(n)$ be the largest integer such that if $1... - [#1011](https://erdosproblems.com/1011) — OPEN | graph theory — edited: 06 December 2025 — Let $f_r(n)$ be minimal such that every graph on $n$ vertices with $\geq f_r(n)$ edges and chromatic number $\geq r$ contains a triangle.... - [#1013](https://erdosproblems.com/1013) — OPEN | graph theory, chromatic number — edited: 21 January 2026 — Let $h_3(k)$ be the minimal $n$ such that there exists a triangle-free graph on $n$ vertices with chromatic number $k$. Find an asymptoti... -- [#1014](https://erdosproblems.com/1014) — OPEN | graph theory, ramsey theory — Let $R(k,l)$ be the Ramsey number, so the minimal $n$ such that every graph on at least $n$ vertices contains either a $K_k$ or an indepe... - [#1016](https://erdosproblems.com/1016) — OPEN | graph theory, cycles — edited: 27 December 2025 — Let $h(n)$ be minimal such that there is a graph on $n$ vertices with $n+h(n)$ edges which contains a cycle on $k$ vertices, for all $3\l... - [#1017](https://erdosproblems.com/1017) — OPEN | graph theory — edited: 28 December 2025 — Let $f(n,k)$ be such that every graph on $n$ vertices and $k$ edges can be partitioned into at most $f(n,k)$ edge-disjoint complete graph... - [#1020](https://erdosproblems.com/1020) — FALSIFIABLE | graph theory, hypergraphs — edited: 28 December 2025 — Let $f(n;r,k)$ be the maximal number of edges in an $r$-uniform hypergraph which contains no set of $k$ many independent edges. For all $... - [#1029](https://erdosproblems.com/1029) — OPEN | $100 | graph theory, ramsey theory — If $R(k)$ is the Ramsey number for $K_k$, the minimal $n$ such that every $2$-colouring of the edges of $K_n$ contains a monochromatic co... -- [#1030](https://erdosproblems.com/1030) — OPEN | graph theory, ramsey theory — edited: 03 December 2025 — If $R(k,l)$ is the Ramsey number then prove the existence of some $c>0$ such that\[\lim_k \frac{R(k+1,k)}{R(k,k)}> 1+c.\] +- [#1030](https://erdosproblems.com/1030) — OPEN | graph theory, ramsey theory — edited: 23 March 2026 — Let $R(k,l)$ be the usual Ramsey number: the smallest $n$ such that if the edges of $K_n$ are coloured red and blue then there exists eit... - [#1032](https://erdosproblems.com/1032) — OPEN | graph theory, chromatic number — edited: 23 January 2026 — We say that a graph is $4$-chromatic critical if it has chromatic number $4$, and removing any edge decreases the chromatic number to $3$... -- [#1033](https://erdosproblems.com/1033) — OPEN | graph theory — Let $h(n)$ be such that every graph on $n$ vertices with $>n^2/4$ many edges contains a triangle whose vertices have degrees summing to a... +- [#1033](https://erdosproblems.com/1033) — OPEN | graph theory — edited: 03 April 2026 — Let $h(n)$ be such that every graph on $n$ vertices with $>n^2/4$ many edges contains a triangle whose vertices have degrees summing to a... - [#1035](https://erdosproblems.com/1035) — OPEN | graph theory — Is there a constant $c>0$ such that every graph on $2^n$ vertices with minimum degree $>(1-c)2^n$ contains the $n$-dimensional hypercube... - [#1038](https://erdosproblems.com/1038) — OPEN | analysis — edited: 11 January 2026 — Determine the infimum and supremum of\[\lvert \{ x\in \mathbb{R} : \lvert f(x)\rvert < 1\}\rvert\]as $f\in \mathbb{R}[x]$ ranges over all... - [#1039](https://erdosproblems.com/1039) — OPEN | analysis, polynomials — edited: 27 December 2025 — Let $f(z)=\prod_{i=1}^n(z-z_i)\in \mathbb{C}[z]$ with $\lvert z_i\rvert \leq 1$ for all $i$. Let $\rho(f)$ be the radius of the largest d... - [#1040](https://erdosproblems.com/1040) — OPEN | analysis — edited: 01 February 2026 — Let $F\subseteq \mathbb{C}$ be a closed infinite set, and let $\mu(F)$ be the infimum of\[\lvert \{ z: \lvert f(z)\rvert < 1\}\rvert,\]as... - [#1041](https://erdosproblems.com/1041) — FALSIFIABLE | analysis, polynomials — edited: 06 December 2025 — Let $f(z)=\prod_{i=1}^n(z-z_i)\in \mathbb{C}[z]$ with $\lvert z_i\rvert < 1$ for all $i$. Must there always exist a path of length less t... -- [#1045](https://erdosproblems.com/1045) — FALSIFIABLE | analysis — edited: 30 December 2025 — Let $z_1,\ldots,z_n\in \mathbb{C}$ with $\lvert z_i-z_j\rvert\leq 2$ for all $i,j$, and\[\Delta(z_1,\ldots,z_n)=\prod_{i\neq j}\lvert z_i... +- [#1045](https://erdosproblems.com/1045) — OPEN | analysis — edited: 02 April 2026 — Let $z_1,\ldots,z_n\in \mathbb{C}$ with $\lvert z_i-z_j\rvert\leq 2$ for all $i,j$, and\[\Delta(z_1,\ldots,z_n)=\prod_{i\neq j}\lvert z_i... - [#1049](https://erdosproblems.com/1049) — OPEN | irrationality — edited: 28 September 2025 — Let $t>1$ be a rational number. Is\[\sum_{n=1}^\infty\frac{1}{t^n-1}=\sum_{n=1}^\infty \frac{\tau(n)}{t^n}\]irrational, where $\tau(n)$ c... - [#1052](https://erdosproblems.com/1052) — OPEN | $10 | number theory — edited: 28 September 2025 — A unitary divisor of $n$ is $d\mid n$ such that $(d,n/d)=1$. A number $n\geq 1$ is a unitary perfect number if it is the sum of its unita... - [#1053](https://erdosproblems.com/1053) — OPEN | number theory — Call a number $k$-perfect if $\sigma(n)=kn$, where $\sigma(n)$ is the sum of the divisors of $n$. Must $k=o(\log\log n)$? @@ -627,20 +591,17 @@ - [#1073](https://erdosproblems.com/1073) — OPEN | number theory — edited: 06 October 2025 — Let $A(x)$ count the number of composite $ur^{-r}$ such that, for any $\epsilon>0$, if $n$ is sufficiently large, the following holds. Any $r$-unif... -- [#1082](https://erdosproblems.com/1082) — FALSIFIABLE | geometry, distances — edited: 20 December 2025 — Let $A\subset \mathbb{R}^2$ be a set of $n$ points with no three on a line. Does $A$ determine at least $\lfloor n/2\rfloor$ distinct dis... +- [#1082](https://erdosproblems.com/1082) — FALSIFIABLE | geometry, distances — edited: 11 April 2026 — Let $A\subset \mathbb{R}^2$ be a set of $n$ points with no three on a line. Does $A$ determine at least $\lfloor n/2\rfloor$ distinct dis... - [#1083](https://erdosproblems.com/1083) — OPEN | geometry, distances — edited: 16 October 2025 — Let $d\geq 3$, and let $f_d(n)$ be the minimal $m$ such that every set of $n$ points in $\mathbb{R}^d$ determines at least $m$ distinct d... - [#1084](https://erdosproblems.com/1084) — OPEN | geometry, distances — edited: 08 February 2026 — Let $f_d(n)$ be minimal such that in any collection of $n$ points in $\mathbb{R}^d$, all of distance at least $1$ apart, there are at mos... -- [#1085](https://erdosproblems.com/1085) — OPEN | geometry, distances — edited: 17 October 2025 — Let $f_d(n)$ be minimal such that, in any set of $n$ points in $\mathbb{R}^d$, there exist at most $f_d(n)$ pairs of points which distanc... +- [#1085](https://erdosproblems.com/1085) — OPEN | geometry, distances — edited: 23 May 2026 — Let $f_d(n)$ be minimal such that, in any set of $n$ points in $\mathbb{R}^d$, there exist at most $f_d(n)$ pairs of points which distanc... - [#1086](https://erdosproblems.com/1086) — OPEN | geometry, distances — edited: 16 October 2025 — Let $g(n)$ be minimal such that any set of $n$ points in $\mathbb{R}^2$ contains the vertices of at most $g(n)$ many triangles with the s... - [#1087](https://erdosproblems.com/1087) — OPEN | geometry, distances — Let $f(n)$ be minimal such that every set of $n$ points in $\mathbb{R}^2$ contains at most $f(n)$ many sets of four points which are 'deg... -- [#1088](https://erdosproblems.com/1088) — OPEN | geometry — edited: 16 October 2025 — Let $f_d(n)$ be the minimal $m$ such that any set of $m$ points in $\mathbb{R}^d$ contains a set of $n$ points such that any two determin... -- [#1091](https://erdosproblems.com/1091) — OPEN | graph theory, chromatic number — edited: 06 December 2025 — Let $G$ be a $K_4$-free graph with chromatic number $4$. Must $G$ contain an odd cycle with at least two diagonals? More generally, is th... -- [#1092](https://erdosproblems.com/1092) — OPEN | graph theory, chromatic number — edited: 06 December 2025 — Let $f_r(n)$ be maximal such that, if a graph $G$ has the property that every subgraph $H$ on $m$ vertices is the union of a graph with c... +- [#1088](https://erdosproblems.com/1088) — OPEN | geometry — edited: 08 April 2026 — Let $f_d(n)$ be the minimal $m$ such that any set of $m$ points in $\mathbb{R}^d$ contains a set of $n$ points such that any two determin... - [#1093](https://erdosproblems.com/1093) — OPEN | number theory, binomial coefficients — edited: 27 December 2025 — For $n\geq 2k$ we define the deficiency of $\binom{n}{k}$ as follows. If $\binom{n}{k}$ is divisible by a prime $p\leq k$ then the defici... - [#1094](https://erdosproblems.com/1094) — OPEN | number theory, binomial coefficients — edited: 24 October 2025 — For all $n\geq 2k$ the least prime factor of $\binom{n}{k}$ is $\leq \max(n/k,k)$, with only finitely many exceptions. - [#1095](https://erdosproblems.com/1095) — OPEN | number theory, binomial coefficients — edited: 12 January 2026 — Let $g(k)>k+1$ be the smallest $n$ such that all prime factors of $\binom{n}{k}$ are $>k$. Estimate $g(k)$. -- [#1096](https://erdosproblems.com/1096) — OPEN | number theory — edited: 19 October 2025 — Let $1q\geq 2$ be two coprime integers. We call $n$ representable if it is the sum of integers of the form $p^kq^l$, none of which divid... +- [#1110](https://erdosproblems.com/1110) — OPEN | number theory — edited: 01 April 2026 — Let $p>q\geq 2$ be two coprime integers. We call $n$ representable if it is the sum of integers of the form $p^kq^l$, none of which divid... - [#1111](https://erdosproblems.com/1111) — OPEN | graph theory — edited: 07 December 2025 — If $G$ is a finite graph and $A,B$ are disjoint sets of vertices then we call $A,B$ anticomplete if there are no edges between $A$ and $B... - [#1112](https://erdosproblems.com/1112) — OPEN | additive combinatorics — edited: 28 December 2025 — Let $1\leq d_10$. There exists $\epsilon>0$ such that if $n$ is sufficiently large the following holds. For any $x_1,\ldots,x_n\in [-1,1]$ there... - [#1135](https://erdosproblems.com/1135) — OPEN | $500 | number theory, iterated functions — edited: 12 January 2026 — Define $f:\mathbb{N}\to \mathbb{N}$ by $f(n)=n/2$ if $n$ is even and $f(n)=\frac{3n+1}{2}$ if $n$ is odd. Given any integer $m\geq 1$ doe... - [#1137](https://erdosproblems.com/1137) — OPEN | number theory, primes — edited: 23 January 2026 — Let $d_n=p_{n+1}-p_n$, where $p_n$ denotes the $n$th prime. Is it true that\[\frac{\max_{n1$. If $d=\max_{p_n1$. If $d=\max_{p_n105$) such that $n-2^k$ is prime for all $1<2^k105$) such that $n-2^k$ is prime for all $1<2^kd_s(B)$ for every $B\subset \mathbb{N}$ with $00$ such that, for all large $n$ and all polynomials $P$ of degree $n$ with coefficients $\pm 1$,\[\max_{\l... - [#1151](https://erdosproblems.com/1151) — OPEN | analysis, polynomials — edited: 23 January 2026 — Given $a_1,\ldots,a_n\in [-1,1]$ let\[\mathcal{L}^nf(x) = \sum_{1\leq i\leq n}f(a_i)\ell_i(x)\]be the unique polynomial of degree $n-1$ w... - [#1152](https://erdosproblems.com/1152) — OPEN | analysis, polynomials — edited: 23 January 2026 — For $n\geq 1$ fix some sequence of $n$ distinct numbers $x_{1n},\ldots,x_{nn}\in [-1,1]$. Let $\epsilon=\epsilon(n)\to 0$. Does there alw... -- [#1153](https://erdosproblems.com/1153) — PROVED | analysis, polynomials — edited: 01 February 2026 — For $x_1,\ldots,x_n\in [-1,1]$ let\[l_k(x)=\frac{\prod_{i\neq k}(x-x_i)}{\prod_{i\neq k}(x_k-x_i)},\]which are such that $l_k(x_k)=1$ and... - [#1154](https://erdosproblems.com/1154) — NOT DISPROVABLE | analysis — edited: 25 January 2026 — Does there exist, for every $\alpha \in [0,1]$, a ring or field in $\mathbb{R}$ with Hausdorff dimension $\alpha$? - [#1155](https://erdosproblems.com/1155) — OPEN | graph theory — edited: 25 January 2026 — Construct a random graph on $n$ vertices in the following way: begin with the complete graph $K_n$. At each stage, choose uniformly a ran... - [#1156](https://erdosproblems.com/1156) — OPEN | graph theory, chromatic number — edited: 27 January 2026 — Let $G$ be a random graph on $n$ vertices, in which every edge is included independently with probability $1/2$. Is there some constant $... - [#1157](https://erdosproblems.com/1157) — OPEN | hypergraphs, turan number — edited: 24 January 2026 — Let $t,k,r\geq 2$. Let $\mathcal{F}$ be the family of all $r$-uniform hypergraphs with $k$ vertices and $s$ edges. Determine\[\mathrm{ex}... - [#1158](https://erdosproblems.com/1158) — OPEN | hypergraphs, turan number — edited: 23 January 2026 — Let $K_{t}(r)$ be the complete $t$-partite $t$-uniform hypergraph with $r$ vertices in each class. Is it true that\[\mathrm{ex}_t(n,K_t(r... -- [#1159](https://erdosproblems.com/1159) — OPEN | combinatorics — edited: 23 January 2026 — Determine whether there exists a constant $C>1$ such that the following holds. Let $P$ be a finite projective plane . Must there exist a... +- [#1159](https://erdosproblems.com/1159) — OPEN | combinatorics — edited: 10 April 2026 — Determine whether there exists a constant $C>1$ such that the following holds. Let $P$ be a finite projective plane . Must there exist a... - [#1160](https://erdosproblems.com/1160) — OPEN | group theory — edited: 26 January 2026 — Let $g(n)$ denote the number of groups of order $n$. If $n\leq 2^m$ then $g(n)\leq g(2^m)$. - [#1162](https://erdosproblems.com/1162) — OPEN | group theory — edited: 23 January 2026 — Give an asymptotic formula for the number of subgroups of $S_n$. Is there a statistical theorem on their order? - [#1163](https://erdosproblems.com/1163) — OPEN | group theory — Describe (by statistical means) the arithmetic structure of the orders of subgroups of $S_n$. @@ -688,10 +646,32 @@ - [#1169](https://erdosproblems.com/1169) — NOT DISPROVABLE | set theory, ramsey theory — edited: 25 January 2026 — Is it true that, for all finite $k<\omega$,\[\omega_1^2 \not\to (\omega_1^2, 3)^2?\] - [#1170](https://erdosproblems.com/1170) — OPEN | set theory, ramsey theory — edited: 23 January 2026 — Is it consistent that\[\omega_2\to (\alpha)_2^2\]for every $\alpha <\omega_2$? - [#1171](https://erdosproblems.com/1171) — OPEN | set theory, ramsey theory — edited: 26 January 2026 — Is it true that, for all finite $k<\omega$,\[\omega_1^2\to (\omega_1\omega, 3,\ldots,3)_{k+1}^2?\] -- [#1172](https://erdosproblems.com/1172) — OPEN | set theory, ramsey theory — edited: 23 January 2026 — Establish whether the following are true assuming the generalised continuum hypothesis:\[\omega_3 \to (\omega_2,\omega_1+2)^2,\]\[\omega_... +- [#1172](https://erdosproblems.com/1172) — OPEN | set theory, ramsey theory — edited: 11 April 2026 — Establish whether the following are true assuming the generalised continuum hypothesis:\[\omega_3 \to (\omega_2,\omega_1+2)^2,\]\[\omega_... - [#1173](https://erdosproblems.com/1173) — OPEN | set theory, combinatorics — edited: 25 January 2026 — Assume the generalised continuum hypothesis. Let\[f: \omega_{\omega+1}\to [\omega_{\omega+1}]^{\leq \aleph_\omega}\]be a set mapping such... -- [#1174](https://erdosproblems.com/1174) — OPEN | set theory, ramsey theory — Does there exist a graph $G$ with no $K_4$ such that every edge colouring of $G$ with countably many colours contains a monochromatic $K_... +- [#1174](https://erdosproblems.com/1174) — NOT DISPROVABLE | set theory, ramsey theory — Does there exist a graph $G$ with no $K_4$ such that every edge colouring of $G$ with countably many colours contains a monochromatic $K_... - [#1175](https://erdosproblems.com/1175) — OPEN | set theory, chromatic number — Let $\kappa$ be an uncountable cardinal. Must there exist a cardinal $\lambda$ such that every graph with chromatic number $\lambda$ cont... -- [#1176](https://erdosproblems.com/1176) — OPEN | set theory, ramsey theory — edited: 24 January 2026 — Let $G$ be a graph with chromatic number $\aleph_1$. Is it true that there is a colouring of the edges with $\aleph_1$ many colours such... +- [#1176](https://erdosproblems.com/1176) — NOT DISPROVABLE | set theory, ramsey theory — edited: 24 January 2026 — Let $G$ be a graph with chromatic number $\aleph_1$. Is it true that there is a colouring of the edges with $\aleph_1$ many colours such... - [#1177](https://erdosproblems.com/1177) — OPEN | set theory, chromatic number, hypergraphs — edited: 23 January 2026 — Let $G$ be a finite $3$-uniform hypergraph, and let $F_G(\kappa)$ denote the collection of $3$-uniform hypergraphs with chromatic number... - [#1178](https://erdosproblems.com/1178) — OPEN | graph theory, hypergraphs — edited: 26 January 2026 — For $r\geq 3$ let $d_r(e)$ be the minimal $d$ such that\[\mathrm{ex}_r(n,\mathcal{F})=o(n^2),\]where $\mathcal{F}$ is the family of $r$-u... +- [#1181](https://erdosproblems.com/1181) — OPEN | number theory — edited: 07 March 2026 — Let $q(n,k)$ denote the least prime which does not divide $\prod_{1\leq i\leq k}(n+i)$. Is it true that there exists some $c>0$ such that... +- [#1182](https://erdosproblems.com/1182) — OPEN | graph theory, ramsey theory — edited: 11 April 2026 — Let $f(n)$ be maximal such that there is a connected graph $G$ with $n$ vertices and $f(n)$ edges such that\[R(K_3,G)= 2n-1.\]Let $F(n)$... +- [#1183](https://erdosproblems.com/1183) — OPEN | combinatorics, ramsey theory — Let $f(n)$ be maximal such that in any $2$-colouring of the subsets of $\{1,\ldots,n\}$ there is always a monochromatic family of at leas... +- [#1184](https://erdosproblems.com/1184) — OPEN | number theory, primes — edited: 06 April 2026 — Let $f(n,k)$ count the number of $1\leq i\leq k$ such that $P(n+i)>k$ (where $P(m)$ is the largest prime divisor of $m$). Is it true that... +- [#1186](https://erdosproblems.com/1186) — OPEN | additive combinatorics, arithmetic progressions — edited: 08 April 2026 — Let $\delta_k$ be such that in any $2$-colouring of $\{1,\ldots,n\}$ there exist at least $(\delta_k+o(1))n^2$ many monochromatic $k$-ter... +- [#1188](https://erdosproblems.com/1188) — OPEN | number theory, covering systems — edited: 17 April 2026 — Call a set of distinct integers $10$ there exists a $k$ such that the density of $n$ for which\[P(n(n+1)\cdots(n+k))>n^{1-\epsilon... +- [#1203](https://erdosproblems.com/1203) — OPEN | number theory — edited: 07 April 2026 — If $\omega(n)$ counts the number of distinct prime divisors of $n$ then let\[F(n)=\max_k \omega(n+k)\frac{\log\log k}{\log k}.\]Prove tha... +- [#1204](https://erdosproblems.com/1204) — OPEN | number theory — edited: 07 April 2026 — We call a sequence of integers $0\leq a_1<\cdots 0$ for $0<\\alpha <1 $? The Schnirelmann density is defined by\\[d_s(A) = \\inf_{N\\geq 1}\\frac{\\lvert A\\cap\\{1,\\ldots,N\\}\\rvert}{N}.\\]", - "tags": [ - "number theory" - ], - "last_edited": "16 September 2025", - "latex_path": "/latex/38", - "formalized": true, - "formalized_url": "https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/ErdosProblems/38.lean", - "oeis_urls": [], - "comments_problem_id": 38, - "comments_count": 1 + "comments_count": 5 }, { "problem_id": 39, @@ -1211,7 +1170,7 @@ "sidon sets", "additive combinatorics" ], - "last_edited": "23 January 2026", + "last_edited": "06 April 2026", "latex_path": "/latex/39", "formalized": true, "formalized_url": "https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/ErdosProblems/39.lean", @@ -1254,7 +1213,7 @@ "sidon sets", "additive combinatorics" ], - "last_edited": "23 January 2026", + "last_edited": "06 April 2026", "latex_path": "/latex/41", "formalized": true, "formalized_url": "https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/ErdosProblems/41.lean", @@ -1262,54 +1221,6 @@ "comments_problem_id": 41, "comments_count": 0 }, - { - "problem_id": 42, - "problem_url": "/42", - "status_bucket": "open", - "status_dom_id": "open", - "status_label": "OPEN", - "status_detail": "This is open, and cannot be resolved with a finite computation.", - "prize_amount": "", - "statement": "Let $M\\geq 1$ and $N$ be sufficiently large in terms of $M$. Is it true that for every Sidon set $A\\subset \\{1,\\ldots,N\\}$ there is another Sidon set $B\\subset \\{1,\\ldots,N\\}$ of size $M$ such that $(A-A)\\cap(B-B)=\\{0\\}$?", - "tags": [ - "number theory", - "sidon sets", - "additive combinatorics" - ], - "last_edited": "23 January 2026", - "latex_path": "/latex/42", - "formalized": true, - "formalized_url": "https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/ErdosProblems/42.lean", - "oeis_urls": [], - "comments_problem_id": 42, - "comments_count": 6 - }, - { - "problem_id": 43, - "problem_url": "/43", - "status_bucket": "open", - "status_dom_id": "open", - "status_label": "OPEN", - "status_detail": "This is open, and cannot be resolved with a finite computation.", - "prize_amount": "$100", - "statement": "If $A,B\\subset \\{1,\\ldots,N\\}$ are two Sidon sets such that $(A-A)\\cap(B-B)=\\{0\\}$ then is it true that\\[ \\binom{\\lvert A\\rvert}{2}+\\binom{\\lvert B\\rvert}{2}\\leq\\binom{f(N)}{2}+O(1),\\]where $f(N)$ is the maximum possible size of a Sidon set in $\\{1,\\ldots,N\\}$? If $\\lvert A\\rvert=\\lvert B\\rvert$ then can this bound be improved to\\[\\binom{\\lvert A\\rvert}{2}+\\binom{\\lvert B\\rvert}{2}\\leq (1-c+o(1))\\binom{f(N)}{2}\\]for some constant $c>0$?", - "tags": [ - "number theory", - "sidon sets", - "additive combinatorics" - ], - "last_edited": "20 December 2025", - "latex_path": "/latex/43", - "formalized": false, - "formalized_url": "", - "oeis_urls": [ - "https://oeis.org/A003022", - "https://oeis.org/A143824", - "https://oeis.org/A227590" - ], - "comments_problem_id": 43, - "comments_count": 9 - }, { "problem_id": 44, "problem_url": "/44", @@ -1346,8 +1257,8 @@ ], "last_edited": "", "latex_path": "/latex/50", - "formalized": false, - "formalized_url": "", + "formalized": true, + "formalized_url": "https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/ErdosProblems/50.lean", "oeis_urls": [], "comments_problem_id": 50, "comments_count": 0 @@ -1388,10 +1299,10 @@ "number theory", "additive combinatorics" ], - "last_edited": "23 January 2026", + "last_edited": "08 April 2026", "latex_path": "/latex/52", - "formalized": false, - "formalized_url": "", + "formalized": true, + "formalized_url": "https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/ErdosProblems/52.lean", "oeis_urls": [ "https://oeis.org/A263996" ], @@ -1415,7 +1326,9 @@ "latex_path": "/latex/60", "formalized": false, "formalized_url": "", - "oeis_urls": [], + "oeis_urls": [ + "https://oeis.org/A006855" + ], "comments_problem_id": 60, "comments_count": 0 }, @@ -1431,13 +1344,13 @@ "tags": [ "graph theory" ], - "last_edited": "23 January 2026", + "last_edited": "10 April 2026", "latex_path": "/latex/61", "formalized": true, "formalized_url": "https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/ErdosProblems/61.lean", "oeis_urls": [], "comments_problem_id": 61, - "comments_count": 3 + "comments_count": 6 }, { "problem_id": 62, @@ -1472,7 +1385,7 @@ "graph theory", "cycles" ], - "last_edited": "18 January 2026", + "last_edited": "10 April 2026", "latex_path": "/latex/64", "formalized": true, "formalized_url": "https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/ErdosProblems/64.lean", @@ -1514,7 +1427,7 @@ "number theory", "additive basis" ], - "last_edited": "23 January 2026", + "last_edited": "06 April 2026", "latex_path": "/latex/66", "formalized": true, "formalized_url": "https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/ErdosProblems/66.lean", @@ -1604,11 +1517,11 @@ ], "last_edited": "29 January 2026", "latex_path": "/latex/75", - "formalized": false, - "formalized_url": "", + "formalized": true, + "formalized_url": "https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/ErdosProblems/75.lean", "oeis_urls": [], "comments_problem_id": 75, - "comments_count": 0 + "comments_count": 3 }, { "problem_id": 77, @@ -1631,7 +1544,7 @@ "https://oeis.org/A059442" ], "comments_problem_id": 77, - "comments_count": 2 + "comments_count": 3 }, { "problem_id": 78, @@ -1654,7 +1567,7 @@ "https://oeis.org/A059442" ], "comments_problem_id": 78, - "comments_count": 3 + "comments_count": 5 }, { "problem_id": 80, @@ -1669,13 +1582,13 @@ "graph theory", "ramsey theory" ], - "last_edited": "", + "last_edited": "07 April 2026", "latex_path": "/latex/80", "formalized": false, "formalized_url": "", "oeis_urls": [], "comments_problem_id": 80, - "comments_count": 0 + "comments_count": 1 }, { "problem_id": 81, @@ -1709,17 +1622,28 @@ "tags": [ "graph theory" ], - "last_edited": "06 October 2025", + "last_edited": "10 April 2026", "latex_path": "/latex/82", "formalized": true, "formalized_url": "https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/ErdosProblems/82.lean", "oeis_urls": [ "https://oeis.org/A120414", "https://oeis.org/A390256", - "https://oeis.org/A390257" + "https://oeis.org/A390257", + "https://oeis.org/A390919", + "https://oeis.org/A392636", + "https://oeis.org/A394400", + "https://oeis.org/A394462", + "https://oeis.org/A394539", + "https://oeis.org/A394563", + "https://oeis.org/A394564", + "https://oeis.org/A394573", + "https://oeis.org/A394574", + "https://oeis.org/A394930", + "https://oeis.org/A394933" ], "comments_problem_id": 82, - "comments_count": 16 + "comments_count": 19 }, { "problem_id": 84, @@ -1747,8 +1671,8 @@ "problem_url": "/85", "status_bucket": "open", "status_dom_id": "open", - "status_label": "FALSIFIABLE", - "status_detail": "Open, but could be disproved with a finite counterexample.", + "status_label": "OPEN", + "status_detail": "This is open, and cannot be resolved with a finite computation.", "prize_amount": "", "statement": "Let $n\\geq 4$ and $f(n)$ be minimal such that every graph on $n$ vertices with minimal degree $\\geq f(n)$ contains a $C_4$. Is it true that, for all large $n$, $f(n+1)\\geq f(n)$?", "tags": [ @@ -1785,7 +1709,7 @@ "https://oeis.org/A245762" ], "comments_problem_id": 86, - "comments_count": 1 + "comments_count": 2 }, { "problem_id": 87, @@ -1834,29 +1758,6 @@ "comments_problem_id": 89, "comments_count": 0 }, - { - "problem_id": 90, - "problem_url": "/90", - "status_bucket": "open", - "status_dom_id": "open", - "status_label": "OPEN", - "status_detail": "This is open, and cannot be resolved with a finite computation.", - "prize_amount": "$500", - "statement": "Does every set of $n$ distinct points in $\\mathbb{R}^2$ contain at most $n^{1+O(1/\\log\\log n)}$ many pairs which are distance 1 apart?", - "tags": [ - "geometry", - "distances" - ], - "last_edited": "23 January 2026", - "latex_path": "/latex/90", - "formalized": true, - "formalized_url": "https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/ErdosProblems/90.lean", - "oeis_urls": [ - "https://oeis.org/A186705" - ], - "comments_problem_id": 90, - "comments_count": 0 - }, { "problem_id": 91, "problem_url": "/91", @@ -1865,41 +1766,20 @@ "status_label": "OPEN", "status_detail": "This is open, and cannot be resolved with a finite computation.", "prize_amount": "", - "statement": "Let $n$ be a sufficently large integer. Suppose $A\\subset \\mathbb{R}^2$ has $\\lvert A\\rvert=n$ and minimises the number of distinct distances between points in $A$. Prove that there are at least two (and probably many) such $A$ which are non-similar.", + "statement": "Let $n$ be a sufficiently large integer. Suppose $A\\subset \\mathbb{R}^2$ has $\\lvert A\\rvert=n$ and minimises the number of distinct distances between points in $A$. Prove that there are at least two (and probably many) such $A$ which are non-similar.", "tags": [ "geometry", "distances" ], - "last_edited": "16 January 2026", + "last_edited": "13 April 2026", "latex_path": "/latex/91", - "formalized": false, - "formalized_url": "", + "formalized": true, + "formalized_url": "https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/ErdosProblems/91.lean", "oeis_urls": [ "https://oeis.org/A186704" ], "comments_problem_id": 91, - "comments_count": 5 - }, - { - "problem_id": 92, - "problem_url": "/92", - "status_bucket": "open", - "status_dom_id": "open", - "status_label": "OPEN", - "status_detail": "This is open, and cannot be resolved with a finite computation.", - "prize_amount": "$500", - "statement": "Let $f(n)$ be maximal such that there exists a set $A$ of $n$ points in $\\mathbb{R}^2$ in which every $x\\in A$ has at least $f(n)$ points in $A$ equidistant from $x$. Is it true that $f(n)\\leq n^{o(1)}$? Or even $f(n) < n^{O(1/\\log\\log n)}$?", - "tags": [ - "geometry", - "distances" - ], - "last_edited": "28 December 2025", - "latex_path": "/latex/92", - "formalized": true, - "formalized_url": "https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/ErdosProblems/92.lean", - "oeis_urls": [], - "comments_problem_id": 92, - "comments_count": 1 + "comments_count": 7 }, { "problem_id": 96, @@ -1917,8 +1797,8 @@ ], "last_edited": "23 January 2026", "latex_path": "/latex/96", - "formalized": false, - "formalized_url": "", + "formalized": true, + "formalized_url": "https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/ErdosProblems/96.lean", "oeis_urls": [], "comments_problem_id": 96, "comments_count": 5 @@ -1943,7 +1823,7 @@ "formalized_url": "https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/ErdosProblems/97.lean", "oeis_urls": [], "comments_problem_id": 97, - "comments_count": 0 + "comments_count": 2 }, { "problem_id": 98, @@ -1960,8 +1840,8 @@ ], "last_edited": "15 October 2025", "latex_path": "/latex/98", - "formalized": false, - "formalized_url": "", + "formalized": true, + "formalized_url": "https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/ErdosProblems/98.lean", "oeis_urls": [], "comments_problem_id": 98, "comments_count": 0 @@ -2022,8 +1902,8 @@ ], "last_edited": "27 December 2025", "latex_path": "/latex/101", - "formalized": false, - "formalized_url": "", + "formalized": true, + "formalized_url": "https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/ErdosProblems/101.lean", "oeis_urls": [ "https://oeis.org/A006065" ], @@ -2105,7 +1985,7 @@ "tags": [ "geometry" ], - "last_edited": "", + "last_edited": "06 March 2026", "latex_path": "/latex/106", "formalized": false, "formalized_url": "", @@ -2126,7 +2006,7 @@ "geometry", "convex" ], - "last_edited": "23 January 2026", + "last_edited": "11 April 2026", "latex_path": "/latex/107", "formalized": true, "formalized_url": "https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/ErdosProblems/107.lean", @@ -2134,7 +2014,7 @@ "https://oeis.org/A000051" ], "comments_problem_id": 107, - "comments_count": 3 + "comments_count": 2 }, { "problem_id": 108, @@ -2206,8 +2086,8 @@ "problem_url": "/114", "status_bucket": "open", "status_dom_id": "open", - "status_label": "OPEN", - "status_detail": "This is open, and cannot be resolved with a finite computation.", + "status_label": "FALSIFIABLE", + "status_detail": "Open, but could be disproved with a finite counterexample.", "prize_amount": "$250", "statement": "If $p(z)\\in\\mathbb{C}[z]$ is a monic polynomial of degree $n$ then is the length of the curve $\\{ z\\in \\mathbb{C} : \\lvert p(z)\\rvert=1\\}$ maximised when $p(z)=z^n-1$?", "tags": [ @@ -2220,7 +2100,7 @@ "formalized_url": "", "oeis_urls": [], "comments_problem_id": 114, - "comments_count": 3 + "comments_count": 6 }, { "problem_id": 117, @@ -2281,7 +2161,7 @@ "formalized_url": "https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/ErdosProblems/120.lean", "oeis_urls": [], "comments_problem_id": 120, - "comments_count": 1 + "comments_count": 3 }, { "problem_id": 122, @@ -2291,17 +2171,17 @@ "status_label": "OPEN", "status_detail": "This is open, and cannot be resolved with a finite computation.", "prize_amount": "", - "statement": "For which number theoretic functions $f$ is it true that, for any $F(n)$ such that $f(n)/F(n)\\to 0$ for almost all $n$, there are infinitely many $x$ such that\\[\\frac{\\#\\{ n\\in \\mathbb{N} : n+f(n)\\in (x,x+F(x))\\}}{F(x)}\\to \\infty?\\]", + "statement": "For which number theoretic functions $f$ is it true that, for any $F(n)$ such that $F(n)/f(n)\\to 0$ for almost all $n$, there are infinitely many $x$ such that\\[\\frac{\\#\\{ n\\in \\mathbb{N} : n+f(n)\\in (x,x+F(x))\\}}{F(x)}\\to \\infty?\\]", "tags": [ "number theory" ], - "last_edited": "01 February 2026", + "last_edited": "01 April 2026", "latex_path": "/latex/122", "formalized": false, "formalized_url": "", "oeis_urls": [], "comments_problem_id": 122, - "comments_count": 0 + "comments_count": 3 }, { "problem_id": 123, @@ -2321,7 +2201,7 @@ "formalized_url": "https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/ErdosProblems/123.lean", "oeis_urls": [], "comments_problem_id": 123, - "comments_count": 1 + "comments_count": 4 }, { "problem_id": 124, @@ -2343,30 +2223,7 @@ "formalized_url": "https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/ErdosProblems/124.lean", "oeis_urls": [], "comments_problem_id": 124, - "comments_count": 13 - }, - { - "problem_id": 125, - "problem_url": "/125", - "status_bucket": "open", - "status_dom_id": "open", - "status_label": "OPEN", - "status_detail": "This is open, and cannot be resolved with a finite computation.", - "prize_amount": "", - "statement": "Let $A = \\{ \\sum\\epsilon_k3^k : \\epsilon_k\\in \\{0,1\\}\\}$ be the set of integers which have only the digits $0,1$ when written base $3$, and $B=\\{ \\sum\\epsilon_k4^k : \\epsilon_k\\in \\{0,1\\}\\}$ be the set of integers which have only the digits $0,1$ when written base $4$. Does $A+B$ have positive density?", - "tags": [ - "number theory", - "base representations" - ], - "last_edited": "", - "latex_path": "/latex/125", - "formalized": true, - "formalized_url": "https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/ErdosProblems/125.lean", - "oeis_urls": [ - "https://oeis.org/A367090" - ], - "comments_problem_id": 125, - "comments_count": 13 + "comments_count": 14 }, { "problem_id": 126, @@ -2525,7 +2382,7 @@ "tags": [ "additive combinatorics" ], - "last_edited": "28 December 2025", + "last_edited": "10 April 2026", "latex_path": "/latex/138", "formalized": true, "formalized_url": "https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/ErdosProblems/138.lean", @@ -2533,7 +2390,7 @@ "https://oeis.org/A005346" ], "comments_problem_id": 138, - "comments_count": 4 + "comments_count": 7 }, { "problem_id": 141, @@ -2572,7 +2429,7 @@ "additive combinatorics", "arithmetic progressions" ], - "last_edited": "23 January 2026", + "last_edited": "04 April 2026", "latex_path": "/latex/142", "formalized": true, "formalized_url": "https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/ErdosProblems/142.lean", @@ -2597,13 +2454,13 @@ "tags": [ "primitive sets" ], - "last_edited": "", + "last_edited": "24 April 2026", "latex_path": "/latex/143", "formalized": true, "formalized_url": "https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/ErdosProblems/143.lean", "oeis_urls": [], "comments_problem_id": 143, - "comments_count": 0 + "comments_count": 1 }, { "problem_id": 145, @@ -2680,11 +2537,11 @@ "status_label": "OPEN", "status_detail": "This is open, and cannot be resolved with a finite computation.", "prize_amount": "", - "statement": "Let $G$ be a graph with maximum degree $\\Delta$. Is $G$ the union of at most $\\tfrac{5}{4}\\Delta^2$ sets of strongly independent edges (sets such that the induced subgraph is the union of vertex-disjoint edges)?", + "statement": "The strong chromatic index of a graph $G$, denoted by $\\mathrm{sq}(G)$, is the minimum $k$ such that the edges of $G$ can be partitioned into $k$ sets of 'strongly independent' edges, that is, such that the subgraph of $G$ induced by each set is the union of vertex-disjoint edges. Is it true that, for any graph $G$ with maximum degree $\\Delta$,\\[\\mathrm{sq}(G)\\leq\\frac{5}{4}\\Delta^2?\\]", "tags": [ "graph theory" ], - "last_edited": "01 February 2026", + "last_edited": "10 April 2026", "latex_path": "/latex/149", "formalized": false, "formalized_url": "", @@ -2712,26 +2569,6 @@ "comments_problem_id": 151, "comments_count": 1 }, - { - "problem_id": 152, - "problem_url": "/152", - "status_bucket": "open", - "status_dom_id": "open", - "status_label": "OPEN", - "status_detail": "This is open, and cannot be resolved with a finite computation.", - "prize_amount": "", - "statement": "For any $M\\geq 1$, if $A\\subset \\mathbb{N}$ is a sufficiently large finite Sidon set then there are at least $M$ many $a\\in A+A$ such that $a+1,a-1\\not\\in A+A$.", - "tags": [ - "sidon sets" - ], - "last_edited": "", - "latex_path": "/latex/152", - "formalized": true, - "formalized_url": "https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/ErdosProblems/152.lean", - "oeis_urls": [], - "comments_problem_id": 152, - "comments_count": 0 - }, { "problem_id": 153, "problem_url": "/153", @@ -2750,7 +2587,7 @@ "formalized_url": "https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/ErdosProblems/153.lean", "oeis_urls": [], "comments_problem_id": 153, - "comments_count": 2 + "comments_count": 7 }, { "problem_id": 155, @@ -2826,13 +2663,13 @@ "status_dom_id": "open", "status_label": "OPEN", "status_detail": "This is open, and cannot be resolved with a finite computation.", - "prize_amount": "", + "prize_amount": "$100", "statement": "There exists some constant $c>0$ such that $$R(C_4,K_n) \\ll n^{2-c}.$$", "tags": [ "graph theory", "ramsey theory" ], - "last_edited": "", + "last_edited": "07 March 2026", "latex_path": "/latex/159", "formalized": false, "formalized_url": "", @@ -2904,7 +2741,7 @@ "formalized_url": "", "oeis_urls": [], "comments_problem_id": 162, - "comments_count": 3 + "comments_count": 4 }, { "problem_id": 165, @@ -2919,7 +2756,7 @@ "graph theory", "ramsey theory" ], - "last_edited": "28 December 2025", + "last_edited": "07 March 2026", "latex_path": "/latex/165", "formalized": false, "formalized_url": "", @@ -2961,7 +2798,7 @@ "tags": [ "additive combinatorics" ], - "last_edited": "24 October 2025", + "last_edited": "23 March 2026", "latex_path": "/latex/168", "formalized": true, "formalized_url": "https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/ErdosProblems/168.lean", @@ -2987,7 +2824,7 @@ "additive combinatorics", "arithmetic progressions" ], - "last_edited": "", + "last_edited": "04 April 2026", "latex_path": "/latex/169", "formalized": false, "formalized_url": "", @@ -3032,7 +2869,7 @@ "additive combinatorics", "ramsey theory" ], - "last_edited": "01 February 2026", + "last_edited": "06 April 2026", "latex_path": "/latex/172", "formalized": true, "formalized_url": "https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/ErdosProblems/172.lean", @@ -3096,13 +2933,13 @@ "arithmetic progressions", "discrepancy" ], - "last_edited": "28 December 2025", + "last_edited": "04 April 2026", "latex_path": "/latex/176", "formalized": false, "formalized_url": "", "oeis_urls": [], "comments_problem_id": 176, - "comments_count": 1 + "comments_count": 8 }, { "problem_id": 177, @@ -3144,7 +2981,7 @@ "formalized_url": "", "oeis_urls": [], "comments_problem_id": 180, - "comments_count": 1 + "comments_count": 4 }, { "problem_id": 181, @@ -3180,7 +3017,7 @@ "graph theory", "ramsey theory" ], - "last_edited": "", + "last_edited": "10 April 2026", "latex_path": "/latex/183", "formalized": false, "formalized_url": "", @@ -3203,13 +3040,13 @@ "graph theory", "cycles" ], - "last_edited": "", + "last_edited": "01 April 2026", "latex_path": "/latex/184", - "formalized": false, - "formalized_url": "", + "formalized": true, + "formalized_url": "https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/ErdosProblems/184.lean", "oeis_urls": [], "comments_problem_id": 184, - "comments_count": 0 + "comments_count": 1 }, { "problem_id": 187, @@ -3225,7 +3062,7 @@ "ramsey theory", "arithmetic progressions" ], - "last_edited": "01 January 2026", + "last_edited": "04 April 2026", "latex_path": "/latex/187", "formalized": false, "formalized_url": "", @@ -3273,7 +3110,7 @@ "formalized_url": "", "oeis_urls": [], "comments_problem_id": 190, - "comments_count": 0 + "comments_count": 4 }, { "problem_id": 193, @@ -3289,8 +3126,8 @@ ], "last_edited": "", "latex_path": "/latex/193", - "formalized": false, - "formalized_url": "", + "formalized": true, + "formalized_url": "https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/ErdosProblems/193.lean", "oeis_urls": [ "https://oeis.org/A231255" ], @@ -3393,7 +3230,7 @@ "additive combinatorics", "arithmetic progressions" ], - "last_edited": "", + "last_edited": "08 April 2026", "latex_path": "/latex/201", "formalized": false, "formalized_url": "", @@ -3418,7 +3255,7 @@ "tags": [ "covering systems" ], - "last_edited": "23 January 2026", + "last_edited": "06 April 2026", "latex_path": "/latex/202", "formalized": false, "formalized_url": "", @@ -3426,7 +3263,7 @@ "https://oeis.org/A389975" ], "comments_problem_id": 202, - "comments_count": 0 + "comments_count": 5 }, { "problem_id": 203, @@ -3527,13 +3364,13 @@ "geometry", "distances" ], - "last_edited": "02 October 2025", + "last_edited": "13 April 2026", "latex_path": "/latex/217", "formalized": false, "formalized_url": "", "oeis_urls": [], "comments_problem_id": 217, - "comments_count": 0 + "comments_count": 1 }, { "problem_id": 218, @@ -3685,7 +3522,7 @@ "additive combinatorics", "sidon sets" ], - "last_edited": "30 September 2025", + "last_edited": "06 April 2026", "latex_path": "/latex/241", "formalized": true, "formalized_url": "https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/ErdosProblems/241.lean", @@ -3693,7 +3530,7 @@ "https://oeis.org/A387704" ], "comments_problem_id": 241, - "comments_count": 2 + "comments_count": 3 }, { "problem_id": 242, @@ -3708,7 +3545,7 @@ "number theory", "unit fractions" ], - "last_edited": "28 January 2026", + "last_edited": "07 May 2026", "latex_path": "/latex/242", "formalized": true, "formalized_url": "https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/ErdosProblems/242.lean", @@ -3744,7 +3581,7 @@ "https://oeis.org/A000058" ], "comments_problem_id": 243, - "comments_count": 6 + "comments_count": 7 }, { "problem_id": 244, @@ -3809,7 +3646,7 @@ "https://oeis.org/A256936" ], "comments_problem_id": 249, - "comments_count": 0 + "comments_count": 2 }, { "problem_id": 251, @@ -3832,7 +3669,7 @@ "https://oeis.org/A098990" ], "comments_problem_id": 251, - "comments_count": 1 + "comments_count": 4 }, { "problem_id": 252, @@ -3856,7 +3693,7 @@ "https://oeis.org/A227989" ], "comments_problem_id": 252, - "comments_count": 2 + "comments_count": 3 }, { "problem_id": 254, @@ -3872,8 +3709,8 @@ ], "last_edited": "07 December 2025", "latex_path": "/latex/254", - "formalized": false, - "formalized_url": "", + "formalized": true, + "formalized_url": "https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/ErdosProblems/254.lean", "oeis_urls": [], "comments_problem_id": 254, "comments_count": 3 @@ -3910,7 +3747,7 @@ "tags": [ "irrationality" ], - "last_edited": "02 December 2025", + "last_edited": "15 April 2026", "latex_path": "/latex/257", "formalized": true, "formalized_url": "https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/ErdosProblems/257.lean", @@ -3918,26 +3755,6 @@ "comments_problem_id": 257, "comments_count": 6 }, - { - "problem_id": 258, - "problem_url": "/258", - "status_bucket": "open", - "status_dom_id": "open", - "status_label": "OPEN", - "status_detail": "This is open, and cannot be resolved with a finite computation.", - "prize_amount": "", - "statement": "Let $a_1,a_2,\\ldots$ be a sequence of positive integers with $a_n\\to \\infty$. Is\\[\\sum_{n} \\frac{\\tau(n)}{a_1\\cdots a_n}\\]irrational, where $\\tau(n)$ is the number of divisors of $n$?", - "tags": [ - "irrationality" - ], - "last_edited": "20 January 2026", - "latex_path": "/latex/258", - "formalized": true, - "formalized_url": "https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/ErdosProblems/258.lean", - "oeis_urls": [], - "comments_problem_id": 258, - "comments_count": 0 - }, { "problem_id": 260, "problem_url": "/260", @@ -3952,8 +3769,8 @@ ], "last_edited": "01 February 2026", "latex_path": "/latex/260", - "formalized": false, - "formalized_url": "", + "formalized": true, + "formalized_url": "https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/ErdosProblems/260.lean", "oeis_urls": [], "comments_problem_id": 260, "comments_count": 2 @@ -3976,7 +3793,7 @@ "formalized_url": "", "oeis_urls": [], "comments_problem_id": 261, - "comments_count": 1 + "comments_count": 3 }, { "problem_id": 263, @@ -3986,17 +3803,17 @@ "status_label": "OPEN", "status_detail": "This is open, and cannot be resolved with a finite computation.", "prize_amount": "", - "statement": "Let $a_n$ be a sequence of positive integers such that for every sequence of positive integers $b_n$ with $b_n/a_n\\to 1$ the sum\\[\\sum\\frac{1}{b_n}\\]is irrational. Is $a_n=2^{2^n}$ such a sequence? Must such a sequence satisfy $a_n^{1/n}\\to \\infty$?", + "statement": "Let $a_n$ be an increasing sequence of positive integers such that for every sequence of positive integers $b_n$ with $b_n/a_n\\to 1$ the sum\\[\\sum\\frac{1}{b_n}\\]is irrational. Is $a_n=2^{2^n}$ such a sequence? Must such a sequence satisfy $a_n^{1/n}\\to \\infty$?", "tags": [ "irrationality" ], - "last_edited": "20 January 2026", + "last_edited": "02 April 2026", "latex_path": "/latex/263", "formalized": true, "formalized_url": "https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/ErdosProblems/263.lean", "oeis_urls": [], "comments_problem_id": 263, - "comments_count": 3 + "comments_count": 6 }, { "problem_id": 264, @@ -4056,7 +3873,7 @@ "formalized_url": "https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/ErdosProblems/267.lean", "oeis_urls": [], "comments_problem_id": 267, - "comments_count": 2 + "comments_count": 4 }, { "problem_id": 269, @@ -4099,7 +3916,7 @@ "https://oeis.org/A005487" ], "comments_problem_id": 271, - "comments_count": 6 + "comments_count": 7 }, { "problem_id": 272, @@ -4116,8 +3933,8 @@ ], "last_edited": "", "latex_path": "/latex/272", - "formalized": false, - "formalized_url": "", + "formalized": true, + "formalized_url": "https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/ErdosProblems/272.lean", "oeis_urls": [], "comments_problem_id": 272, "comments_count": 7 @@ -4162,7 +3979,7 @@ "formalized_url": "https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/ErdosProblems/274.lean", "oeis_urls": [], "comments_problem_id": 274, - "comments_count": 2 + "comments_count": 3 }, { "problem_id": 276, @@ -4220,13 +4037,13 @@ "covering systems", "primes" ], - "last_edited": "20 January 2026", + "last_edited": "17 April 2026", "latex_path": "/latex/279", - "formalized": false, - "formalized_url": "", + "formalized": true, + "formalized_url": "https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/ErdosProblems/279.lean", "oeis_urls": [], "comments_problem_id": 279, - "comments_count": 0 + "comments_count": 10 }, { "problem_id": 282, @@ -4243,35 +4060,12 @@ ], "last_edited": "", "latex_path": "/latex/282", - "formalized": false, - "formalized_url": "", + "formalized": true, + "formalized_url": "https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/ErdosProblems/282.lean", "oeis_urls": [], "comments_problem_id": 282, "comments_count": 2 }, - { - "problem_id": 283, - "problem_url": "/283", - "status_bucket": "open", - "status_dom_id": "open", - "status_label": "OPEN", - "status_detail": "This is open, and cannot be resolved with a finite computation.", - "prize_amount": "", - "statement": "Let $p:\\mathbb{Z}\\to \\mathbb{Z}$ be a polynomial whose leading coefficient is positive and such that there exists no $d\\geq 2$ with $d\\mid p(n)$ for all $n\\geq 1$. Is it true that, for all sufficiently large $m$, there exist integers $1\\leq n_1<\\cdots 0$ and $k\\geq 2$. Is it true that, for all sufficiently large $n$, there is a sequence of $k$ consecutive integers in $\\{1,\\ldots,n\\}$ all of which are $n^\\epsilon$-smooth?", - "tags": [ - "number theory" - ], - "last_edited": "", - "latex_path": "/latex/369", - "formalized": false, - "formalized_url": "", - "oeis_urls": [], - "comments_problem_id": 369, - "comments_count": 0 - }, { "problem_id": 371, "problem_url": "/371", @@ -5467,7 +5165,7 @@ "https://oeis.org/A389148" ], "comments_problem_id": 374, - "comments_count": 0 + "comments_count": 1 }, { "problem_id": 375, @@ -5535,31 +5233,6 @@ "comments_problem_id": 377, "comments_count": 1 }, - { - "problem_id": 380, - "problem_url": "/380", - "status_bucket": "open", - "status_dom_id": "open", - "status_label": "OPEN", - "status_detail": "This is open, and cannot be resolved with a finite computation.", - "prize_amount": "", - "statement": "We call an interval $[u,v]$ 'bad' if the greatest prime factor of $\\prod_{u\\leq m\\leq v}m$ occurs with an exponent greater than $1$. Let $B(x)$ count the number of $n\\leq x$ which are contained in at least one bad interval. Is it true that\\[B(x)\\sim \\#\\{ n\\leq x: P(n)^2\\mid n\\},\\]where $P(n)$ is the largest prime factor of $n$?", - "tags": [ - "number theory" - ], - "last_edited": "", - "latex_path": "/latex/380", - "formalized": false, - "formalized_url": "", - "oeis_urls": [ - "https://oeis.org/A070003", - "https://oeis.org/A387054", - "https://oeis.org/A388654", - "https://oeis.org/A389100" - ], - "comments_problem_id": 380, - "comments_count": 5 - }, { "problem_id": 382, "problem_url": "/382", @@ -5645,7 +5318,7 @@ "https://oeis.org/A280992" ], "comments_problem_id": 386, - "comments_count": 12 + "comments_count": 13 }, { "problem_id": 387, @@ -5666,7 +5339,7 @@ "formalized_url": "https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/ErdosProblems/387.lean", "oeis_urls": [], "comments_problem_id": 387, - "comments_count": 6 + "comments_count": 8 }, { "problem_id": 388, @@ -5680,13 +5353,13 @@ "tags": [ "number theory" ], - "last_edited": "", + "last_edited": "11 April 2026", "latex_path": "/latex/388", "formalized": false, "formalized_url": "", "oeis_urls": [], "comments_problem_id": 388, - "comments_count": 5 + "comments_count": 10 }, { "problem_id": 389, @@ -5708,7 +5381,7 @@ "https://oeis.org/A375071" ], "comments_problem_id": 389, - "comments_count": 6 + "comments_count": 8 }, { "problem_id": 390, @@ -5731,7 +5404,7 @@ "https://oeis.org/A193429" ], "comments_problem_id": 390, - "comments_count": 1 + "comments_count": 3 }, { "problem_id": 393, @@ -5754,7 +5427,7 @@ "https://oeis.org/A388302" ], "comments_problem_id": 393, - "comments_count": 6 + "comments_count": 8 }, { "problem_id": 394, @@ -5799,7 +5472,7 @@ "https://oeis.org/A375077" ], "comments_problem_id": 396, - "comments_count": 1 + "comments_count": 35 }, { "problem_id": 398, @@ -5814,15 +5487,16 @@ "number theory", "factorials" ], - "last_edited": "", + "last_edited": "01 April 2026", "latex_path": "/latex/398", "formalized": true, "formalized_url": "https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/ErdosProblems/398.lean", "oeis_urls": [ + "https://oeis.org/A141399", "https://oeis.org/A146968" ], "comments_problem_id": 398, - "comments_count": 6 + "comments_count": 10 }, { "problem_id": 400, @@ -5839,11 +5513,11 @@ ], "last_edited": "", "latex_path": "/latex/400", - "formalized": false, - "formalized_url": "", + "formalized": true, + "formalized_url": "https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/ErdosProblems/400.lean", "oeis_urls": [], "comments_problem_id": 400, - "comments_count": 0 + "comments_count": 1 }, { "problem_id": 404, @@ -5879,13 +5553,13 @@ "number theory", "base representations" ], - "last_edited": "30 September 2025", + "last_edited": "13 April 2026", "latex_path": "/latex/406", "formalized": true, "formalized_url": "https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/ErdosProblems/406.lean", "oeis_urls": [], "comments_problem_id": 406, - "comments_count": 4 + "comments_count": 5 }, { "problem_id": 408, @@ -5978,7 +5652,7 @@ "https://oeis.org/A383044" ], "comments_problem_id": 411, - "comments_count": 1 + "comments_count": 4 }, { "problem_id": 412, @@ -6017,7 +5691,7 @@ "number theory", "iterated functions" ], - "last_edited": "28 December 2025", + "last_edited": "17 April 2026", "latex_path": "/latex/413", "formalized": true, "formalized_url": "https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/ErdosProblems/413.lean", @@ -6025,7 +5699,7 @@ "https://oeis.org/A005236" ], "comments_problem_id": 413, - "comments_count": 2 + "comments_count": 3 }, { "problem_id": 414, @@ -6058,17 +5732,17 @@ "status_label": "OPEN", "status_detail": "This is open, and cannot be resolved with a finite computation.", "prize_amount": "", - "statement": "For any $n$ let $F(n)$ be the largest $k$ such that any of the $k!$ possible ordering patterns appears in some sequence of $\\phi(m+1),\\ldots,\\phi(m+k)$ with $m+k\\leq n$. Is it true that\\[F(n)=(c+o(1))\\log\\log\\log n\\]for some constant $c$? Is the first pattern which fails to appear always\\[\\phi(m+1)>\\phi(m+2)>\\cdots \\phi(m+k)?\\]Is it true that 'natural' ordering which mimics what happens to $\\phi(1),\\ldots,\\phi(k)$ is the most likely to appear?", + "statement": "For any $n$ let $F(n)$ be the largest $k$ such that any of the $k!$ possible ordering patterns appears in some sequence of $\\phi(m+1),\\ldots,\\phi(m+k)$ with $m+k\\leq n$. Is it true that\\[F(n)=(c+o(1))\\log\\log\\log n\\]for some constant $c$? Is the first pattern which fails to appear always\\[\\phi(m+1)>\\phi(m+2)>\\cdots >\\phi(m+k)?\\]Is it true that the 'natural' ordering which mimics what happens to $\\phi(1),\\ldots,\\phi(k)$ is the most likely to appear?", "tags": [ "number theory" ], - "last_edited": "", + "last_edited": "19 April 2026", "latex_path": "/latex/415", "formalized": false, "formalized_url": "", "oeis_urls": [], "comments_problem_id": 415, - "comments_count": 0 + "comments_count": 4 }, { "problem_id": 416, @@ -6113,7 +5787,7 @@ "https://oeis.org/A264810" ], "comments_problem_id": 417, - "comments_count": 3 + "comments_count": 5 }, { "problem_id": 420, @@ -6192,7 +5866,7 @@ "tags": [ "number theory" ], - "last_edited": "16 January 2026", + "last_edited": "23 March 2026", "latex_path": "/latex/423", "formalized": false, "formalized_url": "", @@ -6200,7 +5874,7 @@ "https://oeis.org/A005243" ], "comments_problem_id": 423, - "comments_count": 18 + "comments_count": 38 }, { "problem_id": 424, @@ -6214,7 +5888,7 @@ "tags": [ "number theory" ], - "last_edited": "", + "last_edited": "31 March 2026", "latex_path": "/latex/424", "formalized": true, "formalized_url": "https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/ErdosProblems/424.lean", @@ -6237,13 +5911,13 @@ "number theory", "sidon sets" ], - "last_edited": "28 December 2025", + "last_edited": "06 April 2026", "latex_path": "/latex/425", "formalized": false, "formalized_url": "", "oeis_urls": [], "comments_problem_id": 425, - "comments_count": 0 + "comments_count": 5 }, { "problem_id": 428, @@ -6299,7 +5973,7 @@ "number theory", "primes" ], - "last_edited": "", + "last_edited": "08 April 2026", "latex_path": "/latex/431", "formalized": false, "formalized_url": "", @@ -6363,8 +6037,8 @@ ], "last_edited": "27 December 2025", "latex_path": "/latex/445", - "formalized": false, - "formalized_url": "", + "formalized": true, + "formalized_url": "https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/ErdosProblems/445.lean", "oeis_urls": [], "comments_problem_id": 445, "comments_count": 1 @@ -6410,7 +6084,7 @@ "https://oeis.org/A386620" ], "comments_problem_id": 451, - "comments_count": 2 + "comments_count": 4 }, { "problem_id": 452, @@ -6494,29 +6168,7 @@ "formalized_url": "", "oeis_urls": [], "comments_problem_id": 456, - "comments_count": 2 - }, - { - "problem_id": 457, - "problem_url": "/457", - "status_bucket": "open", - "status_dom_id": "open", - "status_label": "OPEN", - "status_detail": "This is open, and cannot be resolved with a finite computation.", - "prize_amount": "", - "statement": "Is there some $\\epsilon>0$ such that there are infinitely many $n$ where all primes $p\\leq (2+\\epsilon)\\log n$ divide\\[\\prod_{1\\leq i\\leq \\log n}(n+i)?\\]", - "tags": [ - "number theory" - ], - "last_edited": "07 October 2025", - "latex_path": "/latex/457", - "formalized": true, - "formalized_url": "https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/ErdosProblems/457.lean", - "oeis_urls": [ - "https://oeis.org/A391668" - ], - "comments_problem_id": 457, - "comments_count": 7 + "comments_count": 3 }, { "problem_id": 458, @@ -6539,7 +6191,7 @@ "https://oeis.org/A056604" ], "comments_problem_id": 458, - "comments_count": 1 + "comments_count": 2 }, { "problem_id": 460, @@ -6580,7 +6232,7 @@ "formalized_url": "", "oeis_urls": [], "comments_problem_id": 461, - "comments_count": 0 + "comments_count": 1 }, { "problem_id": 462, @@ -6669,7 +6321,7 @@ "https://oeis.org/A387503" ], "comments_problem_id": 468, - "comments_count": 7 + "comments_count": 1 }, { "problem_id": 469, @@ -6775,7 +6427,7 @@ "number theory", "additive combinatorics" ], - "last_edited": "", + "last_edited": "05 March 2026", "latex_path": "/latex/475", "formalized": false, "formalized_url": "", @@ -6791,11 +6443,12 @@ "status_label": "OPEN", "status_detail": "This is open, and cannot be resolved with a finite computation.", "prize_amount": "", - "statement": "Is there a polynomial $f:\\mathbb{Z}\\to \\mathbb{Z}$ of degree at least $2$ and a set $A\\subset \\mathbb{Z}$ such that for any $n\\in \\mathbb{Z}$ there is exactly one $a\\in A$ and $b\\in \\{ f(n) : n\\in\\mathbb{Z}\\}$ such that $n=a+b$?", + "statement": "Is there a polynomial $f:\\mathbb{Z}\\to \\mathbb{Z}$ of degree at least $2$ and a set $A\\subset \\mathbb{Z}$ such that for any $n\\in \\mathbb{Z}$ there is exactly one $a\\in A$ and $b\\in \\{ f(k) : k\\in\\mathbb{Z}\\}$ such that $n=a+b$?", "tags": [ - "number theory" + "number theory", + "sidon sets" ], - "last_edited": "29 December 2025", + "last_edited": "11 April 2026", "latex_path": "/latex/477", "formalized": true, "formalized_url": "https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/ErdosProblems/477.lean", @@ -6816,7 +6469,7 @@ "number theory", "factorials" ], - "last_edited": "04 October 2025", + "last_edited": "12 April 2026", "latex_path": "/latex/478", "formalized": false, "formalized_url": "", @@ -6868,7 +6521,7 @@ "additive combinatorics", "ramsey theory" ], - "last_edited": "14 October 2025", + "last_edited": "10 April 2026", "latex_path": "/latex/483", "formalized": false, "formalized_url": "", @@ -6876,7 +6529,7 @@ "https://oeis.org/A030126" ], "comments_problem_id": 483, - "comments_count": 3 + "comments_count": 4 }, { "problem_id": 486, @@ -6891,7 +6544,7 @@ "number theory", "primitive sets" ], - "last_edited": "12 January 2026", + "last_edited": "08 April 2026", "latex_path": "/latex/486", "formalized": true, "formalized_url": "https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/ErdosProblems/486.lean", @@ -6904,20 +6557,20 @@ "problem_url": "/488", "status_bucket": "open", "status_dom_id": "open", - "status_label": "OPEN", - "status_detail": "This is open, and cannot be resolved with a finite computation.", + "status_label": "FALSIFIABLE", + "status_detail": "Open, but could be disproved with a finite counterexample.", "prize_amount": "", "statement": "Let $A$ be a finite set and\\[B=\\{ n \\geq 1 : a\\mid n\\textrm{ for some }a\\in A\\}.\\]Is it true that, for every $m>n\\geq \\max(A)$,\\[\\frac{\\lvert B\\cap [1,m]\\rvert }{m}< 2\\frac{\\lvert B\\cap [1,n]\\rvert}{n}?\\]", "tags": [ "number theory" ], - "last_edited": "31 December 2025", + "last_edited": "08 April 2026", "latex_path": "/latex/488", "formalized": true, "formalized_url": "https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/ErdosProblems/488.lean", "oeis_urls": [], "comments_problem_id": 488, - "comments_count": 14 + "comments_count": 29 }, { "problem_id": 489, @@ -6937,7 +6590,7 @@ "formalized_url": "https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/ErdosProblems/489.lean", "oeis_urls": [], "comments_problem_id": 489, - "comments_count": 0 + "comments_count": 3 }, { "problem_id": 495, @@ -6999,8 +6652,8 @@ ], "last_edited": "25 January 2026", "latex_path": "/latex/501", - "formalized": false, - "formalized_url": "", + "formalized": true, + "formalized_url": "https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/ErdosProblems/501.lean", "oeis_urls": [], "comments_problem_id": 501, "comments_count": 8 @@ -7026,7 +6679,7 @@ "https://oeis.org/A175769" ], "comments_problem_id": 503, - "comments_count": 1 + "comments_count": 6 }, { "problem_id": 507, @@ -7122,13 +6775,13 @@ "tags": [ "analysis" ], - "last_edited": "28 December 2025", + "last_edited": "02 April 2026", "latex_path": "/latex/513", "formalized": true, "formalized_url": "https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/ErdosProblems/513.lean", "oeis_urls": [], "comments_problem_id": 513, - "comments_count": 5 + "comments_count": 6 }, { "problem_id": 514, @@ -7148,7 +6801,7 @@ "formalized_url": "", "oeis_urls": [], "comments_problem_id": 514, - "comments_count": 1 + "comments_count": 12 }, { "problem_id": 517, @@ -7168,7 +6821,7 @@ "formalized_url": "https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/ErdosProblems/517.lean", "oeis_urls": [], "comments_problem_id": 517, - "comments_count": 0 + "comments_count": 1 }, { "problem_id": 520, @@ -7211,7 +6864,7 @@ "formalized_url": "", "oeis_urls": [], "comments_problem_id": 521, - "comments_count": 13 + "comments_count": 20 }, { "problem_id": 522, @@ -7233,7 +6886,7 @@ "formalized_url": "https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/ErdosProblems/522.lean", "oeis_urls": [], "comments_problem_id": 522, - "comments_count": 2 + "comments_count": 24 }, { "problem_id": 524, @@ -7255,7 +6908,7 @@ "formalized_url": "", "oeis_urls": [], "comments_problem_id": 524, - "comments_count": 10 + "comments_count": 11 }, { "problem_id": 528, @@ -7314,7 +6967,7 @@ "number theory", "sidon sets" ], - "last_edited": "16 October 2025", + "last_edited": "08 April 2026", "latex_path": "/latex/530", "formalized": false, "formalized_url": "", @@ -7357,13 +7010,13 @@ "tags": [ "number theory" ], - "last_edited": "", + "last_edited": "29 April 2026", "latex_path": "/latex/535", - "formalized": false, - "formalized_url": "", + "formalized": true, + "formalized_url": "https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/ErdosProblems/535.lean", "oeis_urls": [], "comments_problem_id": 535, - "comments_count": 1 + "comments_count": 9 }, { "problem_id": 536, @@ -7373,17 +7026,17 @@ "status_label": "OPEN", "status_detail": "This is open, and cannot be resolved with a finite computation.", "prize_amount": "", - "statement": "Let $\\epsilon>0$ and $N$ be sufficiently large. Is it true that if $A\\subseteq \\{1,\\ldots,N\\}$ has size at least $\\epsilon N$ then there must be distinct $a,b,c\\in A$ such that\\[[a,b]=[b,c]=[a,c],\\]where $[a,b]$ denotes the least common multiple?", + "statement": "Let $f(N)$ be the largest size of $A\\subseteq \\{1,\\ldots,N\\}$ with the property that there are no distinct $a,b,c\\in A$ such that\\[[a,b]=[b,c]=[a,c],\\]where $[a,b]$ denotes the least common multiple. Estimate $f(N)$ - in particular, is it true that $f(N)=o(N)$?", "tags": [ "number theory" ], - "last_edited": "12 January 2026", + "last_edited": "29 April 2026", "latex_path": "/latex/536", "formalized": true, "formalized_url": "https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/ErdosProblems/536.lean", "oeis_urls": [], "comments_problem_id": 536, - "comments_count": 3 + "comments_count": 6 }, { "problem_id": 538, @@ -7419,11 +7072,11 @@ ], "last_edited": "22 January 2026", "latex_path": "/latex/539", - "formalized": false, - "formalized_url": "", + "formalized": true, + "formalized_url": "https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/ErdosProblems/539.lean", "oeis_urls": [], "comments_problem_id": 539, - "comments_count": 1 + "comments_count": 2 }, { "problem_id": 544, @@ -7438,7 +7091,7 @@ "graph theory", "ramsey theory" ], - "last_edited": "", + "last_edited": "24 April 2026", "latex_path": "/latex/544", "formalized": false, "formalized_url": "", @@ -7446,7 +7099,7 @@ "https://oeis.org/A000791" ], "comments_problem_id": 544, - "comments_count": 0 + "comments_count": 2 }, { "problem_id": 545, @@ -7478,12 +7131,12 @@ "status_dom_id": "open", "status_label": "FALSIFIABLE", "status_detail": "Open, but could be disproved with a finite counterexample.", - "prize_amount": "", + "prize_amount": "$100", "statement": "Let $n\\geq k+1$. Every graph on $n$ vertices with at least $\\frac{k-1}{2}n+1$ edges contains every tree on $k+1$ vertices.", "tags": [ "graph theory" ], - "last_edited": "23 January 2026", + "last_edited": "07 March 2026", "latex_path": "/latex/548", "formalized": false, "formalized_url": "", @@ -7573,9 +7226,11 @@ "latex_path": "/latex/555", "formalized": false, "formalized_url": "", - "oeis_urls": [], + "oeis_urls": [ + "https://oeis.org/A389313" + ], "comments_problem_id": 555, - "comments_count": 1 + "comments_count": 0 }, { "problem_id": 557, @@ -7809,7 +7464,7 @@ "formalized_url": "", "oeis_urls": [], "comments_problem_id": 569, - "comments_count": 0 + "comments_count": 2 }, { "problem_id": 571, @@ -7824,7 +7479,7 @@ "graph theory", "turan number" ], - "last_edited": "18 January 2026", + "last_edited": "07 March 2026", "latex_path": "/latex/571", "formalized": false, "formalized_url": "", @@ -7852,7 +7507,7 @@ "formalized_url": "", "oeis_urls": [], "comments_problem_id": 572, - "comments_count": 1 + "comments_count": 2 }, { "problem_id": 573, @@ -7877,27 +7532,6 @@ "comments_problem_id": 573, "comments_count": 0 }, - { - "problem_id": 574, - "problem_url": "/574", - "status_bucket": "open", - "status_dom_id": "open", - "status_label": "OPEN", - "status_detail": "This is open, and cannot be resolved with a finite computation.", - "prize_amount": "", - "statement": "Is it true that, for $k\\geq 2$,\\[\\mathrm{ex}(n;\\{C_{2k-1},C_{2k}\\})=(1+o(1))(n/2)^{1+\\frac{1}{k}}.\\]", - "tags": [ - "graph theory", - "turan number" - ], - "last_edited": "18 January 2026", - "latex_path": "/latex/574", - "formalized": false, - "formalized_url": "", - "oeis_urls": [], - "comments_problem_id": 574, - "comments_count": 0 - }, { "problem_id": 575, "problem_url": "/575", @@ -7993,13 +7627,13 @@ "tags": [ "graph theory" ], - "last_edited": "05 March 2026", + "last_edited": "01 April 2026", "latex_path": "/latex/583", "formalized": false, "formalized_url": "", "oeis_urls": [], "comments_problem_id": 583, - "comments_count": 1 + "comments_count": 3 }, { "problem_id": 584, @@ -8020,7 +7654,7 @@ "formalized_url": "", "oeis_urls": [], "comments_problem_id": 584, - "comments_count": 1 + "comments_count": 3 }, { "problem_id": 585, @@ -8059,7 +7693,10 @@ "latex_path": "/latex/588", "formalized": false, "formalized_url": "", - "oeis_urls": [], + "oeis_urls": [ + "https://oeis.org/A006065", + "https://oeis.org/A008997" + ], "comments_problem_id": 588, "comments_count": 0 }, @@ -8121,8 +7758,8 @@ ], "last_edited": "", "latex_path": "/latex/593", - "formalized": false, - "formalized_url": "", + "formalized": true, + "formalized_url": "https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/ErdosProblems/593.lean", "oeis_urls": [], "comments_problem_id": 593, "comments_count": 0 @@ -8142,8 +7779,8 @@ ], "last_edited": "", "latex_path": "/latex/595", - "formalized": false, - "formalized_url": "", + "formalized": true, + "formalized_url": "https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/ErdosProblems/595.lean", "oeis_urls": [], "comments_problem_id": 595, "comments_count": 0 @@ -8168,7 +7805,7 @@ "formalized_url": "", "oeis_urls": [], "comments_problem_id": 596, - "comments_count": 0 + "comments_count": 1 }, { "problem_id": 597, @@ -8211,7 +7848,7 @@ "formalized_url": "https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/ErdosProblems/598.lean", "oeis_urls": [], "comments_problem_id": 598, - "comments_count": 0 + "comments_count": 6 }, { "problem_id": 600, @@ -8269,11 +7906,11 @@ ], "last_edited": "", "latex_path": "/latex/602", - "formalized": false, - "formalized_url": "", + "formalized": true, + "formalized_url": "https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/ErdosProblems/602.lean", "oeis_urls": [], "comments_problem_id": 602, - "comments_count": 0 + "comments_count": 1 }, { "problem_id": 603, @@ -8294,7 +7931,7 @@ "formalized_url": "", "oeis_urls": [], "comments_problem_id": 603, - "comments_count": 2 + "comments_count": 12 }, { "problem_id": 604, @@ -8309,7 +7946,7 @@ "geometry", "distances" ], - "last_edited": "15 October 2025", + "last_edited": "23 March 2026", "latex_path": "/latex/604", "formalized": false, "formalized_url": "", @@ -8356,7 +7993,7 @@ "formalized_url": "", "oeis_urls": [], "comments_problem_id": 610, - "comments_count": 0 + "comments_count": 4 }, { "problem_id": 611, @@ -8450,13 +8087,13 @@ "tags": [ "graph theory" ], - "last_edited": "", + "last_edited": "01 April 2026", "latex_path": "/latex/617", "formalized": true, "formalized_url": "https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/ErdosProblems/617.lean", "oeis_urls": [], "comments_problem_id": 617, - "comments_count": 0 + "comments_count": 1 }, { "problem_id": 619, @@ -8476,7 +8113,7 @@ "formalized_url": "", "oeis_urls": [], "comments_problem_id": 619, - "comments_count": 0 + "comments_count": 1 }, { "problem_id": 620, @@ -8516,7 +8153,7 @@ "formalized_url": "https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/ErdosProblems/623.lean", "oeis_urls": [], "comments_problem_id": 623, - "comments_count": 0 + "comments_count": 1 }, { "problem_id": 624, @@ -8642,27 +8279,7 @@ "formalized_url": "", "oeis_urls": [], "comments_problem_id": 629, - "comments_count": 1 - }, - { - "problem_id": 633, - "problem_url": "/633", - "status_bucket": "open", - "status_dom_id": "open", - "status_label": "OPEN", - "status_detail": "This is open, and cannot be resolved with a finite computation.", - "prize_amount": "$25", - "statement": "Classify those triangles which can only be cut into a square number of congruent triangles.", - "tags": [ - "geometry" - ], - "last_edited": "", - "latex_path": "/latex/633", - "formalized": false, - "formalized_url": "", - "oeis_urls": [], - "comments_problem_id": 633, - "comments_count": 2 + "comments_count": 3 }, { "problem_id": 634, @@ -8717,13 +8334,13 @@ "graph theory", "ramsey theory" ], - "last_edited": "", + "last_edited": "10 April 2026", "latex_path": "/latex/638", "formalized": false, "formalized_url": "", "oeis_urls": [], "comments_problem_id": 638, - "comments_count": 1 + "comments_count": 8 }, { "problem_id": 640, @@ -8765,7 +8382,7 @@ "formalized_url": "", "oeis_urls": [], "comments_problem_id": 642, - "comments_count": 2 + "comments_count": 3 }, { "problem_id": 643, @@ -8820,7 +8437,7 @@ "tags": [ "number theory" ], - "last_edited": "05 October 2025", + "last_edited": "07 April 2026", "latex_path": "/latex/647", "formalized": true, "formalized_url": "https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/ErdosProblems/647.lean", @@ -8831,26 +8448,6 @@ "comments_problem_id": 647, "comments_count": 6 }, - { - "problem_id": 650, - "problem_url": "/650", - "status_bucket": "open", - "status_dom_id": "open", - "status_label": "OPEN", - "status_detail": "This is open, and cannot be resolved with a finite computation.", - "prize_amount": "", - "statement": "Let $f(m)$ be such that if $A\\subseteq \\{1,\\ldots,N\\}$ has $\\lvert A\\rvert=m$ then every interval in $[1,\\infty)$ of length $2N$ contains $\\geq f(m)$ many distinct integers $b_1,\\ldots,b_r$ where each $b_i$ is divisible by some $a_i\\in A$, where $a_1,\\ldots,a_r$ are distinct. Estimate $f(m)$. In particular is it true that $f(m)\\ll m^{1/2}$?", - "tags": [ - "number theory" - ], - "last_edited": "", - "latex_path": "/latex/650", - "formalized": false, - "formalized_url": "", - "oeis_urls": [], - "comments_problem_id": 650, - "comments_count": 0 - }, { "problem_id": 653, "problem_url": "/653", @@ -8866,8 +8463,8 @@ ], "last_edited": "", "latex_path": "/latex/653", - "formalized": false, - "formalized_url": "", + "formalized": true, + "formalized_url": "https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/ErdosProblems/653.lean", "oeis_urls": [], "comments_problem_id": 653, "comments_count": 0 @@ -8908,11 +8505,11 @@ ], "last_edited": "", "latex_path": "/latex/655", - "formalized": false, - "formalized_url": "", + "formalized": true, + "formalized_url": "https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/ErdosProblems/655.lean", "oeis_urls": [], "comments_problem_id": 655, - "comments_count": 4 + "comments_count": 5 }, { "problem_id": 657, @@ -8997,7 +8594,7 @@ "formalized_url": "", "oeis_urls": [], "comments_problem_id": 662, - "comments_count": 2 + "comments_count": 4 }, { "problem_id": 663, @@ -9060,7 +8657,7 @@ "formalized_url": "", "oeis_urls": [], "comments_problem_id": 667, - "comments_count": 0 + "comments_count": 1 }, { "problem_id": 668, @@ -9079,7 +8676,9 @@ "latex_path": "/latex/668", "formalized": false, "formalized_url": "", - "oeis_urls": [], + "oeis_urls": [ + "https://oeis.org/A385657" + ], "comments_problem_id": 668, "comments_count": 3 }, @@ -9120,13 +8719,13 @@ "geometry", "distances" ], - "last_edited": "", + "last_edited": "17 April 2026", "latex_path": "/latex/670", "formalized": false, "formalized_url": "", "oeis_urls": [], "comments_problem_id": 670, - "comments_count": 0 + "comments_count": 4 }, { "problem_id": 671, @@ -9186,7 +8785,7 @@ "formalized_url": "", "oeis_urls": [], "comments_problem_id": 675, - "comments_count": 0 + "comments_count": 7 }, { "problem_id": 676, @@ -9200,7 +8799,7 @@ "tags": [ "number theory" ], - "last_edited": "", + "last_edited": "07 April 2026", "latex_path": "/latex/676", "formalized": false, "formalized_url": "", @@ -9208,7 +8807,7 @@ "https://oeis.org/A390181" ], "comments_problem_id": 676, - "comments_count": 10 + "comments_count": 9 }, { "problem_id": 677, @@ -9228,7 +8827,7 @@ "formalized_url": "https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/ErdosProblems/677.lean", "oeis_urls": [], "comments_problem_id": 677, - "comments_count": 0 + "comments_count": 8 }, { "problem_id": 679, @@ -9242,7 +8841,7 @@ "tags": [ "number theory" ], - "last_edited": "12 January 2026", + "last_edited": "17 April 2026", "latex_path": "/latex/679", "formalized": false, "formalized_url": "", @@ -9310,8 +8909,8 @@ ], "last_edited": "31 December 2025", "latex_path": "/latex/683", - "formalized": false, - "formalized_url": "", + "formalized": true, + "formalized_url": "https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/ErdosProblems/683.lean", "oeis_urls": [ "https://oeis.org/A006530", "https://oeis.org/A074399", @@ -9334,7 +8933,7 @@ "primes", "binomial coefficients" ], - "last_edited": "23 January 2026", + "last_edited": "01 April 2026", "latex_path": "/latex/684", "formalized": false, "formalized_url": "", @@ -9342,7 +8941,7 @@ "https://oeis.org/A392019" ], "comments_problem_id": 684, - "comments_count": 23 + "comments_count": 27 }, { "problem_id": 685, @@ -9364,7 +8963,7 @@ "formalized_url": "", "oeis_urls": [], "comments_problem_id": 685, - "comments_count": 1 + "comments_count": 4 }, { "problem_id": 686, @@ -9378,13 +8977,13 @@ "tags": [ "number theory" ], - "last_edited": "", + "last_edited": "11 April 2026", "latex_path": "/latex/686", "formalized": true, "formalized_url": "https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/ErdosProblems/686.lean", "oeis_urls": [], "comments_problem_id": 686, - "comments_count": 16 + "comments_count": 36 }, { "problem_id": 687, @@ -9421,10 +9020,10 @@ "tags": [ "number theory" ], - "last_edited": "", + "last_edited": "07 April 2026", "latex_path": "/latex/688", - "formalized": false, - "formalized_url": "", + "formalized": true, + "formalized_url": "https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/ErdosProblems/688.lean", "oeis_urls": [], "comments_problem_id": 688, "comments_count": 0 @@ -9441,43 +9040,23 @@ "tags": [ "number theory" ], - "last_edited": "06 December 2025", + "last_edited": "08 April 2026", "latex_path": "/latex/689", "formalized": true, "formalized_url": "https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/ErdosProblems/689.lean", "oeis_urls": [], "comments_problem_id": 689, - "comments_count": 17 + "comments_count": 24 }, { - "problem_id": 690, - "problem_url": "/690", + "problem_id": 691, + "problem_url": "/691", "status_bucket": "open", "status_dom_id": "open", "status_label": "OPEN", "status_detail": "This is open, and cannot be resolved with a finite computation.", "prize_amount": "", - "statement": "Let $d_k(p)$ be the density of those integers whose $k$th smallest prime factor is $p$ (i.e. if $p_10$ and let $N$ be large. Let $A\\subseteq \\{2,\\ldots,N\\}$ be such that $(a,b)=1$ for all $a\\neq b\\in A$ and $\\sum_{n\\in A}\\frac{1}{n}\\leq C$. What choice of such an $A$ minimises the number of integers $m\\leq N$ not divisible by any $a\\in A$?", - "tags": [ - "number theory" - ], - "last_edited": "08 February 2026", - "latex_path": "/latex/783", - "formalized": false, - "formalized_url": "", - "oeis_urls": [], - "comments_problem_id": 783, - "comments_count": 28 - }, { "problem_id": 786, "problem_url": "/786", @@ -10480,7 +9980,7 @@ "tags": [ "number theory" ], - "last_edited": "02 February 2026", + "last_edited": "11 April 2026", "latex_path": "/latex/786", "formalized": true, "formalized_url": "https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/ErdosProblems/786.lean", @@ -10544,8 +10044,8 @@ ], "last_edited": "", "latex_path": "/latex/789", - "formalized": false, - "formalized_url": "", + "formalized": true, + "formalized_url": "https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/ErdosProblems/789.lean", "oeis_urls": [], "comments_problem_id": 789, "comments_count": 0 @@ -10700,18 +10200,18 @@ "status_label": "OPEN", "status_detail": "This is open, and cannot be resolved with a finite computation.", "prize_amount": "", - "statement": "Let $k\\geq 3$ and define $F_k(n)$ to be the minimal $r$ such that there is a graph $G$ on $n$ vertices with $\\lfloor n^2/4\\rfloor+1$ many edges such that the edges can be $r$-coloured so that every subgraph isomorphic to $C_{2k+1}$ has no colour repeating on the edges. Is it true that\\[F_k(n)\\sim n^2/8?\\]", + "statement": "Define the anti-Ramsey number $\\chi_S(n,e,G)$ as the smallest $r$ such that there is a graph with $n$ vertices and $e$ edges with an $r$-colouring of its edges in which every copy of $G$ has entirely distinct edge colours. Is it true that, for all $k\\geq 3$,\\[\\chi_S(n, \\lfloor n^2/4\\rfloor+1,C_{2k+1})\\sim n^2/8?\\]", "tags": [ "graph theory", "ramsey theory" ], - "last_edited": "", + "last_edited": "01 April 2026", "latex_path": "/latex/809", "formalized": false, "formalized_url": "", "oeis_urls": [], "comments_problem_id": 809, - "comments_count": 0 + "comments_count": 1 }, { "problem_id": 810, @@ -10726,13 +10226,13 @@ "graph theory", "ramsey theory" ], - "last_edited": "", + "last_edited": "01 April 2026", "latex_path": "/latex/810", "formalized": false, "formalized_url": "", "oeis_urls": [], "comments_problem_id": 810, - "comments_count": 7 + "comments_count": 9 }, { "problem_id": 811, @@ -10770,9 +10270,11 @@ ], "last_edited": "", "latex_path": "/latex/812", - "formalized": false, - "formalized_url": "", - "oeis_urls": [], + "formalized": true, + "formalized_url": "https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/ErdosProblems/812.lean", + "oeis_urls": [ + "https://oeis.org/A059442" + ], "comments_problem_id": 812, "comments_count": 0 }, @@ -10834,7 +10336,7 @@ "formalized_url": "", "oeis_urls": [], "comments_problem_id": 819, - "comments_count": 0 + "comments_count": 2 }, { "problem_id": 820, @@ -10912,13 +10414,13 @@ "tags": [ "number theory" ], - "last_edited": "", + "last_edited": "17 April 2026", "latex_path": "/latex/826", "formalized": true, "formalized_url": "https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/ErdosProblems/826.lean", "oeis_urls": [], "comments_problem_id": 826, - "comments_count": 1 + "comments_count": 2 }, { "problem_id": 827, @@ -10932,13 +10434,13 @@ "tags": [ "geometry" ], - "last_edited": "24 October 2025", + "last_edited": "11 May 2026", "latex_path": "/latex/827", "formalized": false, "formalized_url": "", "oeis_urls": [], "comments_problem_id": 827, - "comments_count": 1 + "comments_count": 3 }, { "problem_id": 828, @@ -11041,7 +10543,7 @@ "formalized_url": "https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/ErdosProblems/835.lean", "oeis_urls": [], "comments_problem_id": 835, - "comments_count": 4 + "comments_count": 6 }, { "problem_id": 836, @@ -11063,7 +10565,7 @@ "formalized_url": "", "oeis_urls": [], "comments_problem_id": 836, - "comments_count": 0 + "comments_count": 3 }, { "problem_id": 837, @@ -11105,7 +10607,7 @@ "formalized_url": "", "oeis_urls": [], "comments_problem_id": 838, - "comments_count": 0 + "comments_count": 3 }, { "problem_id": 839, @@ -11125,7 +10627,7 @@ "formalized_url": "", "oeis_urls": [], "comments_problem_id": 839, - "comments_count": 1 + "comments_count": 2 }, { "problem_id": 840, @@ -11148,26 +10650,6 @@ "comments_problem_id": 840, "comments_count": 0 }, - { - "problem_id": 846, - "problem_url": "/846", - "status_bucket": "open", - "status_dom_id": "open", - "status_label": "DISPROVED (LEAN)", - "status_detail": "This has been solved in the negative and the proof verified in Lean.", - "prize_amount": "", - "statement": "Let $A\\subset \\mathbb{R}^2$ be an infinite set for which there exists some $\\epsilon>0$ such that in any subset of $A$ of size $n$ there are always at least $\\epsilon n$ with no three on a line. Is it true that $A$ is the union of a finite number of sets where no three are on a line?", - "tags": [ - "geometry" - ], - "last_edited": "", - "latex_path": "/latex/846", - "formalized": true, - "formalized_url": "https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/ErdosProblems/846.lean", - "oeis_urls": [], - "comments_problem_id": 846, - "comments_count": 8 - }, { "problem_id": 849, "problem_url": "/849", @@ -11220,26 +10702,6 @@ "comments_problem_id": 850, "comments_count": 4 }, - { - "problem_id": 851, - "problem_url": "/851", - "status_bucket": "open", - "status_dom_id": "open", - "status_label": "PROVED", - "status_detail": "This has been solved in the affirmative.", - "prize_amount": "", - "statement": "Let $\\epsilon>0$. Is there some $r\\ll_\\epsilon 1$ such that the density of integers of the form $2^k+n$, where $k\\geq 0$ and $n$ has at most $r$ prime divisors, is at least $1-\\epsilon$?", - "tags": [ - "number theory" - ], - "last_edited": "", - "latex_path": "/latex/851", - "formalized": true, - "formalized_url": "https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/ErdosProblems/851.lean", - "oeis_urls": [], - "comments_problem_id": 851, - "comments_count": 5 - }, { "problem_id": 852, "problem_url": "/852", @@ -11263,7 +10725,7 @@ "https://oeis.org/A078515" ], "comments_problem_id": 852, - "comments_count": 1 + "comments_count": 6 }, { "problem_id": 853, @@ -11287,7 +10749,7 @@ "https://oeis.org/A390769" ], "comments_problem_id": 853, - "comments_count": 2 + "comments_count": 3 }, { "problem_id": 854, @@ -11317,15 +10779,15 @@ "problem_url": "/855", "status_bucket": "open", "status_dom_id": "open", - "status_label": "FALSIFIABLE", - "status_detail": "Open, but could be disproved with a finite counterexample.", + "status_label": "OPEN", + "status_detail": "This is open, and cannot be resolved with a finite computation.", "prize_amount": "", "statement": "If $\\pi(x)$ counts the number of primes in $[1,x]$ then is it true that (for large $x$ and $y$)\\[\\pi(x+y) \\leq \\pi(x)+\\pi(y)?\\]", "tags": [ "number theory", "primes" ], - "last_edited": "12 January 2026", + "last_edited": "08 April 2026", "latex_path": "/latex/855", "formalized": true, "formalized_url": "https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/ErdosProblems/855.lean", @@ -11353,7 +10815,7 @@ "formalized_url": "", "oeis_urls": [], "comments_problem_id": 856, - "comments_count": 12 + "comments_count": 17 }, { "problem_id": 857, @@ -11369,33 +10831,12 @@ ], "last_edited": "", "latex_path": "/latex/857", - "formalized": false, - "formalized_url": "", + "formalized": true, + "formalized_url": "https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/ErdosProblems/857.lean", "oeis_urls": [], "comments_problem_id": 857, "comments_count": 1 }, - { - "problem_id": 858, - "problem_url": "/858", - "status_bucket": "open", - "status_dom_id": "open", - "status_label": "OPEN", - "status_detail": "This is open, and cannot be resolved with a finite computation.", - "prize_amount": "", - "statement": "Let $A\\subseteq \\{1,\\ldots,N\\}$ be such that there is no solution to $at=b$ with $a,b\\in A$ and the smallest prime factor of $t$ is $>a$. Estimate the maximum of\\[\\frac{1}{\\log N}\\sum_{n\\in A}\\frac{1}{n}.\\]", - "tags": [ - "number theory", - "primitive sets" - ], - "last_edited": "", - "latex_path": "/latex/858", - "formalized": false, - "formalized_url": "", - "oeis_urls": [], - "comments_problem_id": 858, - "comments_count": 0 - }, { "problem_id": 859, "problem_url": "/859", @@ -11441,28 +10882,6 @@ "comments_problem_id": 860, "comments_count": 3 }, - { - "problem_id": 863, - "problem_url": "/863", - "status_bucket": "open", - "status_dom_id": "open", - "status_label": "OPEN", - "status_detail": "This is open, and cannot be resolved with a finite computation.", - "prize_amount": "", - "statement": "Let $r\\geq 2$ and let $A\\subseteq \\{1,\\ldots,N\\}$ be a set of maximal size such that there are at most $r$ solutions to $n=a+b$ with $a\\leq b$ for any $n$. (That is, $A$ is a $B_2[r]$ set.) Similarly, let $B\\subseteq \\{1,\\ldots,N\\}$ be a set of maximal size such that there are at most $r$ solutions to $n=a-b$ for any $n$. If $\\lvert A\\rvert\\sim c_rN^{1/2}$ as $N\\to \\infty$ and $\\lvert B\\rvert \\sim c_r'N^{1/2}$ as $N\\to \\infty$ then is it true that $c_r\\neq c_r'$ for $r\\geq 2$? Is it true that $c_r'k$, then is it true that, for every $k\\geq 1$,\\[\\liminf_{n\\to \\infty}\\sum_{0\\leq ik$. Estimate $k(n)$.", + "statement": "Let $k(n)$ be the maximal $k$ such that there exists $m\\leq n$ such that each of the integers\\[m+1,\\ldots,m+k\\]are divisible by at least one prime $>k$. Estimate $k(n)$ - in particular, is it true that\\[\\log k(n) \\leq (\\log n)^{1/2+o(1)}?\\]", "tags": [ "number theory" ], - "last_edited": "30 December 2025", + "last_edited": "03 April 2026", "latex_path": "/latex/962", - "formalized": false, - "formalized_url": "", + "formalized": true, + "formalized_url": "https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/ErdosProblems/962.lean", "oeis_urls": [ "https://oeis.org/A327909" ], "comments_problem_id": 962, - "comments_count": 3 + "comments_count": 5 }, { "problem_id": 963, @@ -12830,7 +12167,7 @@ "tags": [ "number theory" ], - "last_edited": "", + "last_edited": "31 March 2026", "latex_path": "/latex/968", "formalized": true, "formalized_url": "https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/ErdosProblems/968.lean", @@ -12944,7 +12281,7 @@ "formalized_url": "", "oeis_urls": [], "comments_problem_id": 973, - "comments_count": 7 + "comments_count": 6 }, { "problem_id": 975, @@ -12988,7 +12325,7 @@ "formalized_url": "", "oeis_urls": [], "comments_problem_id": 976, - "comments_count": 1 + "comments_count": 3 }, { "problem_id": 978, @@ -12998,17 +12335,17 @@ "status_label": "OPEN", "status_detail": "This is open, and cannot be resolved with a finite computation.", "prize_amount": "", - "statement": "Let $f\\in \\mathbb{Z}[x]$ be an irreducible polynomial of degree $k>2$ (and suppose that $k\\neq 2^l$ for any $l\\geq 1$) such that the leading coefficient of $f$ is positive. Does the set of integers $n\\geq 1$ for which $f(n)$ is $(k-1)$-power-free have positive density? If $k>3$ then are there infinitely many $n$ for which $f(n)$ is $(k-2)$-power-free? In particular, does\\[n^4+2\\]represent infinitely many squarefree numbers?", + "statement": "Let $f\\in \\mathbb{Z}[x]$ be an irreducible polynomial of degree $k>2$ (and suppose that $k\\neq 2^l$ for any $l\\geq 1$) such that the leading coefficient of $f$ is positive. Does the set of integers $n\\geq 1$ for which $f(n)$ is $(k-1)$-power-free have positive density? If $k>3$, and for all primes $p$ there exists $n$ such that $p^{k-2}\\nmid f(n)$, then are there infinitely many $n$ for which $f(n)$ is $(k-2)$-power-free? In particular, does\\[n^4+2\\]represent infinitely many squarefree numbers?", "tags": [ "number theory" ], - "last_edited": "05 March 2026", + "last_edited": "31 March 2026", "latex_path": "/latex/978", "formalized": true, "formalized_url": "https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/ErdosProblems/978.lean", "oeis_urls": [], "comments_problem_id": 978, - "comments_count": 8 + "comments_count": 16 }, { "problem_id": 979, @@ -13030,7 +12367,7 @@ "https://oeis.org/A385316" ], "comments_problem_id": 979, - "comments_count": 12 + "comments_count": 11 }, { "problem_id": 982, @@ -13074,7 +12411,7 @@ "formalized_url": "", "oeis_urls": [], "comments_problem_id": 983, - "comments_count": 4 + "comments_count": 7 }, { "problem_id": 985, @@ -13098,7 +12435,7 @@ "https://oeis.org/A219429" ], "comments_problem_id": 985, - "comments_count": 1 + "comments_count": 2 }, { "problem_id": 986, @@ -13124,47 +12461,6 @@ "comments_problem_id": 986, "comments_count": 0 }, - { - "problem_id": 987, - "problem_url": "/987", - "status_bucket": "open", - "status_dom_id": "open", - "status_label": "OPEN", - "status_detail": "This is open, and cannot be resolved with a finite computation.", - "prize_amount": "", - "statement": "Let $x_1,x_2,\\ldots \\in (0,1)$ be an infinite sequence and let\\[A_k=\\limsup_{n\\to \\infty}\\left\\lvert \\sum_{j\\leq n} e(kx_j)\\right\\rvert,\\]where $e(x)=e^{2\\pi ix}$. Is it true that\\[\\limsup_{k\\to \\infty} A_k=\\infty?\\]Is it possible for $A_k=o(k)$?", - "tags": [ - "analysis", - "discrepancy" - ], - "last_edited": "29 December 2025", - "latex_path": "/latex/987", - "formalized": false, - "formalized_url": "", - "oeis_urls": [], - "comments_problem_id": 987, - "comments_count": 7 - }, - { - "problem_id": 990, - "problem_url": "/990", - "status_bucket": "open", - "status_dom_id": "open", - "status_label": "OPEN", - "status_detail": "This is open, and cannot be resolved with a finite computation.", - "prize_amount": "", - "statement": "Let $f=a_0+\\cdots+a_dx^d\\in \\mathbb{C}[x]$ be a polynomial. Is it true that, if $f$ has roots $z_1,\\ldots,z_d$ with corresponding arguments $\\theta_1,\\ldots,\\theta_d\\in [0,2\\pi]$, then for all intervals $I\\subseteq [0,2\\pi]$\\[\\left\\lvert (\\# \\theta_i \\in I) - \\frac{\\lvert I\\rvert}{2\\pi}d\\right\\rvert \\ll \\left(n\\log M\\right)^{1/2},\\]where $n$ is the number of non-zero coefficients of $f$ and\\[M=\\frac{\\lvert a_0\\rvert+\\cdots +\\lvert a_d\\rvert}{(\\lvert a_0\\rvert\\lvert a_d\\rvert)^{1/2}}.\\]", - "tags": [ - "analysis" - ], - "last_edited": "", - "latex_path": "/latex/990", - "formalized": false, - "formalized_url": "", - "oeis_urls": [], - "comments_problem_id": 990, - "comments_count": 1 - }, { "problem_id": 993, "problem_url": "/993", @@ -13183,7 +12479,7 @@ "formalized_url": "", "oeis_urls": [], "comments_problem_id": 993, - "comments_count": 1 + "comments_count": 3 }, { "problem_id": 995, @@ -13204,7 +12500,7 @@ "formalized_url": "", "oeis_urls": [], "comments_problem_id": 995, - "comments_count": 0 + "comments_count": 7 }, { "problem_id": 996, @@ -13224,29 +12520,7 @@ "formalized_url": "https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/ErdosProblems/996.lean", "oeis_urls": [], "comments_problem_id": 996, - "comments_count": 0 - }, - { - "problem_id": 997, - "problem_url": "/997", - "status_bucket": "open", - "status_dom_id": "open", - "status_label": "OPEN", - "status_detail": "This is open, and cannot be resolved with a finite computation.", - "prize_amount": "", - "statement": "Call $x_1,x_2,\\ldots \\in (0,1)$ well-distributed if, for every $\\epsilon>0$, if $k$ is sufficiently large then, for all $n>0$ and intervals $I\\subseteq [0,1]$,\\[\\lvert \\# \\{ n0$ such that\\[\\lim_k \\frac{R(k+1,k)}{R(k,k)}> 1+c.\\]", + "statement": "Let $R(k,l)$ be the usual Ramsey number: the smallest $n$ such that if the edges of $K_n$ are coloured red and blue then there exists either a red $K_k$ or a blue $K_l$. Prove the existence of some $c>0$ such that\\[\\lim_{k\\to \\infty}\\frac{R(k+1,k)}{R(k,k)}> 1+c.\\]", "tags": [ "graph theory", "ramsey theory" ], - "last_edited": "03 December 2025", + "last_edited": "23 March 2026", "latex_path": "/latex/1030", "formalized": false, "formalized_url": "", @@ -13504,7 +12759,7 @@ "https://oeis.org/A059442" ], "comments_problem_id": 1030, - "comments_count": 1 + "comments_count": 2 }, { "problem_id": 1032, @@ -13525,7 +12780,7 @@ "formalized_url": "", "oeis_urls": [], "comments_problem_id": 1032, - "comments_count": 1 + "comments_count": 3 }, { "problem_id": 1033, @@ -13539,7 +12794,7 @@ "tags": [ "graph theory" ], - "last_edited": "", + "last_edited": "03 April 2026", "latex_path": "/latex/1033", "formalized": false, "formalized_url": "", @@ -13585,7 +12840,7 @@ "formalized_url": "https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/ErdosProblems/1038.lean", "oeis_urls": [], "comments_problem_id": 1038, - "comments_count": 127 + "comments_count": 137 }, { "problem_id": 1039, @@ -13606,7 +12861,7 @@ "formalized_url": "", "oeis_urls": [], "comments_problem_id": 1039, - "comments_count": 1 + "comments_count": 15 }, { "problem_id": 1040, @@ -13626,7 +12881,7 @@ "formalized_url": "", "oeis_urls": [], "comments_problem_id": 1040, - "comments_count": 0 + "comments_count": 1 }, { "problem_id": 1041, @@ -13647,27 +12902,27 @@ "formalized_url": "https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/ErdosProblems/1041.lean", "oeis_urls": [], "comments_problem_id": 1041, - "comments_count": 2 + "comments_count": 46 }, { "problem_id": 1045, "problem_url": "/1045", "status_bucket": "open", "status_dom_id": "open", - "status_label": "FALSIFIABLE", - "status_detail": "Open, but could be disproved with a finite counterexample.", + "status_label": "OPEN", + "status_detail": "This is open, and cannot be resolved with a finite computation.", "prize_amount": "", "statement": "Let $z_1,\\ldots,z_n\\in \\mathbb{C}$ with $\\lvert z_i-z_j\\rvert\\leq 2$ for all $i,j$, and\\[\\Delta(z_1,\\ldots,z_n)=\\prod_{i\\neq j}\\lvert z_i-z_j\\rvert.\\]What is the maximum possible value of $\\Delta$? Is it maximised by taking the $z_i$ to be the vertices of a regular polygon?", "tags": [ "analysis" ], - "last_edited": "30 December 2025", + "last_edited": "02 April 2026", "latex_path": "/latex/1045", "formalized": false, "formalized_url": "", "oeis_urls": [], "comments_problem_id": 1045, - "comments_count": 43 + "comments_count": 47 }, { "problem_id": 1049, @@ -13687,7 +12942,7 @@ "formalized_url": "https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/ErdosProblems/1049.lean", "oeis_urls": [], "comments_problem_id": 1049, - "comments_count": 1 + "comments_count": 3 }, { "problem_id": 1052, @@ -13754,7 +13009,7 @@ "https://oeis.org/A167485" ], "comments_problem_id": 1054, - "comments_count": 5 + "comments_count": 22 }, { "problem_id": 1055, @@ -13910,7 +13165,7 @@ "https://oeis.org/A038372" ], "comments_problem_id": 1062, - "comments_count": 4 + "comments_count": 13 }, { "problem_id": 1063, @@ -13976,7 +13231,7 @@ "formalized_url": "", "oeis_urls": [], "comments_problem_id": 1066, - "comments_count": 1 + "comments_count": 2 }, { "problem_id": 1068, @@ -14087,7 +13342,7 @@ "https://oeis.org/A064164" ], "comments_problem_id": 1074, - "comments_count": 3 + "comments_count": 5 }, { "problem_id": 1075, @@ -14122,7 +13377,7 @@ "geometry", "distances" ], - "last_edited": "20 December 2025", + "last_edited": "11 April 2026", "latex_path": "/latex/1082", "formalized": true, "formalized_url": "https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/ErdosProblems/1082.lean", @@ -14147,7 +13402,9 @@ "latex_path": "/latex/1083", "formalized": false, "formalized_url": "", - "oeis_urls": [], + "oeis_urls": [ + "https://oeis.org/A186704" + ], "comments_problem_id": 1083, "comments_count": 0 }, @@ -14187,13 +13444,15 @@ "geometry", "distances" ], - "last_edited": "17 October 2025", + "last_edited": "23 May 2026", "latex_path": "/latex/1085", "formalized": true, "formalized_url": "https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/ErdosProblems/1085.lean", - "oeis_urls": [], + "oeis_urls": [ + "https://oeis.org/A186705" + ], "comments_problem_id": 1085, - "comments_count": 0 + "comments_count": 1 }, { "problem_id": 1086, @@ -14249,7 +13508,7 @@ "tags": [ "geometry" ], - "last_edited": "16 October 2025", + "last_edited": "08 April 2026", "latex_path": "/latex/1088", "formalized": false, "formalized_url": "", @@ -14258,88 +13517,46 @@ "comments_count": 0 }, { - "problem_id": 1091, - "problem_url": "/1091", + "problem_id": 1093, + "problem_url": "/1093", "status_bucket": "open", "status_dom_id": "open", "status_label": "OPEN", "status_detail": "This is open, and cannot be resolved with a finite computation.", "prize_amount": "", - "statement": "Let $G$ be a $K_4$-free graph with chromatic number $4$. Must $G$ contain an odd cycle with at least two diagonals? More generally, is there some $f(r)\\to \\infty$ such that every graph with chromatic number $4$, in which every subgraph on $\\leq r$ vertices has chromatic number $\\leq 3$, contains an odd cycle with at least $f(r)$ diagonals?", + "statement": "For $n\\geq 2k$ we define the deficiency of $\\binom{n}{k}$ as follows. If $\\binom{n}{k}$ is divisible by a prime $p\\leq k$ then the deficiency is undefined. Otherwise, the deficiency is the number of $0\\leq i1$?", "tags": [ - "graph theory", - "chromatic number" + "number theory", + "binomial coefficients" ], - "last_edited": "06 December 2025", - "latex_path": "/latex/1091", - "formalized": false, - "formalized_url": "", + "last_edited": "27 December 2025", + "latex_path": "/latex/1093", + "formalized": true, + "formalized_url": "https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/ErdosProblems/1093.lean", "oeis_urls": [], - "comments_problem_id": 1091, + "comments_problem_id": 1093, "comments_count": 2 }, { - "problem_id": 1092, - "problem_url": "/1092", + "problem_id": 1094, + "problem_url": "/1094", "status_bucket": "open", "status_dom_id": "open", "status_label": "OPEN", "status_detail": "This is open, and cannot be resolved with a finite computation.", "prize_amount": "", - "statement": "Let $f_r(n)$ be maximal such that, if a graph $G$ has the property that every subgraph $H$ on $m$ vertices is the union of a graph with chromatic number $r$ and a graph with $\\leq f_r(m)$ edges, then $G$ has chromatic number $\\leq r+1$. Is it true that $f_2(n) \\gg n$? More generally, is $f_r(n)\\gg_r n$?", + "statement": "For all $n\\geq 2k$ the least prime factor of $\\binom{n}{k}$ is $\\leq \\max(n/k,k)$, with only finitely many exceptions.", "tags": [ - "graph theory", - "chromatic number" + "number theory", + "binomial coefficients" ], - "last_edited": "06 December 2025", - "latex_path": "/latex/1092", + "last_edited": "24 October 2025", + "latex_path": "/latex/1094", "formalized": true, - "formalized_url": "https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/ErdosProblems/1092.lean", + "formalized_url": "https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/ErdosProblems/1094.lean", "oeis_urls": [], - "comments_problem_id": 1092, - "comments_count": 2 - }, - { - "problem_id": 1093, - "problem_url": "/1093", - "status_bucket": "open", - "status_dom_id": "open", - "status_label": "OPEN", - "status_detail": "This is open, and cannot be resolved with a finite computation.", - "prize_amount": "", - "statement": "For $n\\geq 2k$ we define the deficiency of $\\binom{n}{k}$ as follows. If $\\binom{n}{k}$ is divisible by a prime $p\\leq k$ then the deficiency is undefined. Otherwise, the deficiency is the number of $0\\leq i1$?", - "tags": [ - "number theory", - "binomial coefficients" - ], - "last_edited": "27 December 2025", - "latex_path": "/latex/1093", - "formalized": true, - "formalized_url": "https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/ErdosProblems/1093.lean", - "oeis_urls": [], - "comments_problem_id": 1093, - "comments_count": 2 - }, - { - "problem_id": 1094, - "problem_url": "/1094", - "status_bucket": "open", - "status_dom_id": "open", - "status_label": "OPEN", - "status_detail": "This is open, and cannot be resolved with a finite computation.", - "prize_amount": "", - "statement": "For all $n\\geq 2k$ the least prime factor of $\\binom{n}{k}$ is $\\leq \\max(n/k,k)$, with only finitely many exceptions.", - "tags": [ - "number theory", - "binomial coefficients" - ], - "last_edited": "24 October 2025", - "latex_path": "/latex/1094", - "formalized": true, - "formalized_url": "https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/ErdosProblems/1094.lean", - "oeis_urls": [], - "comments_problem_id": 1094, - "comments_count": 1 + "comments_problem_id": 1094, + "comments_count": 1 }, { "problem_id": 1095, @@ -14362,27 +13579,7 @@ "https://oeis.org/A003458" ], "comments_problem_id": 1095, - "comments_count": 6 - }, - { - "problem_id": 1096, - "problem_url": "/1096", - "status_bucket": "open", - "status_dom_id": "open", - "status_label": "OPEN", - "status_detail": "This is open, and cannot be resolved with a finite computation.", - "prize_amount": "", - "statement": "Let $10$ is sufficiently small, $x_{k+1}-x_k \\to 0$?", - "tags": [ - "number theory" - ], - "last_edited": "19 October 2025", - "latex_path": "/latex/1096", - "formalized": false, - "formalized_url": "", - "oeis_urls": [], - "comments_problem_id": 1096, - "comments_count": 0 + "comments_count": 8 }, { "problem_id": 1097, @@ -14392,18 +13589,18 @@ "status_label": "OPEN", "status_detail": "This is open, and cannot be resolved with a finite computation.", "prize_amount": "", - "statement": "Let $A$ be a set of $n$ integers. How many distinct $d$ can occur as the common difference of a three-term arithmetic progression in $A$? Are there always $O(n^{3/2})$ many such $d$?", + "statement": "Let $A$ be a set of $n$ integers. How many distinct $d$ can occur as the common difference of a three-term arithmetic progression in $A$? In particular, are there always $O(n^{3/2})$ many such $d$?", "tags": [ "number theory", "additive combinatorics" ], - "last_edited": "03 December 2025", + "last_edited": "01 April 2026", "latex_path": "/latex/1097", "formalized": true, "formalized_url": "https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/ErdosProblems/1097.lean", "oeis_urls": [], "comments_problem_id": 1097, - "comments_count": 11 + "comments_count": 17 }, { "problem_id": 1100, @@ -14426,7 +13623,7 @@ "https://oeis.org/A325864" ], "comments_problem_id": 1100, - "comments_count": 2 + "comments_count": 5 }, { "problem_id": 1101, @@ -14446,7 +13643,7 @@ "formalized_url": "https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/ErdosProblems/1101.lean", "oeis_urls": [], "comments_problem_id": 1101, - "comments_count": 1 + "comments_count": 3 }, { "problem_id": 1103, @@ -14468,7 +13665,7 @@ "https://oeis.org/A392164" ], "comments_problem_id": 1103, - "comments_count": 4 + "comments_count": 5 }, { "problem_id": 1104, @@ -14487,7 +13684,9 @@ "latex_path": "/latex/1104", "formalized": true, "formalized_url": "https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/ErdosProblems/1104.lean", - "oeis_urls": [], + "oeis_urls": [ + "https://oeis.org/A292528" + ], "comments_problem_id": 1104, "comments_count": 2 }, @@ -14599,13 +13798,13 @@ "tags": [ "number theory" ], - "last_edited": "22 January 2026", + "last_edited": "01 April 2026", "latex_path": "/latex/1110", "formalized": false, "formalized_url": "", "oeis_urls": [], "comments_problem_id": 1110, - "comments_count": 3 + "comments_count": 4 }, { "problem_id": 1111, @@ -14662,8 +13861,8 @@ ], "last_edited": "29 December 2025", "latex_path": "/latex/1113", - "formalized": false, - "formalized_url": "", + "formalized": true, + "formalized_url": "https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/ErdosProblems/1113.lean", "oeis_urls": [ "https://oeis.org/A076336" ], @@ -14722,13 +13921,13 @@ "tags": [ "number theory" ], - "last_edited": "30 December 2025", + "last_edited": "01 April 2026", "latex_path": "/latex/1122", "formalized": false, "formalized_url": "", "oeis_urls": [], "comments_problem_id": 1122, - "comments_count": 0 + "comments_count": 1 }, { "problem_id": 1131, @@ -14749,7 +13948,7 @@ "formalized_url": "", "oeis_urls": [], "comments_problem_id": 1131, - "comments_count": 2 + "comments_count": 5 }, { "problem_id": 1132, @@ -14764,13 +13963,13 @@ "analysis", "polynomials" ], - "last_edited": "23 January 2026", + "last_edited": "01 April 2026", "latex_path": "/latex/1132", "formalized": false, "formalized_url": "", "oeis_urls": [], "comments_problem_id": 1132, - "comments_count": 1 + "comments_count": 2 }, { "problem_id": 1133, @@ -14791,7 +13990,7 @@ "formalized_url": "", "oeis_urls": [], "comments_problem_id": 1133, - "comments_count": 0 + "comments_count": 2 }, { "problem_id": 1135, @@ -14846,8 +14045,8 @@ "problem_url": "/1138", "status_bucket": "open", "status_dom_id": "open", - "status_label": "OPEN", - "status_detail": "This is open, and cannot be resolved with a finite computation.", + "status_label": "DISPROVED (LEAN)", + "status_detail": "This has been solved in the negative and the proof verified in Lean.", "prize_amount": "", "statement": "Let $x/21$. If $d=\\max_{p_n \\left(\\frac{2}{\\pi}-o(1)\\right)\\log n?\\]", - "tags": [ - "analysis", - "polynomials" - ], - "last_edited": "01 February 2026", - "latex_path": "/latex/1153", - "formalized": false, - "formalized_url": "", - "oeis_urls": [], - "comments_problem_id": 1153, - "comments_count": 7 - }, { "problem_id": 1154, "problem_url": "/1154", @@ -15234,7 +14366,7 @@ "tags": [ "combinatorics" ], - "last_edited": "23 January 2026", + "last_edited": "10 April 2026", "latex_path": "/latex/1159", "formalized": false, "formalized_url": "", @@ -15302,7 +14434,7 @@ "formalized_url": "", "oeis_urls": [], "comments_problem_id": 1163, - "comments_count": 1 + "comments_count": 2 }, { "problem_id": 1167, @@ -15319,8 +14451,8 @@ ], "last_edited": "23 January 2026", "latex_path": "/latex/1167", - "formalized": false, - "formalized_url": "", + "formalized": true, + "formalized_url": "https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/ErdosProblems/1167.lean", "oeis_urls": [], "comments_problem_id": 1167, "comments_count": 3 @@ -15344,7 +14476,7 @@ "formalized_url": "", "oeis_urls": [], "comments_problem_id": 1168, - "comments_count": 0 + "comments_count": 1 }, { "problem_id": 1169, @@ -15417,18 +14549,18 @@ "status_label": "OPEN", "status_detail": "This is open, and cannot be resolved with a finite computation.", "prize_amount": "", - "statement": "Establish whether the following are true assuming the generalised continuum hypothesis:\\[\\omega_3 \\to (\\omega_2,\\omega_1+2)^2,\\]\\[\\omega_3\\to (\\omega_2+\\omega_1,\\omega_2+\\omega)^2,\\]\\[\\omega_2\\to (\\omega_1^{\\omega+2}+2, \\omega_1+2)^2.\\]Establish whether the following is true assuming the continuum hypothesis:\\[\\omega_2\\to (\\omega_1+\\omega)_2^2.\\]", + "statement": "Establish whether the following are true assuming the generalised continuum hypothesis:\\[\\omega_3 \\to (\\omega_2,\\omega_1+2)^2,\\]\\[\\omega_3\\to (\\omega_2+\\omega_1,\\omega_2+\\omega)^2,\\]\\[\\omega_2\\to (\\omega_1^{\\omega+2}+2, \\omega_1+2)^2.\\]Establish whether the following is consistent with the generalised continuum hypothesis:\\[\\omega_2\\to (\\omega_1+\\omega)_2^2,\\]or even $\\omega_2 \\to (\\xi)_2^2$ for all $\\xi<\\omega_2$.", "tags": [ "set theory", "ramsey theory" ], - "last_edited": "23 January 2026", + "last_edited": "11 April 2026", "latex_path": "/latex/1172", "formalized": false, "formalized_url": "", "oeis_urls": [], "comments_problem_id": 1172, - "comments_count": 1 + "comments_count": 2 }, { "problem_id": 1173, @@ -15456,8 +14588,8 @@ "problem_url": "/1174", "status_bucket": "open", "status_dom_id": "open", - "status_label": "OPEN", - "status_detail": "This is open, and cannot be resolved with a finite computation.", + "status_label": "NOT DISPROVABLE", + "status_detail": "Open in general, but there exist models of set theory where the result is true.", "prize_amount": "", "statement": "Does there exist a graph $G$ with no $K_4$ such that every edge colouring of $G$ with countably many colours contains a monochromatic $K_3$? Does there exist a graph $G$ with no $K_{\\aleph_1}$ such that every edge colouring of $G$ with countably many colours contains a monochromatic $K_{\\aleph_0}$?", "tags": [ @@ -15470,7 +14602,7 @@ "formalized_url": "", "oeis_urls": [], "comments_problem_id": 1174, - "comments_count": 1 + "comments_count": 2 }, { "problem_id": 1175, @@ -15487,8 +14619,8 @@ ], "last_edited": "", "latex_path": "/latex/1175", - "formalized": false, - "formalized_url": "", + "formalized": true, + "formalized_url": "https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/ErdosProblems/1175.lean", "oeis_urls": [], "comments_problem_id": 1175, "comments_count": 1 @@ -15498,8 +14630,8 @@ "problem_url": "/1176", "status_bucket": "open", "status_dom_id": "open", - "status_label": "OPEN", - "status_detail": "This is open, and cannot be resolved with a finite computation.", + "status_label": "NOT DISPROVABLE", + "status_detail": "Open in general, but there exist models of set theory where the result is true.", "prize_amount": "", "statement": "Let $G$ be a graph with chromatic number $\\aleph_1$. Is it true that there is a colouring of the edges with $\\aleph_1$ many colours such that, in any countable colouring of the vertices, there exists a vertex colour containing all edge colours?", "tags": [ @@ -15512,7 +14644,7 @@ "formalized_url": "https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/ErdosProblems/1176.lean", "oeis_urls": [], "comments_problem_id": 1176, - "comments_count": 0 + "comments_count": 1 }, { "problem_id": 1177, @@ -15556,6 +14688,468 @@ "oeis_urls": [], "comments_problem_id": 1178, "comments_count": 1 + }, + { + "problem_id": 1181, + "problem_url": "/1181", + "status_bucket": "open", + "status_dom_id": "open", + "status_label": "OPEN", + "status_detail": "This is open, and cannot be resolved with a finite computation.", + "prize_amount": "", + "statement": "Let $q(n,k)$ denote the least prime which does not divide $\\prod_{1\\leq i\\leq k}(n+i)$. Is it true that there exists some $c>0$ such that, for all large $n$,\\[q(n,\\log n)<(1-c)(\\log n)^2?\\]", + "tags": [ + "number theory" + ], + "last_edited": "07 March 2026", + "latex_path": "/latex/1181", + "formalized": false, + "formalized_url": "", + "oeis_urls": [], + "comments_problem_id": 1181, + "comments_count": 0 + }, + { + "problem_id": 1182, + "problem_url": "/1182", + "status_bucket": "open", + "status_dom_id": "open", + "status_label": "OPEN", + "status_detail": "This is open, and cannot be resolved with a finite computation.", + "prize_amount": "", + "statement": "Let $f(n)$ be maximal such that there is a connected graph $G$ with $n$ vertices and $f(n)$ edges such that\\[R(K_3,G)= 2n-1.\\]Let $F(n)$ be maximal such that every connected graph $G$ with $n$ vertices and $\\leq F(n)$ edges has\\[R(K_3,G)= 2n-1.\\]Estimate $f(n)$ and $F(n)$. In particular, is it true that $F(n)/n\\to \\infty$?", + "tags": [ + "graph theory", + "ramsey theory" + ], + "last_edited": "11 April 2026", + "latex_path": "/latex/1182", + "formalized": false, + "formalized_url": "", + "oeis_urls": [], + "comments_problem_id": 1182, + "comments_count": 3 + }, + { + "problem_id": 1183, + "problem_url": "/1183", + "status_bucket": "open", + "status_dom_id": "open", + "status_label": "OPEN", + "status_detail": "This is open, and cannot be resolved with a finite computation.", + "prize_amount": "", + "statement": "Let $f(n)$ be maximal such that in any $2$-colouring of the subsets of $\\{1,\\ldots,n\\}$ there is always a monochromatic family of at least $f(n)$ sets which is closed under taking unions and intersections. Estimate $f(n)$. Let $F(n)$ be defined similarly, except that we only require the family be closed under taking unions. Estimate $F(n)$. In particular, is it true that $F(n)\\geq n^{\\omega(n)}$ for some $\\omega(n)\\to \\infty$ as $n\\to \\infty$, and $F(n)<(1+o(1))^n$?", + "tags": [ + "combinatorics", + "ramsey theory" + ], + "last_edited": "", + "latex_path": "/latex/1183", + "formalized": false, + "formalized_url": "", + "oeis_urls": [], + "comments_problem_id": 1183, + "comments_count": 10 + }, + { + "problem_id": 1184, + "problem_url": "/1184", + "status_bucket": "open", + "status_dom_id": "open", + "status_label": "OPEN", + "status_detail": "This is open, and cannot be resolved with a finite computation.", + "prize_amount": "", + "statement": "Let $f(n,k)$ count the number of $1\\leq i\\leq k$ such that $P(n+i)>k$ (where $P(m)$ is the largest prime divisor of $m$). Is it true that, if $\\alpha>1$ is such that $n=k^{\\alpha+o(1)}$, then\\[f(n,k)=(1-\\rho(\\alpha)+o(1))k,\\]where $\\rho$ is the Dickman function ?", + "tags": [ + "number theory", + "primes" + ], + "last_edited": "06 April 2026", + "latex_path": "/latex/1184", + "formalized": false, + "formalized_url": "", + "oeis_urls": [], + "comments_problem_id": 1184, + "comments_count": 1 + }, + { + "problem_id": 1186, + "problem_url": "/1186", + "status_bucket": "open", + "status_dom_id": "open", + "status_label": "OPEN", + "status_detail": "This is open, and cannot be resolved with a finite computation.", + "prize_amount": "", + "statement": "Let $\\delta_k$ be such that in any $2$-colouring of $\\{1,\\ldots,n\\}$ there exist at least $(\\delta_k+o(1))n^2$ many monochromatic $k$-term arithmetic progressions. Give reasonable bounds (or even an asymptotic formula) for $\\delta_k$.", + "tags": [ + "additive combinatorics", + "arithmetic progressions" + ], + "last_edited": "08 April 2026", + "latex_path": "/latex/1186", + "formalized": false, + "formalized_url": "", + "oeis_urls": [], + "comments_problem_id": 1186, + "comments_count": 0 + }, + { + "problem_id": 1188, + "problem_url": "/1188", + "status_bucket": "open", + "status_dom_id": "open", + "status_label": "OPEN", + "status_detail": "This is open, and cannot be resolved with a finite computation.", + "prize_amount": "", + "statement": "Call a set of distinct integers $11$) form an irreducible covering set?", + "tags": [ + "number theory", + "covering systems" + ], + "last_edited": "08 April 2026", + "latex_path": "/latex/1189", + "formalized": false, + "formalized_url": "", + "oeis_urls": [], + "comments_problem_id": 1189, + "comments_count": 6 + }, + { + "problem_id": 1190, + "problem_url": "/1190", + "status_bucket": "open", + "status_dom_id": "open", + "status_label": "OPEN", + "status_detail": "This is open, and cannot be resolved with a finite computation.", + "prize_amount": "", + "statement": "Let\\[\\epsilon_m=\\max \\sum \\frac{1}{n_i}\\]where the maximum is taken over all finite sequences $m0\\]for some $c>0$?", + "tags": [ + "additive combinatorics", + "sidon sets" + ], + "last_edited": "06 April 2026", + "latex_path": "/latex/1191", + "formalized": false, + "formalized_url": "", + "oeis_urls": [], + "comments_problem_id": 1191, + "comments_count": 0 + }, + { + "problem_id": 1192, + "problem_url": "/1192", + "status_bucket": "open", + "status_dom_id": "open", + "status_label": "OPEN", + "status_detail": "This is open, and cannot be resolved with a finite computation.", + "prize_amount": "", + "statement": "For $A\\subset \\mathbb{N}$ let $f_r(n)$ count the number of solutions to $n=a_1+\\cdots+a_r$ with $a_i\\in A$. Does there exist, for all $r\\geq 2$, a basis $A$ of order $r$ (so that $f_r(n)>0$ for all large $n$) such that\\[\\sum_{n\\leq x}f_r(n)^2 \\ll x\\]for all $x$?", + "tags": [ + "additive combinatorics", + "additive basis" + ], + "last_edited": "06 April 2026", + "latex_path": "/latex/1192", + "formalized": false, + "formalized_url": "", + "oeis_urls": [], + "comments_problem_id": 1192, + "comments_count": 0 + }, + { + "problem_id": 1194, + "problem_url": "/1194", + "status_bucket": "open", + "status_dom_id": "open", + "status_label": "OPEN", + "status_detail": "This is open, and cannot be resolved with a finite computation.", + "prize_amount": "", + "statement": "Let $A\\subset\\mathbb{N}$ be such that every integer $n\\geq 1$ can be written uniquely as $a_n-b_n$ for some $a_n,b_n\\in A$. How fast must $a_n/n$ increase?", + "tags": [ + "additive combinatorics", + "additive basis", + "sidon sets" + ], + "last_edited": "24 April 2026", + "latex_path": "/latex/1194", + "formalized": false, + "formalized_url": "", + "oeis_urls": [], + "comments_problem_id": 1194, + "comments_count": 8 + }, + { + "problem_id": 1199, + "problem_url": "/1199", + "status_bucket": "open", + "status_dom_id": "open", + "status_label": "OPEN", + "status_detail": "This is open, and cannot be resolved with a finite computation.", + "prize_amount": "", + "statement": "Is it true that in any $2$-colouring of $\\mathbb{N}$ there exists an infinite set $A$ such that all elements of $A+A$ are the same colour?", + "tags": [ + "additive combinatorics", + "ramsey theory" + ], + "last_edited": "", + "latex_path": "/latex/1199", + "formalized": true, + "formalized_url": "https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/ErdosProblems/1199.lean", + "oeis_urls": [], + "comments_problem_id": 1199, + "comments_count": 3 + }, + { + "problem_id": 1200, + "problem_url": "/1200", + "status_bucket": "open", + "status_dom_id": "open", + "status_label": "OPEN", + "status_detail": "This is open, and cannot be resolved with a finite computation.", + "prize_amount": "", + "statement": "There exists a constant $C$ such that for all large $x$ there is a collection of primes $p_1<\\ldots0$ there exists a $k$ such that the density of $n$ for which\\[P(n(n+1)\\cdots(n+k))>n^{1-\\epsilon}\\]is at least $1-\\eta$ (where $P(m)$ is the greatest prime divisor of $m$)?", + "tags": [ + "number theory", + "primes" + ], + "last_edited": "", + "latex_path": "/latex/1201", + "formalized": false, + "formalized_url": "", + "oeis_urls": [], + "comments_problem_id": 1201, + "comments_count": 10 + }, + { + "problem_id": 1203, + "problem_url": "/1203", + "status_bucket": "open", + "status_dom_id": "open", + "status_label": "OPEN", + "status_detail": "This is open, and cannot be resolved with a finite computation.", + "prize_amount": "", + "statement": "If $\\omega(n)$ counts the number of distinct prime divisors of $n$ then let\\[F(n)=\\max_k \\omega(n+k)\\frac{\\log\\log k}{\\log k}.\\]Prove that $F(n)\\to \\infty$ as $n\\to \\infty$.", + "tags": [ + "number theory" + ], + "last_edited": "07 April 2026", + "latex_path": "/latex/1203", + "formalized": true, + "formalized_url": "https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/ErdosProblems/1203.lean", + "oeis_urls": [], + "comments_problem_id": 1203, + "comments_count": 0 + }, + { + "problem_id": 1204, + "problem_url": "/1204", + "status_bucket": "open", + "status_dom_id": "open", + "status_label": "OPEN", + "status_detail": "This is open, and cannot be resolved with a finite computation.", + "prize_amount": "", + "statement": "We call a sequence of integers $0\\leq a_1<\\cdots 0$?", + "tags": [ + "geometry", + "distances" + ], + "last_edited": "08 April 2026", + "latex_path": "/latex/1207", + "formalized": false, + "formalized_url": "", + "oeis_urls": [], + "comments_problem_id": 1207, + "comments_count": 0 + }, + { + "problem_id": 1208, + "problem_url": "/1208", + "status_bucket": "open", + "status_dom_id": "open", + "status_label": "OPEN", + "status_detail": "This is open, and cannot be resolved with a finite computation.", + "prize_amount": "", + "statement": "For $d\\geq 2$ let $F_d(n)$ be minimal such that every set of $n$ points in $\\mathbb{R}^d$ contains a set of $F_d(n)$ points with distinct distances. Estimate $F_d(n)$ for fixed $d$ as $n\\to \\infty$.", + "tags": [ + "geometry", + "distances" + ], + "last_edited": "08 April 2026", + "latex_path": "/latex/1208", + "formalized": false, + "formalized_url": "", + "oeis_urls": [], + "comments_problem_id": 1208, + "comments_count": 0 + }, + { + "problem_id": 1209, + "problem_url": "/1209", + "status_bucket": "open", + "status_dom_id": "open", + "status_label": "OPEN", + "status_detail": "This is open, and cannot be resolved with a finite computation.", + "prize_amount": "", + "statement": "Let $A=\\{a_11$ and at least one of $x$ or $y$ is composite?", + "tags": [ + "number theory", + "primes" + ], + "last_edited": "08 April 2026", + "latex_path": "/latex/1212", + "formalized": false, + "formalized_url": "", + "oeis_urls": [], + "comments_problem_id": 1212, + "comments_count": 0 } ] } diff --git a/packs/erdos-open-problems/data/erdos_problems.all.json b/packs/erdos-open-problems/data/erdos_problems.all.json index 2570e3c..ffd5d0c 100644 --- a/packs/erdos-open-problems/data/erdos_problems.all.json +++ b/packs/erdos-open-problems/data/erdos_problems.all.json @@ -2,35 +2,35 @@ "schema_version": "1.0.0", "subset": "all", "active_status": "open", - "generated_at_utc": "2026-03-05T15:52:31Z", + "generated_at_utc": "2026-05-27T07:44:18Z", "source": { "site": "erdosproblems.com", "url": "https://erdosproblems.com/range/1-end", - "source_sha256": "72366d21b9530355396bef95b3ead90176a9c634b7ecd68fd30c85bad703a0c1", + "source_sha256": "8a599b708e4f1d2a8ac913674e68a7628982fff732e9f52966807a8fdd2b92c1", "solve_count": { - "raw": "488 solved out of 1179 shown", - "solved": 488, - "shown": 1179 + "raw": "546 solved out of 1217 shown", + "solved": 546, + "shown": 1217 } }, "summary": { - "total": 1179, - "open": 691, - "closed": 488, + "total": 1217, + "open": 671, + "closed": 546, "unknown": 0, "status_label_counts": { "DECIDABLE": 9, "DISPROVED": 76, - "DISPROVED (LEAN)": 40, - "FALSIFIABLE": 29, + "DISPROVED (LEAN)": 56, + "FALSIFIABLE": 27, "INDEPENDENT": 3, - "NOT DISPROVABLE": 2, - "NOT PROVABLE": 4, - "OPEN": 645, - "PROVED": 239, - "PROVED (LEAN)": 63, - "SOLVED": 52, - "SOLVED (LEAN)": 10, + "NOT DISPROVABLE": 4, + "NOT PROVABLE": 3, + "OPEN": 627, + "PROVED": 216, + "PROVED (LEAN)": 107, + "SOLVED": 62, + "SOLVED (LEAN)": 20, "VERIFIABLE": 7 } }, @@ -1213,7 +1213,45 @@ 1176, 1177, 1178, - 1179 + 1179, + 1180, + 1181, + 1182, + 1183, + 1184, + 1185, + 1186, + 1187, + 1188, + 1189, + 1190, + 1191, + 1192, + 1193, + 1194, + 1195, + 1196, + 1197, + 1198, + 1199, + 1200, + 1201, + 1202, + 1203, + 1204, + 1205, + 1206, + 1207, + 1208, + 1209, + 1210, + 1211, + 1212, + 1213, + 1214, + 1215, + 1216, + 1217 ], "problems": [ { @@ -1229,7 +1267,7 @@ "number theory", "additive combinatorics" ], - "last_edited": "23 January 2026", + "last_edited": "06 April 2026", "latex_path": "/latex/1", "formalized": true, "formalized_url": "https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/ErdosProblems/1.lean", @@ -1237,7 +1275,7 @@ "https://oeis.org/A276661" ], "comments_problem_id": 1, - "comments_count": 4 + "comments_count": 3 }, { "problem_id": 2, @@ -1252,7 +1290,7 @@ "number theory", "covering systems" ], - "last_edited": "23 January 2026", + "last_edited": "05 April 2026", "latex_path": "/latex/2", "formalized": false, "formalized_url": "", @@ -1276,7 +1314,7 @@ "additive combinatorics", "arithmetic progressions" ], - "last_edited": "23 January 2026", + "last_edited": "04 April 2026", "latex_path": "/latex/3", "formalized": true, "formalized_url": "https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/ErdosProblems/3.lean", @@ -1333,7 +1371,7 @@ "https://oeis.org/A001223" ], "comments_problem_id": 5, - "comments_count": 2 + "comments_count": 3 }, { "problem_id": 6, @@ -1373,11 +1411,11 @@ ], "last_edited": "22 January 2026", "latex_path": "/latex/7", - "formalized": false, - "formalized_url": "", + "formalized": true, + "formalized_url": "https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/ErdosProblems/7.lean", "oeis_urls": [], "comments_problem_id": 7, - "comments_count": 12 + "comments_count": 21 }, { "problem_id": 8, @@ -1392,7 +1430,7 @@ "number theory", "covering systems" ], - "last_edited": "28 December 2025", + "last_edited": "05 April 2026", "latex_path": "/latex/8", "formalized": false, "formalized_url": "", @@ -1414,7 +1452,7 @@ "additive basis", "primes" ], - "last_edited": "20 January 2026", + "last_edited": "07 April 2026", "latex_path": "/latex/9", "formalized": true, "formalized_url": "https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/ErdosProblems/9.lean", @@ -1438,7 +1476,7 @@ "additive basis", "primes" ], - "last_edited": "24 January 2026", + "last_edited": "11 April 2026", "latex_path": "/latex/10", "formalized": true, "formalized_url": "https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/ErdosProblems/10.lean", @@ -1446,22 +1484,22 @@ "https://oeis.org/A387053" ], "comments_problem_id": 10, - "comments_count": 0 + "comments_count": 1 }, { "problem_id": 11, "problem_url": "/11", "status_bucket": "open", "status_dom_id": "open", - "status_label": "FALSIFIABLE", - "status_detail": "Open, but could be disproved with a finite counterexample.", + "status_label": "OPEN", + "status_detail": "This is open, and cannot be resolved with a finite computation.", "prize_amount": "", "statement": "Is every large odd integer $n$ the sum of a squarefree number and a power of 2?", "tags": [ "number theory", "additive basis" ], - "last_edited": "20 January 2026", + "last_edited": "05 April 2026", "latex_path": "/latex/11", "formalized": true, "formalized_url": "https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/ErdosProblems/11.lean", @@ -1484,13 +1522,13 @@ "tags": [ "number theory" ], - "last_edited": "", + "last_edited": "08 April 2026", "latex_path": "/latex/12", "formalized": true, "formalized_url": "https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/ErdosProblems/12.lean", "oeis_urls": [], "comments_problem_id": 12, - "comments_count": 0 + "comments_count": 13 }, { "problem_id": 13, @@ -1504,7 +1542,7 @@ "tags": [ "number theory" ], - "last_edited": "", + "last_edited": "08 April 2026", "latex_path": "/latex/13", "formalized": true, "formalized_url": "https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/ErdosProblems/13.lean", @@ -1553,8 +1591,8 @@ ], "last_edited": "", "latex_path": "/latex/15", - "formalized": false, - "formalized_url": "", + "formalized": true, + "formalized_url": "https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/ErdosProblems/15.lean", "oeis_urls": [], "comments_problem_id": 15, "comments_count": 2 @@ -1573,10 +1611,10 @@ "additive basis", "primes" ], - "last_edited": "28 December 2025", + "last_edited": "05 April 2026", "latex_path": "/latex/16", - "formalized": false, - "formalized_url": "", + "formalized": true, + "formalized_url": "https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/ErdosProblems/16.lean", "oeis_urls": [ "https://oeis.org/A006285" ], @@ -1620,10 +1658,10 @@ "divisors", "factorials" ], - "last_edited": "20 January 2026", + "last_edited": "11 April 2026", "latex_path": "/latex/18", - "formalized": false, - "formalized_url": "", + "formalized": true, + "formalized_url": "https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/ErdosProblems/18.lean", "oeis_urls": [ "https://oeis.org/A005153" ], @@ -1643,7 +1681,7 @@ "graph theory", "chromatic number" ], - "last_edited": "23 January 2026", + "last_edited": "07 March 2026", "latex_path": "/latex/19", "formalized": false, "formalized_url": "", @@ -1663,7 +1701,7 @@ "tags": [ "combinatorics" ], - "last_edited": "25 January 2026", + "last_edited": "07 March 2026", "latex_path": "/latex/20", "formalized": true, "formalized_url": "https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/ErdosProblems/20.lean", @@ -1671,7 +1709,7 @@ "https://oeis.org/A332077" ], "comments_problem_id": 20, - "comments_count": 6 + "comments_count": 9 }, { "problem_id": 21, @@ -1743,8 +1781,8 @@ "problem_url": "/24", "status_bucket": "closed", "status_dom_id": "solved", - "status_label": "PROVED", - "status_detail": "This has been solved in the affirmative.", + "status_label": "PROVED (LEAN)", + "status_detail": "This has been solved in the affirmative and the proof verified in Lean.", "prize_amount": "", "statement": "Does every triangle-free graph on $5n$ vertices contain at most $n^5$ copies of $C_5$?", "tags": [ @@ -1756,7 +1794,7 @@ "formalized_url": "", "oeis_urls": [], "comments_problem_id": 24, - "comments_count": 0 + "comments_count": 2 }, { "problem_id": 25, @@ -1776,7 +1814,7 @@ "formalized_url": "https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/ErdosProblems/25.lean", "oeis_urls": [], "comments_problem_id": 25, - "comments_count": 2 + "comments_count": 7 }, { "problem_id": 26, @@ -1791,13 +1829,13 @@ "number theory", "divisors" ], - "last_edited": "07 December 2025", + "last_edited": "08 April 2026", "latex_path": "/latex/26", "formalized": true, "formalized_url": "https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/ErdosProblems/26.lean", "oeis_urls": [], "comments_problem_id": 26, - "comments_count": 6 + "comments_count": 11 }, { "problem_id": 27, @@ -1833,13 +1871,13 @@ "number theory", "additive basis" ], - "last_edited": "23 January 2026", + "last_edited": "06 April 2026", "latex_path": "/latex/28", "formalized": true, "formalized_url": "https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/ErdosProblems/28.lean", "oeis_urls": [], "comments_problem_id": 28, - "comments_count": 4 + "comments_count": 5 }, { "problem_id": 29, @@ -1876,7 +1914,7 @@ "sidon sets", "additive combinatorics" ], - "last_edited": "23 January 2026", + "last_edited": "06 April 2026", "latex_path": "/latex/30", "formalized": true, "formalized_url": "https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/ErdosProblems/30.lean", @@ -1949,7 +1987,7 @@ "formalized_url": "https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/ErdosProblems/33.lean", "oeis_urls": [], "comments_problem_id": 33, - "comments_count": 4 + "comments_count": 5 }, { "problem_id": 34, @@ -2017,7 +2055,7 @@ "https://oeis.org/A393584" ], "comments_problem_id": 36, - "comments_count": 4 + "comments_count": 5 }, { "problem_id": 37, @@ -2043,22 +2081,22 @@ { "problem_id": 38, "problem_url": "/38", - "status_bucket": "open", - "status_dom_id": "open", - "status_label": "OPEN", - "status_detail": "This is open, and cannot be resolved with a finite computation.", + "status_bucket": "closed", + "status_dom_id": "solved", + "status_label": "PROVED (LEAN)", + "status_detail": "This has been solved in the affirmative and the proof verified in Lean.", "prize_amount": "", "statement": "Does there exist $B\\subset\\mathbb{N}$ which is not an additive basis, but is such that for every set $A\\subseteq\\mathbb{N}$ of Schnirelmann density $\\alpha$ and every $N$ there exists $b\\in B$ such that\\[\\lvert (A\\cup (A+b))\\cap \\{1,\\ldots,N\\}\\rvert\\geq (\\alpha+f(\\alpha))N\\]where $f(\\alpha)>0$ for $0<\\alpha <1 $? The Schnirelmann density is defined by\\[d_s(A) = \\inf_{N\\geq 1}\\frac{\\lvert A\\cap\\{1,\\ldots,N\\}\\rvert}{N}.\\]", "tags": [ "number theory" ], - "last_edited": "16 September 2025", + "last_edited": "02 May 2026", "latex_path": "/latex/38", "formalized": true, "formalized_url": "https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/ErdosProblems/38.lean", "oeis_urls": [], "comments_problem_id": 38, - "comments_count": 1 + "comments_count": 6 }, { "problem_id": 39, @@ -2074,7 +2112,7 @@ "sidon sets", "additive combinatorics" ], - "last_edited": "23 January 2026", + "last_edited": "06 April 2026", "latex_path": "/latex/39", "formalized": true, "formalized_url": "https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/ErdosProblems/39.lean", @@ -2117,7 +2155,7 @@ "sidon sets", "additive combinatorics" ], - "last_edited": "23 January 2026", + "last_edited": "06 April 2026", "latex_path": "/latex/41", "formalized": true, "formalized_url": "https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/ErdosProblems/41.lean", @@ -2128,10 +2166,10 @@ { "problem_id": 42, "problem_url": "/42", - "status_bucket": "open", - "status_dom_id": "open", - "status_label": "OPEN", - "status_detail": "This is open, and cannot be resolved with a finite computation.", + "status_bucket": "closed", + "status_dom_id": "solved", + "status_label": "SOLVED (LEAN)", + "status_detail": "This has been resolved in some other way than a proof or disproof, and that resolution verified in Lean.", "prize_amount": "", "statement": "Let $M\\geq 1$ and $N$ be sufficiently large in terms of $M$. Is it true that for every Sidon set $A\\subset \\{1,\\ldots,N\\}$ there is another Sidon set $B\\subset \\{1,\\ldots,N\\}$ of size $M$ such that $(A-A)\\cap(B-B)=\\{0\\}$?", "tags": [ @@ -2139,21 +2177,21 @@ "sidon sets", "additive combinatorics" ], - "last_edited": "23 January 2026", + "last_edited": "10 May 2026", "latex_path": "/latex/42", "formalized": true, "formalized_url": "https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/ErdosProblems/42.lean", "oeis_urls": [], "comments_problem_id": 42, - "comments_count": 6 + "comments_count": 40 }, { "problem_id": 43, "problem_url": "/43", - "status_bucket": "open", - "status_dom_id": "open", - "status_label": "OPEN", - "status_detail": "This is open, and cannot be resolved with a finite computation.", + "status_bucket": "closed", + "status_dom_id": "solved", + "status_label": "DISPROVED", + "status_detail": "This has been solved in the negative.", "prize_amount": "$100", "statement": "If $A,B\\subset \\{1,\\ldots,N\\}$ are two Sidon sets such that $(A-A)\\cap(B-B)=\\{0\\}$ then is it true that\\[ \\binom{\\lvert A\\rvert}{2}+\\binom{\\lvert B\\rvert}{2}\\leq\\binom{f(N)}{2}+O(1),\\]where $f(N)$ is the maximum possible size of a Sidon set in $\\{1,\\ldots,N\\}$? If $\\lvert A\\rvert=\\lvert B\\rvert$ then can this bound be improved to\\[\\binom{\\lvert A\\rvert}{2}+\\binom{\\lvert B\\rvert}{2}\\leq (1-c+o(1))\\binom{f(N)}{2}\\]for some constant $c>0$?", "tags": [ @@ -2161,17 +2199,17 @@ "sidon sets", "additive combinatorics" ], - "last_edited": "20 December 2025", + "last_edited": "10 May 2026", "latex_path": "/latex/43", - "formalized": false, - "formalized_url": "", + "formalized": true, + "formalized_url": "https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/ErdosProblems/43.lean", "oeis_urls": [ "https://oeis.org/A003022", "https://oeis.org/A143824", "https://oeis.org/A227590" ], "comments_problem_id": 43, - "comments_count": 9 + "comments_count": 11 }, { "problem_id": 44, @@ -2200,8 +2238,8 @@ "problem_url": "/45", "status_bucket": "closed", "status_dom_id": "solved", - "status_label": "PROVED", - "status_detail": "This has been solved in the affirmative.", + "status_label": "PROVED (LEAN)", + "status_detail": "This has been solved in the affirmative and the proof verified in Lean.", "prize_amount": "", "statement": "Let $k\\geq 2$. Is there an integer $n_k$ such that, if $D=\\{ 10$ such that $$R(C_4,K_n) \\ll n^{2-c}.$$", "tags": [ "graph theory", "ramsey theory" ], - "last_edited": "", + "last_edited": "07 March 2026", "latex_path": "/latex/159", "formalized": false, "formalized_url": "", @@ -4741,7 +4792,7 @@ "formalized_url": "", "oeis_urls": [], "comments_problem_id": 162, - "comments_count": 3 + "comments_count": 4 }, { "problem_id": 163, @@ -4769,15 +4820,15 @@ "problem_url": "/164", "status_bucket": "closed", "status_dom_id": "solved", - "status_label": "PROVED", - "status_detail": "This has been solved in the affirmative.", + "status_label": "PROVED (LEAN)", + "status_detail": "This has been solved in the affirmative and the proof verified in Lean.", "prize_amount": "", "statement": "A set $A\\subset \\mathbb{N}$ is primitive if no member of $A$ divides another. Is the sum\\[\\sum_{n\\in A}\\frac{1}{n\\log n}\\]maximised over all primitive sets when $A$ is the set of primes?", "tags": [ "number theory", "primitive sets" ], - "last_edited": "23 January 2026", + "last_edited": "12 May 2026", "latex_path": "/latex/164", "formalized": false, "formalized_url": "", @@ -4785,7 +4836,7 @@ "https://oeis.org/A137245" ], "comments_problem_id": 164, - "comments_count": 0 + "comments_count": 16 }, { "problem_id": 165, @@ -4800,7 +4851,7 @@ "graph theory", "ramsey theory" ], - "last_edited": "28 December 2025", + "last_edited": "07 March 2026", "latex_path": "/latex/165", "formalized": false, "formalized_url": "", @@ -4865,7 +4916,7 @@ "tags": [ "additive combinatorics" ], - "last_edited": "24 October 2025", + "last_edited": "23 March 2026", "latex_path": "/latex/168", "formalized": true, "formalized_url": "https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/ErdosProblems/168.lean", @@ -4891,7 +4942,7 @@ "additive combinatorics", "arithmetic progressions" ], - "last_edited": "", + "last_edited": "04 April 2026", "latex_path": "/latex/169", "formalized": false, "formalized_url": "", @@ -4959,7 +5010,7 @@ "additive combinatorics", "ramsey theory" ], - "last_edited": "01 February 2026", + "last_edited": "06 April 2026", "latex_path": "/latex/172", "formalized": true, "formalized_url": "https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/ErdosProblems/172.lean", @@ -5044,13 +5095,13 @@ "arithmetic progressions", "discrepancy" ], - "last_edited": "28 December 2025", + "last_edited": "04 April 2026", "latex_path": "/latex/176", "formalized": false, "formalized_url": "", "oeis_urls": [], "comments_problem_id": 176, - "comments_count": 1 + "comments_count": 8 }, { "problem_id": 177, @@ -5078,8 +5129,8 @@ "problem_url": "/178", "status_bucket": "closed", "status_dom_id": "solved", - "status_label": "PROVED", - "status_detail": "This has been solved in the affirmative.", + "status_label": "PROVED (LEAN)", + "status_detail": "This has been solved in the affirmative and the proof verified in Lean.", "prize_amount": "", "statement": "Let $A_1,A_2,\\ldots$ be an infinite collection of infinite sets of integers, say $A_i=\\{a_{i1}1$ there is an associated $a_d$ such that every integer is congruent to some $a_d\\pmod{d}$, and if there is some integer $x$ with\\[x\\equiv a_d\\pmod{d}\\textrm{ and }x\\equiv a_{d'}\\pmod{d'}\\]then $(d,d')=1$.", "tags": [ @@ -5646,7 +5697,7 @@ "formalized_url": "https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/ErdosProblems/204.lean", "oeis_urls": [], "comments_problem_id": 204, - "comments_count": 0 + "comments_count": 1 }, { "problem_id": 205, @@ -5660,21 +5711,21 @@ "tags": [ "number theory" ], - "last_edited": "23 January 2026", + "last_edited": "05 April 2026", "latex_path": "/latex/205", "formalized": false, "formalized_url": "", "oeis_urls": [], "comments_problem_id": 205, - "comments_count": 21 + "comments_count": 22 }, { "problem_id": 206, "problem_url": "/206", "status_bucket": "closed", "status_dom_id": "solved", - "status_label": "DISPROVED", - "status_detail": "This has been solved in the negative.", + "status_label": "DISPROVED (LEAN)", + "status_detail": "This has been solved in the negative and the proof verified in Lean.", "prize_amount": "", "statement": "Let $x>0$ be a real number. For any $n\\geq 1$ let\\[R_n(x) = \\sum_{i=1}^n\\frac{1}{m_i}cn$ but there is no covering system whose moduli all divide $n$?", + "statement": "Is it true that, for every $c$, there exists an $n$ such that $\\sigma(n)>cn$ but there is no covering system whose moduli all distinct divisors of $n$ (which are $>1$)?", "tags": [ "number theory", "covering systems" ], - "last_edited": "06 October 2025", + "last_edited": "10 April 2026", "latex_path": "/latex/277", "formalized": true, "formalized_url": "https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/ErdosProblems/277.lean", @@ -7233,21 +7285,21 @@ "covering systems", "primes" ], - "last_edited": "20 January 2026", + "last_edited": "17 April 2026", "latex_path": "/latex/279", - "formalized": false, - "formalized_url": "", + "formalized": true, + "formalized_url": "https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/ErdosProblems/279.lean", "oeis_urls": [], "comments_problem_id": 279, - "comments_count": 0 + "comments_count": 10 }, { "problem_id": 280, "problem_url": "/280", "status_bucket": "closed", "status_dom_id": "solved", - "status_label": "DISPROVED", - "status_detail": "This has been solved in the negative.", + "status_label": "DISPROVED (LEAN)", + "status_detail": "This has been solved in the negative and the proof verified in Lean.", "prize_amount": "", "statement": "Let $n_10$ we have $n_k>(1+\\epsilon)k\\log k$ for all $k$. Then\\[\\#\\{ m0$ and $k\\geq 2$. Is it true that, for all sufficiently large $n$, there is a sequence of $k$ consecutive integers in $\\{1,\\ldots,n\\}$ all of which are $n^\\epsilon$-smooth?", "tags": [ "number theory" ], - "last_edited": "", + "last_edited": "28 April 2026", "latex_path": "/latex/369", "formalized": false, "formalized_url": "", "oeis_urls": [], "comments_problem_id": 369, - "comments_count": 0 + "comments_count": 11 }, { "problem_id": 370, @@ -9287,7 +9350,7 @@ "https://oeis.org/A389148" ], "comments_problem_id": 374, - "comments_count": 0 + "comments_count": 1 }, { "problem_id": 375, @@ -9400,16 +9463,16 @@ { "problem_id": 380, "problem_url": "/380", - "status_bucket": "open", - "status_dom_id": "open", - "status_label": "OPEN", - "status_detail": "This is open, and cannot be resolved with a finite computation.", + "status_bucket": "closed", + "status_dom_id": "solved", + "status_label": "PROVED", + "status_detail": "This has been solved in the affirmative.", "prize_amount": "", "statement": "We call an interval $[u,v]$ 'bad' if the greatest prime factor of $\\prod_{u\\leq m\\leq v}m$ occurs with an exponent greater than $1$. Let $B(x)$ count the number of $n\\leq x$ which are contained in at least one bad interval. Is it true that\\[B(x)\\sim \\#\\{ n\\leq x: P(n)^2\\mid n\\},\\]where $P(n)$ is the largest prime factor of $n$?", "tags": [ "number theory" ], - "last_edited": "", + "last_edited": "10 April 2026", "latex_path": "/latex/380", "formalized": false, "formalized_url": "", @@ -9420,7 +9483,7 @@ "https://oeis.org/A389100" ], "comments_problem_id": 380, - "comments_count": 5 + "comments_count": 10 }, { "problem_id": 381, @@ -9551,7 +9614,7 @@ "https://oeis.org/A280992" ], "comments_problem_id": 386, - "comments_count": 12 + "comments_count": 13 }, { "problem_id": 387, @@ -9572,7 +9635,7 @@ "formalized_url": "https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/ErdosProblems/387.lean", "oeis_urls": [], "comments_problem_id": 387, - "comments_count": 6 + "comments_count": 8 }, { "problem_id": 388, @@ -9586,13 +9649,13 @@ "tags": [ "number theory" ], - "last_edited": "", + "last_edited": "11 April 2026", "latex_path": "/latex/388", "formalized": false, "formalized_url": "", "oeis_urls": [], "comments_problem_id": 388, - "comments_count": 5 + "comments_count": 10 }, { "problem_id": 389, @@ -9614,7 +9677,7 @@ "https://oeis.org/A375071" ], "comments_problem_id": 389, - "comments_count": 6 + "comments_count": 8 }, { "problem_id": 390, @@ -9637,7 +9700,7 @@ "https://oeis.org/A193429" ], "comments_problem_id": 390, - "comments_count": 1 + "comments_count": 3 }, { "problem_id": 391, @@ -9705,7 +9768,7 @@ "https://oeis.org/A388302" ], "comments_problem_id": 393, - "comments_count": 6 + "comments_count": 8 }, { "problem_id": 394, @@ -9770,7 +9833,7 @@ "https://oeis.org/A375077" ], "comments_problem_id": 396, - "comments_count": 1 + "comments_count": 35 }, { "problem_id": 397, @@ -9806,15 +9869,16 @@ "number theory", "factorials" ], - "last_edited": "", + "last_edited": "01 April 2026", "latex_path": "/latex/398", "formalized": true, "formalized_url": "https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/ErdosProblems/398.lean", "oeis_urls": [ + "https://oeis.org/A141399", "https://oeis.org/A146968" ], "comments_problem_id": 398, - "comments_count": 6 + "comments_count": 10 }, { "problem_id": 399, @@ -9852,11 +9916,11 @@ ], "last_edited": "", "latex_path": "/latex/400", - "formalized": false, - "formalized_url": "", + "formalized": true, + "formalized_url": "https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/ErdosProblems/400.lean", "oeis_urls": [], "comments_problem_id": 400, - "comments_count": 0 + "comments_count": 1 }, { "problem_id": 401, @@ -9877,7 +9941,7 @@ "formalized_url": "", "oeis_urls": [], "comments_problem_id": 401, - "comments_count": 18 + "comments_count": 21 }, { "problem_id": 402, @@ -9891,7 +9955,7 @@ "tags": [ "number theory" ], - "last_edited": "", + "last_edited": "08 April 2026", "latex_path": "/latex/402", "formalized": true, "formalized_url": "https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/ErdosProblems/402.lean", @@ -9975,13 +10039,13 @@ "number theory", "base representations" ], - "last_edited": "30 September 2025", + "last_edited": "13 April 2026", "latex_path": "/latex/406", "formalized": true, "formalized_url": "https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/ErdosProblems/406.lean", "oeis_urls": [], "comments_problem_id": 406, - "comments_count": 4 + "comments_count": 5 }, { "problem_id": 407, @@ -10096,7 +10160,7 @@ "https://oeis.org/A383044" ], "comments_problem_id": 411, - "comments_count": 1 + "comments_count": 4 }, { "problem_id": 412, @@ -10135,7 +10199,7 @@ "number theory", "iterated functions" ], - "last_edited": "28 December 2025", + "last_edited": "17 April 2026", "latex_path": "/latex/413", "formalized": true, "formalized_url": "https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/ErdosProblems/413.lean", @@ -10143,7 +10207,7 @@ "https://oeis.org/A005236" ], "comments_problem_id": 413, - "comments_count": 2 + "comments_count": 3 }, { "problem_id": 414, @@ -10176,17 +10240,17 @@ "status_label": "OPEN", "status_detail": "This is open, and cannot be resolved with a finite computation.", "prize_amount": "", - "statement": "For any $n$ let $F(n)$ be the largest $k$ such that any of the $k!$ possible ordering patterns appears in some sequence of $\\phi(m+1),\\ldots,\\phi(m+k)$ with $m+k\\leq n$. Is it true that\\[F(n)=(c+o(1))\\log\\log\\log n\\]for some constant $c$? Is the first pattern which fails to appear always\\[\\phi(m+1)>\\phi(m+2)>\\cdots \\phi(m+k)?\\]Is it true that 'natural' ordering which mimics what happens to $\\phi(1),\\ldots,\\phi(k)$ is the most likely to appear?", + "statement": "For any $n$ let $F(n)$ be the largest $k$ such that any of the $k!$ possible ordering patterns appears in some sequence of $\\phi(m+1),\\ldots,\\phi(m+k)$ with $m+k\\leq n$. Is it true that\\[F(n)=(c+o(1))\\log\\log\\log n\\]for some constant $c$? Is the first pattern which fails to appear always\\[\\phi(m+1)>\\phi(m+2)>\\cdots >\\phi(m+k)?\\]Is it true that the 'natural' ordering which mimics what happens to $\\phi(1),\\ldots,\\phi(k)$ is the most likely to appear?", "tags": [ "number theory" ], - "last_edited": "", + "last_edited": "19 April 2026", "latex_path": "/latex/415", "formalized": false, "formalized_url": "", "oeis_urls": [], "comments_problem_id": 415, - "comments_count": 0 + "comments_count": 4 }, { "problem_id": 416, @@ -10231,7 +10295,7 @@ "https://oeis.org/A264810" ], "comments_problem_id": 417, - "comments_count": 3 + "comments_count": 5 }, { "problem_id": 418, @@ -10354,7 +10418,7 @@ "tags": [ "number theory" ], - "last_edited": "16 January 2026", + "last_edited": "23 March 2026", "latex_path": "/latex/423", "formalized": false, "formalized_url": "", @@ -10362,7 +10426,7 @@ "https://oeis.org/A005243" ], "comments_problem_id": 423, - "comments_count": 18 + "comments_count": 38 }, { "problem_id": 424, @@ -10376,7 +10440,7 @@ "tags": [ "number theory" ], - "last_edited": "", + "last_edited": "31 March 2026", "latex_path": "/latex/424", "formalized": true, "formalized_url": "https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/ErdosProblems/424.lean", @@ -10399,21 +10463,21 @@ "number theory", "sidon sets" ], - "last_edited": "28 December 2025", + "last_edited": "06 April 2026", "latex_path": "/latex/425", "formalized": false, "formalized_url": "", "oeis_urls": [], "comments_problem_id": 425, - "comments_count": 0 + "comments_count": 5 }, { "problem_id": 426, "problem_url": "/426", "status_bucket": "closed", "status_dom_id": "solved", - "status_label": "DISPROVED", - "status_detail": "This has been solved in the negative.", + "status_label": "DISPROVED (LEAN)", + "status_detail": "This has been solved in the negative and the proof verified in Lean.", "prize_amount": "$25", "statement": "We say $H$ is a unique subgraph of $G$ if there is exactly one way to find $H$ as a subgraph (not necessarily induced) of $G$. Is there a graph on $n$ vertices with\\[\\gg \\frac{2^{\\binom{n}{2}}}{n!}\\]many distinct unique subgraphs?", "tags": [ @@ -10425,15 +10489,15 @@ "formalized_url": "", "oeis_urls": [], "comments_problem_id": 426, - "comments_count": 0 + "comments_count": 2 }, { "problem_id": 427, "problem_url": "/427", "status_bucket": "closed", "status_dom_id": "solved", - "status_label": "PROVED", - "status_detail": "This has been solved in the affirmative.", + "status_label": "PROVED (LEAN)", + "status_detail": "This has been solved in the affirmative and the proof verified in Lean.", "prize_amount": "", "statement": "Is it true that, for every $n$ and $d$, there exists $k$ such that\\[d \\mid p_{n+1}+\\cdots+p_{n+k},\\]where $p_r$ denotes the $r$th prime?", "tags": [ @@ -10446,7 +10510,7 @@ "formalized_url": "https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/ErdosProblems/427.lean", "oeis_urls": [], "comments_problem_id": 427, - "comments_count": 1 + "comments_count": 2 }, { "problem_id": 428, @@ -10481,7 +10545,7 @@ "tags": [ "number theory" ], - "last_edited": "", + "last_edited": "08 April 2026", "latex_path": "/latex/429", "formalized": false, "formalized_url": "", @@ -10522,7 +10586,7 @@ "number theory", "primes" ], - "last_edited": "", + "last_edited": "08 April 2026", "latex_path": "/latex/431", "formalized": false, "formalized_url": "", @@ -10666,7 +10730,7 @@ "tags": [ "number theory" ], - "last_edited": "18 November 2025", + "last_edited": "07 April 2026", "latex_path": "/latex/438", "formalized": false, "formalized_url": "", @@ -10689,7 +10753,7 @@ "number theory", "ramsey theory" ], - "last_edited": "", + "last_edited": "07 April 2026", "latex_path": "/latex/439", "formalized": false, "formalized_url": "", @@ -10814,8 +10878,8 @@ ], "last_edited": "27 December 2025", "latex_path": "/latex/445", - "formalized": false, - "formalized_url": "", + "formalized": true, + "formalized_url": "https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/ErdosProblems/445.lean", "oeis_urls": [], "comments_problem_id": 445, "comments_count": 1 @@ -10946,7 +11010,7 @@ "https://oeis.org/A386620" ], "comments_problem_id": 451, - "comments_count": 2 + "comments_count": 4 }, { "problem_id": 452, @@ -10980,7 +11044,7 @@ "tags": [ "number theory" ], - "last_edited": "26 October 2025", + "last_edited": "08 April 2026", "latex_path": "/latex/453", "formalized": false, "formalized_url": "", @@ -11050,21 +11114,21 @@ "formalized_url": "", "oeis_urls": [], "comments_problem_id": 456, - "comments_count": 2 + "comments_count": 3 }, { "problem_id": 457, "problem_url": "/457", - "status_bucket": "open", - "status_dom_id": "open", - "status_label": "OPEN", - "status_detail": "This is open, and cannot be resolved with a finite computation.", + "status_bucket": "closed", + "status_dom_id": "solved", + "status_label": "PROVED (LEAN)", + "status_detail": "This has been solved in the affirmative and the proof verified in Lean.", "prize_amount": "", "statement": "Is there some $\\epsilon>0$ such that there are infinitely many $n$ where all primes $p\\leq (2+\\epsilon)\\log n$ divide\\[\\prod_{1\\leq i\\leq \\log n}(n+i)?\\]", "tags": [ "number theory" ], - "last_edited": "07 October 2025", + "last_edited": "07 March 2026", "latex_path": "/latex/457", "formalized": true, "formalized_url": "https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/ErdosProblems/457.lean", @@ -11072,7 +11136,7 @@ "https://oeis.org/A391668" ], "comments_problem_id": 457, - "comments_count": 7 + "comments_count": 10 }, { "problem_id": 458, @@ -11095,15 +11159,15 @@ "https://oeis.org/A056604" ], "comments_problem_id": 458, - "comments_count": 1 + "comments_count": 2 }, { "problem_id": 459, "problem_url": "/459", "status_bucket": "closed", "status_dom_id": "solved", - "status_label": "SOLVED", - "status_detail": "This has been resolved in some other way than a proof or disproof.", + "status_label": "SOLVED (LEAN)", + "status_detail": "This has been resolved in some other way than a proof or disproof, and that resolution verified in Lean.", "prize_amount": "", "statement": "Let $f(u)$ be the largest $v$ such that no $m\\in (u,v)$ is composed entirely of primes dividing $uv$. Estimate $f(u)$.", "tags": [ @@ -11118,7 +11182,7 @@ "https://oeis.org/A289280" ], "comments_problem_id": 459, - "comments_count": 2 + "comments_count": 3 }, { "problem_id": 460, @@ -11159,7 +11223,7 @@ "formalized_url": "", "oeis_urls": [], "comments_problem_id": 461, - "comments_count": 0 + "comments_count": 1 }, { "problem_id": 462, @@ -11263,7 +11327,7 @@ "formalized_url": "", "oeis_urls": [], "comments_problem_id": 466, - "comments_count": 0 + "comments_count": 1 }, { "problem_id": 467, @@ -11308,7 +11372,7 @@ "https://oeis.org/A387503" ], "comments_problem_id": 468, - "comments_count": 7 + "comments_count": 1 }, { "problem_id": 469, @@ -11456,7 +11520,7 @@ "number theory", "additive combinatorics" ], - "last_edited": "", + "last_edited": "05 March 2026", "latex_path": "/latex/475", "formalized": false, "formalized_url": "", @@ -11493,11 +11557,12 @@ "status_label": "OPEN", "status_detail": "This is open, and cannot be resolved with a finite computation.", "prize_amount": "", - "statement": "Is there a polynomial $f:\\mathbb{Z}\\to \\mathbb{Z}$ of degree at least $2$ and a set $A\\subset \\mathbb{Z}$ such that for any $n\\in \\mathbb{Z}$ there is exactly one $a\\in A$ and $b\\in \\{ f(n) : n\\in\\mathbb{Z}\\}$ such that $n=a+b$?", + "statement": "Is there a polynomial $f:\\mathbb{Z}\\to \\mathbb{Z}$ of degree at least $2$ and a set $A\\subset \\mathbb{Z}$ such that for any $n\\in \\mathbb{Z}$ there is exactly one $a\\in A$ and $b\\in \\{ f(k) : k\\in\\mathbb{Z}\\}$ such that $n=a+b$?", "tags": [ - "number theory" + "number theory", + "sidon sets" ], - "last_edited": "29 December 2025", + "last_edited": "11 April 2026", "latex_path": "/latex/477", "formalized": true, "formalized_url": "https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/ErdosProblems/477.lean", @@ -11518,7 +11583,7 @@ "number theory", "factorials" ], - "last_edited": "04 October 2025", + "last_edited": "12 April 2026", "latex_path": "/latex/478", "formalized": false, "formalized_url": "", @@ -11632,7 +11697,7 @@ "additive combinatorics", "ramsey theory" ], - "last_edited": "14 October 2025", + "last_edited": "10 April 2026", "latex_path": "/latex/483", "formalized": false, "formalized_url": "", @@ -11640,15 +11705,15 @@ "https://oeis.org/A030126" ], "comments_problem_id": 483, - "comments_count": 3 + "comments_count": 4 }, { "problem_id": 484, "problem_url": "/484", "status_bucket": "closed", "status_dom_id": "solved", - "status_label": "PROVED", - "status_detail": "This has been solved in the affirmative.", + "status_label": "PROVED (LEAN)", + "status_detail": "This has been solved in the affirmative and the proof verified in Lean.", "prize_amount": "", "statement": "Prove that there exists an absolute constant $c>0$ such that, whenever $\\{1,\\ldots,N\\}$ is $k$-coloured (and $N$ is large enough depending on $k$) then there are at least $cN$ many integers in $\\{1,\\ldots,N\\}$ which are representable as a monochromatic sum (that is, $a+b$ where $a,b\\in \\{1,\\ldots,N\\}$ are in the same colour class and $a\\neq b$).", "tags": [ @@ -11656,13 +11721,13 @@ "additive combinatorics", "ramsey theory" ], - "last_edited": "", + "last_edited": "08 April 2026", "latex_path": "/latex/484", "formalized": false, "formalized_url": "", "oeis_urls": [], "comments_problem_id": 484, - "comments_count": 0 + "comments_count": 1 }, { "problem_id": 485, @@ -11677,7 +11742,7 @@ "analysis", "polynomials" ], - "last_edited": "29 December 2025", + "last_edited": "08 April 2026", "latex_path": "/latex/485", "formalized": false, "formalized_url": "", @@ -11698,7 +11763,7 @@ "number theory", "primitive sets" ], - "last_edited": "12 January 2026", + "last_edited": "08 April 2026", "latex_path": "/latex/486", "formalized": true, "formalized_url": "https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/ErdosProblems/486.lean", @@ -11731,20 +11796,20 @@ "problem_url": "/488", "status_bucket": "open", "status_dom_id": "open", - "status_label": "OPEN", - "status_detail": "This is open, and cannot be resolved with a finite computation.", + "status_label": "FALSIFIABLE", + "status_detail": "Open, but could be disproved with a finite counterexample.", "prize_amount": "", "statement": "Let $A$ be a finite set and\\[B=\\{ n \\geq 1 : a\\mid n\\textrm{ for some }a\\in A\\}.\\]Is it true that, for every $m>n\\geq \\max(A)$,\\[\\frac{\\lvert B\\cap [1,m]\\rvert }{m}< 2\\frac{\\lvert B\\cap [1,n]\\rvert}{n}?\\]", "tags": [ "number theory" ], - "last_edited": "31 December 2025", + "last_edited": "08 April 2026", "latex_path": "/latex/488", "formalized": true, "formalized_url": "https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/ErdosProblems/488.lean", "oeis_urls": [], "comments_problem_id": 488, - "comments_count": 14 + "comments_count": 29 }, { "problem_id": 489, @@ -11764,7 +11829,7 @@ "formalized_url": "https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/ErdosProblems/489.lean", "oeis_urls": [], "comments_problem_id": 489, - "comments_count": 0 + "comments_count": 3 }, { "problem_id": 490, @@ -11784,7 +11849,7 @@ "formalized_url": "", "oeis_urls": [], "comments_problem_id": 490, - "comments_count": 0 + "comments_count": 3 }, { "problem_id": 491, @@ -11798,13 +11863,13 @@ "tags": [ "number theory" ], - "last_edited": "30 December 2025", + "last_edited": "01 April 2026", "latex_path": "/latex/491", "formalized": false, "formalized_url": "", "oeis_urls": [], "comments_problem_id": 491, - "comments_count": 1 + "comments_count": 0 }, { "problem_id": 492, @@ -11865,7 +11930,7 @@ "formalized_url": "https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/ErdosProblems/494.lean", "oeis_urls": [], "comments_problem_id": 494, - "comments_count": 2 + "comments_count": 3 }, { "problem_id": 495, @@ -12011,8 +12076,8 @@ ], "last_edited": "25 January 2026", "latex_path": "/latex/501", - "formalized": false, - "formalized_url": "", + "formalized": true, + "formalized_url": "https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/ErdosProblems/501.lean", "oeis_urls": [], "comments_problem_id": 501, "comments_count": 8 @@ -12061,7 +12126,7 @@ "https://oeis.org/A175769" ], "comments_problem_id": 503, - "comments_count": 1 + "comments_count": 6 }, { "problem_id": 504, @@ -12097,8 +12162,8 @@ ], "last_edited": "30 December 2025", "latex_path": "/latex/505", - "formalized": false, - "formalized_url": "", + "formalized": true, + "formalized_url": "https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/ErdosProblems/505.lean", "oeis_urls": [], "comments_problem_id": 505, "comments_count": 1 @@ -12257,13 +12322,13 @@ "tags": [ "analysis" ], - "last_edited": "28 December 2025", + "last_edited": "02 April 2026", "latex_path": "/latex/513", "formalized": true, "formalized_url": "https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/ErdosProblems/513.lean", "oeis_urls": [], "comments_problem_id": 513, - "comments_count": 5 + "comments_count": 6 }, { "problem_id": 514, @@ -12283,7 +12348,7 @@ "formalized_url": "", "oeis_urls": [], "comments_problem_id": 514, - "comments_count": 1 + "comments_count": 12 }, { "problem_id": 515, @@ -12343,7 +12408,7 @@ "formalized_url": "https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/ErdosProblems/517.lean", "oeis_urls": [], "comments_problem_id": 517, - "comments_count": 0 + "comments_count": 1 }, { "problem_id": 518, @@ -12371,8 +12436,8 @@ "problem_url": "/519", "status_bucket": "closed", "status_dom_id": "solved", - "status_label": "PROVED", - "status_detail": "This has been solved in the affirmative.", + "status_label": "PROVED (LEAN)", + "status_detail": "This has been solved in the affirmative and the proof verified in Lean.", "prize_amount": "", "statement": "Let $z_1,\\ldots,z_n\\in \\mathbb{C}$ with $z_1=1$. Must there exist an absolute constant $c>0$ such that\\[\\max_{1\\leq k\\leq n}\\left\\lvert \\sum_{i}z_i^k\\right\\rvert>c?\\]", "tags": [ @@ -12384,7 +12449,7 @@ "formalized_url": "", "oeis_urls": [], "comments_problem_id": 519, - "comments_count": 2 + "comments_count": 3 }, { "problem_id": 520, @@ -12427,7 +12492,7 @@ "formalized_url": "", "oeis_urls": [], "comments_problem_id": 521, - "comments_count": 13 + "comments_count": 20 }, { "problem_id": 522, @@ -12449,7 +12514,7 @@ "formalized_url": "https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/ErdosProblems/522.lean", "oeis_urls": [], "comments_problem_id": 522, - "comments_count": 2 + "comments_count": 24 }, { "problem_id": 523, @@ -12493,7 +12558,7 @@ "formalized_url": "", "oeis_urls": [], "comments_problem_id": 524, - "comments_count": 10 + "comments_count": 11 }, { "problem_id": 525, @@ -12616,7 +12681,7 @@ "number theory", "sidon sets" ], - "last_edited": "16 October 2025", + "last_edited": "08 April 2026", "latex_path": "/latex/530", "formalized": false, "formalized_url": "", @@ -12701,7 +12766,7 @@ "number theory", "intersecting family" ], - "last_edited": "", + "last_edited": "08 April 2026", "latex_path": "/latex/534", "formalized": false, "formalized_url": "", @@ -12724,13 +12789,13 @@ "tags": [ "number theory" ], - "last_edited": "", + "last_edited": "29 April 2026", "latex_path": "/latex/535", - "formalized": false, - "formalized_url": "", + "formalized": true, + "formalized_url": "https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/ErdosProblems/535.lean", "oeis_urls": [], "comments_problem_id": 535, - "comments_count": 1 + "comments_count": 9 }, { "problem_id": 536, @@ -12740,17 +12805,17 @@ "status_label": "OPEN", "status_detail": "This is open, and cannot be resolved with a finite computation.", "prize_amount": "", - "statement": "Let $\\epsilon>0$ and $N$ be sufficiently large. Is it true that if $A\\subseteq \\{1,\\ldots,N\\}$ has size at least $\\epsilon N$ then there must be distinct $a,b,c\\in A$ such that\\[[a,b]=[b,c]=[a,c],\\]where $[a,b]$ denotes the least common multiple?", + "statement": "Let $f(N)$ be the largest size of $A\\subseteq \\{1,\\ldots,N\\}$ with the property that there are no distinct $a,b,c\\in A$ such that\\[[a,b]=[b,c]=[a,c],\\]where $[a,b]$ denotes the least common multiple. Estimate $f(N)$ - in particular, is it true that $f(N)=o(N)$?", "tags": [ "number theory" ], - "last_edited": "12 January 2026", + "last_edited": "29 April 2026", "latex_path": "/latex/536", "formalized": true, "formalized_url": "https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/ErdosProblems/536.lean", "oeis_urls": [], "comments_problem_id": 536, - "comments_count": 3 + "comments_count": 6 }, { "problem_id": 537, @@ -12806,25 +12871,25 @@ ], "last_edited": "22 January 2026", "latex_path": "/latex/539", - "formalized": false, - "formalized_url": "", + "formalized": true, + "formalized_url": "https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/ErdosProblems/539.lean", "oeis_urls": [], "comments_problem_id": 539, - "comments_count": 1 + "comments_count": 2 }, { "problem_id": 540, "problem_url": "/540", "status_bucket": "closed", "status_dom_id": "solved", - "status_label": "PROVED", - "status_detail": "This has been solved in the affirmative.", + "status_label": "PROVED (LEAN)", + "status_detail": "This has been solved in the affirmative and the proof verified in Lean.", "prize_amount": "", "statement": "Is it true that if $A\\subseteq \\mathbb{Z}/N\\mathbb{Z}$ has size $\\gg N^{1/2}$ then there exists some non-empty $S\\subseteq A$ such that $\\sum_{n\\in S}n\\equiv 0\\pmod{N}$?", "tags": [ "number theory" ], - "last_edited": "28 December 2025", + "last_edited": "06 March 2026", "latex_path": "/latex/540", "formalized": false, "formalized_url": "", @@ -12832,7 +12897,7 @@ "https://oeis.org/A034463" ], "comments_problem_id": 540, - "comments_count": 1 + "comments_count": 2 }, { "problem_id": 541, @@ -12846,13 +12911,13 @@ "tags": [ "number theory" ], - "last_edited": "", + "last_edited": "08 April 2026", "latex_path": "/latex/541", "formalized": true, "formalized_url": "https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/ErdosProblems/541.lean", "oeis_urls": [], "comments_problem_id": 541, - "comments_count": 4 + "comments_count": 5 }, { "problem_id": 542, @@ -12866,7 +12931,7 @@ "tags": [ "number theory" ], - "last_edited": "17 October 2025", + "last_edited": "08 April 2026", "latex_path": "/latex/542", "formalized": false, "formalized_url": "", @@ -12908,7 +12973,7 @@ "graph theory", "ramsey theory" ], - "last_edited": "", + "last_edited": "24 April 2026", "latex_path": "/latex/544", "formalized": false, "formalized_url": "", @@ -12916,7 +12981,7 @@ "https://oeis.org/A000791" ], "comments_problem_id": 544, - "comments_count": 0 + "comments_count": 2 }, { "problem_id": 545, @@ -12981,7 +13046,7 @@ "formalized_url": "", "oeis_urls": [], "comments_problem_id": 547, - "comments_count": 2 + "comments_count": 3 }, { "problem_id": 548, @@ -12990,12 +13055,12 @@ "status_dom_id": "open", "status_label": "FALSIFIABLE", "status_detail": "Open, but could be disproved with a finite counterexample.", - "prize_amount": "", + "prize_amount": "$100", "statement": "Let $n\\geq k+1$. Every graph on $n$ vertices with at least $\\frac{k-1}{2}n+1$ edges contains every tree on $k+1$ vertices.", "tags": [ "graph theory" ], - "last_edited": "23 January 2026", + "last_edited": "07 March 2026", "latex_path": "/latex/548", "formalized": false, "formalized_url": "", @@ -13150,9 +13215,11 @@ "latex_path": "/latex/555", "formalized": false, "formalized_url": "", - "oeis_urls": [], + "oeis_urls": [ + "https://oeis.org/A389313" + ], "comments_problem_id": 555, - "comments_count": 1 + "comments_count": 0 }, { "problem_id": 556, @@ -13451,7 +13518,7 @@ "formalized_url": "", "oeis_urls": [], "comments_problem_id": 569, - "comments_count": 0 + "comments_count": 2 }, { "problem_id": 570, @@ -13487,7 +13554,7 @@ "graph theory", "turan number" ], - "last_edited": "18 January 2026", + "last_edited": "07 March 2026", "latex_path": "/latex/571", "formalized": false, "formalized_url": "", @@ -13515,7 +13582,7 @@ "formalized_url": "", "oeis_urls": [], "comments_problem_id": 572, - "comments_count": 1 + "comments_count": 2 }, { "problem_id": 573, @@ -13543,23 +13610,23 @@ { "problem_id": 574, "problem_url": "/574", - "status_bucket": "open", - "status_dom_id": "open", - "status_label": "OPEN", - "status_detail": "This is open, and cannot be resolved with a finite computation.", + "status_bucket": "closed", + "status_dom_id": "solved", + "status_label": "DISPROVED", + "status_detail": "This has been solved in the negative.", "prize_amount": "", "statement": "Is it true that, for $k\\geq 2$,\\[\\mathrm{ex}(n;\\{C_{2k-1},C_{2k}\\})=(1+o(1))(n/2)^{1+\\frac{1}{k}}.\\]", "tags": [ "graph theory", "turan number" ], - "last_edited": "18 January 2026", + "last_edited": "01 April 2026", "latex_path": "/latex/574", "formalized": false, "formalized_url": "", "oeis_urls": [], "comments_problem_id": 574, - "comments_count": 0 + "comments_count": 1 }, { "problem_id": 575, @@ -13737,13 +13804,13 @@ "tags": [ "graph theory" ], - "last_edited": "05 March 2026", + "last_edited": "01 April 2026", "latex_path": "/latex/583", "formalized": false, "formalized_url": "", "oeis_urls": [], "comments_problem_id": 583, - "comments_count": 1 + "comments_count": 3 }, { "problem_id": 584, @@ -13764,7 +13831,7 @@ "formalized_url": "", "oeis_urls": [], "comments_problem_id": 584, - "comments_count": 1 + "comments_count": 3 }, { "problem_id": 585, @@ -13846,7 +13913,10 @@ "latex_path": "/latex/588", "formalized": false, "formalized_url": "", - "oeis_urls": [], + "oeis_urls": [ + "https://oeis.org/A006065", + "https://oeis.org/A008997" + ], "comments_problem_id": 588, "comments_count": 0 }, @@ -13950,8 +14020,8 @@ ], "last_edited": "", "latex_path": "/latex/593", - "formalized": false, - "formalized_url": "", + "formalized": true, + "formalized_url": "https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/ErdosProblems/593.lean", "oeis_urls": [], "comments_problem_id": 593, "comments_count": 0 @@ -13992,8 +14062,8 @@ ], "last_edited": "", "latex_path": "/latex/595", - "formalized": false, - "formalized_url": "", + "formalized": true, + "formalized_url": "https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/ErdosProblems/595.lean", "oeis_urls": [], "comments_problem_id": 595, "comments_count": 0 @@ -14018,7 +14088,7 @@ "formalized_url": "", "oeis_urls": [], "comments_problem_id": 596, - "comments_count": 0 + "comments_count": 1 }, { "problem_id": 597, @@ -14061,7 +14131,7 @@ "formalized_url": "https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/ErdosProblems/598.lean", "oeis_urls": [], "comments_problem_id": 598, - "comments_count": 0 + "comments_count": 6 }, { "problem_id": 599, @@ -14140,11 +14210,11 @@ ], "last_edited": "", "latex_path": "/latex/602", - "formalized": false, - "formalized_url": "", + "formalized": true, + "formalized_url": "https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/ErdosProblems/602.lean", "oeis_urls": [], "comments_problem_id": 602, - "comments_count": 0 + "comments_count": 1 }, { "problem_id": 603, @@ -14165,7 +14235,7 @@ "formalized_url": "", "oeis_urls": [], "comments_problem_id": 603, - "comments_count": 2 + "comments_count": 12 }, { "problem_id": 604, @@ -14180,7 +14250,7 @@ "geometry", "distances" ], - "last_edited": "15 October 2025", + "last_edited": "23 March 2026", "latex_path": "/latex/604", "formalized": false, "formalized_url": "", @@ -14308,7 +14378,7 @@ "formalized_url": "", "oeis_urls": [], "comments_problem_id": 610, - "comments_count": 0 + "comments_count": 4 }, { "problem_id": 611, @@ -14365,8 +14435,8 @@ ], "last_edited": "01 December 2025", "latex_path": "/latex/613", - "formalized": false, - "formalized_url": "", + "formalized": true, + "formalized_url": "https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/ErdosProblems/613.lean", "oeis_urls": [], "comments_problem_id": 613, "comments_count": 4 @@ -14444,13 +14514,13 @@ "tags": [ "graph theory" ], - "last_edited": "", + "last_edited": "01 April 2026", "latex_path": "/latex/617", "formalized": true, "formalized_url": "https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/ErdosProblems/617.lean", "oeis_urls": [], "comments_problem_id": 617, - "comments_count": 0 + "comments_count": 1 }, { "problem_id": 618, @@ -14490,7 +14560,7 @@ "formalized_url": "", "oeis_urls": [], "comments_problem_id": 619, - "comments_count": 0 + "comments_count": 1 }, { "problem_id": 620, @@ -14517,8 +14587,8 @@ "problem_url": "/621", "status_bucket": "closed", "status_dom_id": "solved", - "status_label": "PROVED", - "status_detail": "This has been solved in the affirmative.", + "status_label": "PROVED (LEAN)", + "status_detail": "This has been solved in the affirmative and the proof verified in Lean.", "prize_amount": "", "statement": "Let $G$ be a graph on $n$ vertices, $\\alpha_1(G)$ be the maximum number of edges that contain at most one edge from every triangle, and $\\tau_1(G)$ be the minimum number of edges that contain at least one edge from every triangle. Is it true that\\[\\alpha_1(G)+\\tau_1(G) \\leq \\frac{n^2}{4}?\\]", "tags": [ @@ -14530,7 +14600,7 @@ "formalized_url": "", "oeis_urls": [], "comments_problem_id": 621, - "comments_count": 1 + "comments_count": 2 }, { "problem_id": 622, @@ -14570,7 +14640,7 @@ "formalized_url": "https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/ErdosProblems/623.lean", "oeis_urls": [], "comments_problem_id": 623, - "comments_count": 0 + "comments_count": 1 }, { "problem_id": 624, @@ -14696,7 +14766,7 @@ "formalized_url": "", "oeis_urls": [], "comments_problem_id": 629, - "comments_count": 1 + "comments_count": 3 }, { "problem_id": 630, @@ -14764,22 +14834,22 @@ { "problem_id": 633, "problem_url": "/633", - "status_bucket": "open", - "status_dom_id": "open", - "status_label": "OPEN", - "status_detail": "This is open, and cannot be resolved with a finite computation.", + "status_bucket": "closed", + "status_dom_id": "solved", + "status_label": "SOLVED", + "status_detail": "This has been resolved in some other way than a proof or disproof.", "prize_amount": "$25", "statement": "Classify those triangles which can only be cut into a square number of congruent triangles.", "tags": [ "geometry" ], - "last_edited": "", + "last_edited": "08 April 2026", "latex_path": "/latex/633", - "formalized": false, - "formalized_url": "", + "formalized": true, + "formalized_url": "https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/ErdosProblems/633.lean", "oeis_urls": [], "comments_problem_id": 633, - "comments_count": 2 + "comments_count": 7 }, { "problem_id": 634, @@ -14876,21 +14946,21 @@ "graph theory", "ramsey theory" ], - "last_edited": "", + "last_edited": "10 April 2026", "latex_path": "/latex/638", "formalized": false, "formalized_url": "", "oeis_urls": [], "comments_problem_id": 638, - "comments_count": 1 + "comments_count": 8 }, { "problem_id": 639, "problem_url": "/639", "status_bucket": "closed", "status_dom_id": "solved", - "status_label": "PROVED", - "status_detail": "This has been solved in the affirmative.", + "status_label": "PROVED (LEAN)", + "status_detail": "This has been solved in the affirmative and the proof verified in Lean.", "prize_amount": "", "statement": "Is it true that if the edges of $K_n$ are 2-coloured then there are at most $n^2/4$ many edges which do not occur in a monochromatic triangle?", "tags": [ @@ -14903,7 +14973,7 @@ "formalized_url": "", "oeis_urls": [], "comments_problem_id": 639, - "comments_count": 0 + "comments_count": 1 }, { "problem_id": 640, @@ -14965,7 +15035,7 @@ "formalized_url": "", "oeis_urls": [], "comments_problem_id": 642, - "comments_count": 2 + "comments_count": 3 }, { "problem_id": 643, @@ -15020,9 +15090,10 @@ "tags": [ "number theory", "additive combinatorics", - "ramsey theory" + "ramsey theory", + "arithmetic progressions" ], - "last_edited": "24 October 2025", + "last_edited": "04 April 2026", "latex_path": "/latex/645", "formalized": true, "formalized_url": "https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/ErdosProblems/645.lean", @@ -15063,7 +15134,7 @@ "tags": [ "number theory" ], - "last_edited": "05 October 2025", + "last_edited": "07 April 2026", "latex_path": "/latex/647", "formalized": true, "formalized_url": "https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/ErdosProblems/647.lean", @@ -15094,7 +15165,7 @@ "https://oeis.org/A391750" ], "comments_problem_id": 648, - "comments_count": 2 + "comments_count": 3 }, { "problem_id": 649, @@ -15119,22 +15190,24 @@ { "problem_id": 650, "problem_url": "/650", - "status_bucket": "open", - "status_dom_id": "open", - "status_label": "OPEN", - "status_detail": "This is open, and cannot be resolved with a finite computation.", + "status_bucket": "closed", + "status_dom_id": "solved", + "status_label": "SOLVED (LEAN)", + "status_detail": "This has been resolved in some other way than a proof or disproof, and that resolution verified in Lean.", "prize_amount": "", - "statement": "Let $f(m)$ be such that if $A\\subseteq \\{1,\\ldots,N\\}$ has $\\lvert A\\rvert=m$ then every interval in $[1,\\infty)$ of length $2N$ contains $\\geq f(m)$ many distinct integers $b_1,\\ldots,b_r$ where each $b_i$ is divisible by some $a_i\\in A$, where $a_1,\\ldots,a_r$ are distinct. Estimate $f(m)$. In particular is it true that $f(m)\\ll m^{1/2}$?", + "statement": "Let $f(m)$ be such that if $A\\subseteq \\{1,\\ldots,N\\}$ has $\\lvert A\\rvert=m$ then every interval in $[1,\\infty)$ of length $2N$ contains $\\geq f(m)$ many distinct integers $b_1,\\ldots,b_r$ where each $b_i$ is divisible by some $a_i\\in A$, where $a_1,\\ldots,a_r$ are distinct. Estimate $f(m)$. In particular is it true that $f(m)\\leq \\sqrt{m}$?", "tags": [ "number theory" ], - "last_edited": "", + "last_edited": "02 April 2026", "latex_path": "/latex/650", "formalized": false, "formalized_url": "", - "oeis_urls": [], + "oeis_urls": [ + "https://oeis.org/A027434" + ], "comments_problem_id": 650, - "comments_count": 0 + "comments_count": 28 }, { "problem_id": 651, @@ -15193,8 +15266,8 @@ ], "last_edited": "", "latex_path": "/latex/653", - "formalized": false, - "formalized_url": "", + "formalized": true, + "formalized_url": "https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/ErdosProblems/653.lean", "oeis_urls": [], "comments_problem_id": 653, "comments_count": 0 @@ -15235,11 +15308,11 @@ ], "last_edited": "", "latex_path": "/latex/655", - "formalized": false, - "formalized_url": "", + "formalized": true, + "formalized_url": "https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/ErdosProblems/655.lean", "oeis_urls": [], "comments_problem_id": 655, - "comments_count": 4 + "comments_count": 5 }, { "problem_id": 656, @@ -15249,12 +15322,12 @@ "status_label": "PROVED", "status_detail": "This has been solved in the affirmative.", "prize_amount": "", - "statement": "Let $A\\subset \\mathbb{N}$ be a set with positive upper density. Must there exist an infinite set $B$ and integer $t$ such that\\[B+B+t\\subseteq A?\\]", + "statement": "Let $A\\subset \\mathbb{N}$ be a set with positive upper density. Must there exist an infinite set $B\\subseteq A$ and integer $t$ such that\\[\\{b_1+b_2: b_1\\neq b_2\\in B\\}+t\\subseteq A?\\]", "tags": [ "number theory", "additive combinatorics" ], - "last_edited": "", + "last_edited": "07 April 2026", "latex_path": "/latex/656", "formalized": false, "formalized_url": "", @@ -15288,8 +15361,8 @@ "problem_url": "/658", "status_bucket": "closed", "status_dom_id": "solved", - "status_label": "PROVED", - "status_detail": "This has been solved in the affirmative.", + "status_label": "PROVED (LEAN)", + "status_detail": "This has been solved in the affirmative and the proof verified in Lean.", "prize_amount": "", "statement": "Let $\\delta>0$ and $N$ be sufficiently large depending on $\\delta$. Is it true that if $A\\subseteq \\{1,\\ldots,N\\}^2$ has $\\lvert A\\rvert \\geq \\delta N^2$ then $A$ must contain the vertices of a square?", "tags": [ @@ -15301,7 +15374,7 @@ "formalized_url": "", "oeis_urls": [], "comments_problem_id": 658, - "comments_count": 0 + "comments_count": 2 }, { "problem_id": 659, @@ -15386,7 +15459,7 @@ "formalized_url": "", "oeis_urls": [], "comments_problem_id": 662, - "comments_count": 2 + "comments_count": 4 }, { "problem_id": 663, @@ -15489,7 +15562,7 @@ "formalized_url": "", "oeis_urls": [], "comments_problem_id": 667, - "comments_count": 0 + "comments_count": 1 }, { "problem_id": 668, @@ -15508,7 +15581,9 @@ "latex_path": "/latex/668", "formalized": false, "formalized_url": "", - "oeis_urls": [], + "oeis_urls": [ + "https://oeis.org/A385657" + ], "comments_problem_id": 668, "comments_count": 3 }, @@ -15549,13 +15624,13 @@ "geometry", "distances" ], - "last_edited": "", + "last_edited": "17 April 2026", "latex_path": "/latex/670", "formalized": false, "formalized_url": "", "oeis_urls": [], "comments_problem_id": 670, - "comments_count": 0 + "comments_count": 4 }, { "problem_id": 671, @@ -15656,7 +15731,7 @@ "formalized_url": "", "oeis_urls": [], "comments_problem_id": 675, - "comments_count": 0 + "comments_count": 7 }, { "problem_id": 676, @@ -15670,7 +15745,7 @@ "tags": [ "number theory" ], - "last_edited": "", + "last_edited": "07 April 2026", "latex_path": "/latex/676", "formalized": false, "formalized_url": "", @@ -15678,7 +15753,7 @@ "https://oeis.org/A390181" ], "comments_problem_id": 676, - "comments_count": 10 + "comments_count": 9 }, { "problem_id": 677, @@ -15698,7 +15773,7 @@ "formalized_url": "https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/ErdosProblems/677.lean", "oeis_urls": [], "comments_problem_id": 677, - "comments_count": 0 + "comments_count": 8 }, { "problem_id": 678, @@ -15732,7 +15807,7 @@ "tags": [ "number theory" ], - "last_edited": "12 January 2026", + "last_edited": "17 April 2026", "latex_path": "/latex/679", "formalized": false, "formalized_url": "", @@ -15823,8 +15898,8 @@ ], "last_edited": "31 December 2025", "latex_path": "/latex/683", - "formalized": false, - "formalized_url": "", + "formalized": true, + "formalized_url": "https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/ErdosProblems/683.lean", "oeis_urls": [ "https://oeis.org/A006530", "https://oeis.org/A074399", @@ -15847,7 +15922,7 @@ "primes", "binomial coefficients" ], - "last_edited": "23 January 2026", + "last_edited": "01 April 2026", "latex_path": "/latex/684", "formalized": false, "formalized_url": "", @@ -15855,7 +15930,7 @@ "https://oeis.org/A392019" ], "comments_problem_id": 684, - "comments_count": 23 + "comments_count": 27 }, { "problem_id": 685, @@ -15877,7 +15952,7 @@ "formalized_url": "", "oeis_urls": [], "comments_problem_id": 685, - "comments_count": 1 + "comments_count": 4 }, { "problem_id": 686, @@ -15891,13 +15966,13 @@ "tags": [ "number theory" ], - "last_edited": "", + "last_edited": "11 April 2026", "latex_path": "/latex/686", "formalized": true, "formalized_url": "https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/ErdosProblems/686.lean", "oeis_urls": [], "comments_problem_id": 686, - "comments_count": 16 + "comments_count": 36 }, { "problem_id": 687, @@ -15934,10 +16009,10 @@ "tags": [ "number theory" ], - "last_edited": "", + "last_edited": "07 April 2026", "latex_path": "/latex/688", - "formalized": false, - "formalized_url": "", + "formalized": true, + "formalized_url": "https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/ErdosProblems/688.lean", "oeis_urls": [], "comments_problem_id": 688, "comments_count": 0 @@ -15954,33 +16029,33 @@ "tags": [ "number theory" ], - "last_edited": "06 December 2025", + "last_edited": "08 April 2026", "latex_path": "/latex/689", "formalized": true, "formalized_url": "https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/ErdosProblems/689.lean", "oeis_urls": [], "comments_problem_id": 689, - "comments_count": 17 + "comments_count": 24 }, { "problem_id": 690, "problem_url": "/690", - "status_bucket": "open", - "status_dom_id": "open", - "status_label": "OPEN", - "status_detail": "This is open, and cannot be resolved with a finite computation.", + "status_bucket": "closed", + "status_dom_id": "solved", + "status_label": "SOLVED", + "status_detail": "This has been resolved in some other way than a proof or disproof.", "prize_amount": "", "statement": "Let $d_k(p)$ be the density of those integers whose $k$th smallest prime factor is $p$ (i.e. if $p_1n+1$ (i.e. increases until some $m$ then decreases thereafter)? For fixed $n$, where does $\\delta_1(n,m)$ achieve its maximum?", "tags": [ @@ -16021,7 +16096,7 @@ "formalized_url": "", "oeis_urls": [], "comments_problem_id": 692, - "comments_count": 0 + "comments_count": 1 }, { "problem_id": 693, @@ -16049,22 +16124,22 @@ { "problem_id": 694, "problem_url": "/694", - "status_bucket": "open", - "status_dom_id": "open", - "status_label": "OPEN", - "status_detail": "This is open, and cannot be resolved with a finite computation.", + "status_bucket": "closed", + "status_dom_id": "solved", + "status_label": "SOLVED (LEAN)", + "status_detail": "This has been resolved in some other way than a proof or disproof, and that resolution verified in Lean.", "prize_amount": "", - "statement": "Let $f_{\\max}(n)$ be the largest $m$ such that $\\phi(m)=n$, and $f_{\\min}(n)$ be the smallest such $m$, where $\\phi$ is Euler's totient function. Investigate\\[\\max_{n\\leq x}\\frac{f_{\\max}(n)}{f_{\\min}(n)}.\\]", + "statement": "Let $f_{\\max}(n)$ be the largest $m$ such that $\\phi(m)=n$, and $f_{\\min}(n)$ be the smallest such $m$, where $\\phi$ is Euler's totient function. Investigate\\[\\max_{n\\leq x}\\frac{f_{\\max}(n)}{f_{\\min}(n)}\\](where the maximum is restricted to those $n$ of the form $n=\\phi(m)$ for some $m$.)", "tags": [ "number theory" ], - "last_edited": "", + "last_edited": "02 May 2026", "latex_path": "/latex/694", "formalized": true, "formalized_url": "https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/ErdosProblems/694.lean", "oeis_urls": [], "comments_problem_id": 694, - "comments_count": 2 + "comments_count": 13 }, { "problem_id": 695, @@ -16091,23 +16166,23 @@ { "problem_id": 696, "problem_url": "/696", - "status_bucket": "open", - "status_dom_id": "open", - "status_label": "OPEN", - "status_detail": "This is open, and cannot be resolved with a finite computation.", + "status_bucket": "closed", + "status_dom_id": "solved", + "status_label": "SOLVED (LEAN)", + "status_detail": "This has been resolved in some other way than a proof or disproof, and that resolution verified in Lean.", "prize_amount": "", "statement": "Let $h(n)$ be the largest $\\ell$ such that there is a sequence of primes $p_1<\\cdots < p_\\ell$ all dividing $n$ with $p_{i+1}\\equiv 1\\pmod{p_i}$. Let $H(n)$ be the largest $u$ such that there is a sequence of integers $d_1<\\cdots < d_u$ all dividing $n$ with $d_{i+1}\\equiv 1\\pmod{d_i}$. Estimate $h(n)$ and $H(n)$. Is it true that $H(n)/h(n)\\to \\infty$ for almost all $n$?", "tags": [ "number theory", "divisors" ], - "last_edited": "15 October 2025", + "last_edited": "11 May 2026", "latex_path": "/latex/696", "formalized": false, "formalized_url": "", "oeis_urls": [], "comments_problem_id": 696, - "comments_count": 1 + "comments_count": 11 }, { "problem_id": 697, @@ -16170,7 +16245,7 @@ "formalized_url": "https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/ErdosProblems/699.lean", "oeis_urls": [], "comments_problem_id": 699, - "comments_count": 3 + "comments_count": 6 }, { "problem_id": 700, @@ -16210,11 +16285,11 @@ ], "last_edited": "", "latex_path": "/latex/701", - "formalized": false, - "formalized_url": "", + "formalized": true, + "formalized_url": "https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/ErdosProblems/701.lean", "oeis_urls": [], "comments_problem_id": 701, - "comments_count": 0 + "comments_count": 6 }, { "problem_id": 702, @@ -16272,13 +16347,13 @@ "geometry", "chromatic number" ], - "last_edited": "", + "last_edited": "10 April 2026", "latex_path": "/latex/704", "formalized": false, "formalized_url": "", "oeis_urls": [], "comments_problem_id": 704, - "comments_count": 1 + "comments_count": 4 }, { "problem_id": 705, @@ -16320,7 +16395,7 @@ "formalized_url": "", "oeis_urls": [], "comments_problem_id": 706, - "comments_count": 0 + "comments_count": 1 }, { "problem_id": 707, @@ -16361,7 +16436,7 @@ "formalized_url": "", "oeis_urls": [], "comments_problem_id": 708, - "comments_count": 0 + "comments_count": 1 }, { "problem_id": 709, @@ -16375,13 +16450,13 @@ "tags": [ "number theory" ], - "last_edited": "", + "last_edited": "23 March 2026", "latex_path": "/latex/709", "formalized": false, "formalized_url": "", "oeis_urls": [], "comments_problem_id": 709, - "comments_count": 0 + "comments_count": 2 }, { "problem_id": 710, @@ -16460,7 +16535,7 @@ "graph theory", "turan number" ], - "last_edited": "06 October 2025", + "last_edited": "07 March 2026", "latex_path": "/latex/713", "formalized": false, "formalized_url": "", @@ -16604,7 +16679,7 @@ "graph theory", "ramsey theory" ], - "last_edited": "26 October 2025", + "last_edited": "07 March 2026", "latex_path": "/latex/720", "formalized": false, "formalized_url": "", @@ -16626,7 +16701,7 @@ "additive combinatorics", "ramsey theory" ], - "last_edited": "", + "last_edited": "04 April 2026", "latex_path": "/latex/721", "formalized": false, "formalized_url": "", @@ -16654,7 +16729,7 @@ "formalized_url": "", "oeis_urls": [], "comments_problem_id": 722, - "comments_count": 0 + "comments_count": 1 }, { "problem_id": 723, @@ -16718,7 +16793,7 @@ "https://oeis.org/A001009" ], "comments_problem_id": 725, - "comments_count": 0 + "comments_count": 1 }, { "problem_id": 726, @@ -16784,7 +16859,7 @@ "formalized_url": "https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/ErdosProblems/728.lean", "oeis_urls": [], "comments_problem_id": 728, - "comments_count": 84 + "comments_count": 86 }, { "problem_id": 729, @@ -16852,7 +16927,7 @@ "https://oeis.org/A006197" ], "comments_problem_id": 731, - "comments_count": 0 + "comments_count": 1 }, { "problem_id": 732, @@ -16913,7 +16988,7 @@ "formalized_url": "", "oeis_urls": [], "comments_problem_id": 734, - "comments_count": 2 + "comments_count": 3 }, { "problem_id": 735, @@ -16996,7 +17071,7 @@ "formalized_url": "", "oeis_urls": [], "comments_problem_id": 738, - "comments_count": 0 + "comments_count": 1 }, { "problem_id": 739, @@ -17034,8 +17109,8 @@ ], "last_edited": "", "latex_path": "/latex/740", - "formalized": false, - "formalized_url": "", + "formalized": true, + "formalized_url": "https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/ErdosProblems/740.lean", "oeis_urls": [], "comments_problem_id": 740, "comments_count": 0 @@ -17043,22 +17118,22 @@ { "problem_id": 741, "problem_url": "/741", - "status_bucket": "open", - "status_dom_id": "open", - "status_label": "OPEN", - "status_detail": "This is open, and cannot be resolved with a finite computation.", + "status_bucket": "closed", + "status_dom_id": "solved", + "status_label": "SOLVED (LEAN)", + "status_detail": "This has been resolved in some other way than a proof or disproof, and that resolution verified in Lean.", "prize_amount": "", - "statement": "Let $A\\subseteq \\mathbb{N}$ be such that $A+A$ has positive density. Can one always decompose $A=A_1\\sqcup A_2$ such that $A_1+A_1$ and $A_2+A_2$ both have positive density? Is there a basis $A$ of order $2$ such that if $A=A_1\\sqcup A_2$ then $A_1+A_1$ and $A_2+A_2$ cannot both have bounded gaps?", + "statement": "Let $A\\subseteq \\mathbb{N}$ be such that $A+A$ has positive (upper) density. Can one always decompose $A=A_1\\sqcup A_2$ such that $A_1+A_1$ and $A_2+A_2$ both have positive (upper) density? Is there a basis $A$ of order $2$ such that if $A=A_1\\sqcup A_2$ then $A_1+A_1$ and $A_2+A_2$ cannot both have bounded gaps?", "tags": [ "additive combinatorics" ], - "last_edited": "", + "last_edited": "02 May 2026", "latex_path": "/latex/741", "formalized": true, "formalized_url": "https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/ErdosProblems/741.lean", "oeis_urls": [], "comments_problem_id": 741, - "comments_count": 0 + "comments_count": 6 }, { "problem_id": 742, @@ -17216,13 +17291,13 @@ "tags": [ "additive combinatorics" ], - "last_edited": "", + "last_edited": "06 April 2026", "latex_path": "/latex/749", "formalized": true, "formalized_url": "https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/ErdosProblems/749.lean", "oeis_urls": [], "comments_problem_id": 749, - "comments_count": 0 + "comments_count": 12 }, { "problem_id": 750, @@ -17239,11 +17314,11 @@ ], "last_edited": "14 October 2025", "latex_path": "/latex/750", - "formalized": false, - "formalized_url": "", + "formalized": true, + "formalized_url": "https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/ErdosProblems/750.lean", "oeis_urls": [], "comments_problem_id": 750, - "comments_count": 1 + "comments_count": 15 }, { "problem_id": 751, @@ -17292,8 +17367,8 @@ "problem_url": "/753", "status_bucket": "closed", "status_dom_id": "solved", - "status_label": "DISPROVED", - "status_detail": "This has been solved in the negative.", + "status_label": "DISPROVED (LEAN)", + "status_detail": "This has been solved in the negative and the proof verified in Lean.", "prize_amount": "", "statement": "The list chromatic number $\\chi_L(G)$ is defined to be the minimal $k$ such that for any assignment of a list of $k$ colours to each vertex of $G$ (perhaps different lists for different vertices) a colouring of each vertex by a colour on its list can be chosen such that adjacent vertices receive distinct colours. Does there exist some constant $c>0$ such that\\[\\chi_L(G)+\\chi_L(G^c)> n^{1/2+c}\\]for every graph $G$ on $n$ vertices (where $G^c$ is the complement of $G$)?", "tags": [ @@ -17306,7 +17381,7 @@ "formalized_url": "", "oeis_urls": [], "comments_problem_id": 753, - "comments_count": 0 + "comments_count": 1 }, { "problem_id": 754, @@ -17354,8 +17429,8 @@ "problem_url": "/756", "status_bucket": "closed", "status_dom_id": "solved", - "status_label": "PROVED", - "status_detail": "This has been solved in the affirmative.", + "status_label": "PROVED (LEAN)", + "status_detail": "This has been solved in the affirmative and the proof verified in Lean.", "prize_amount": "", "statement": "Let $A\\subset \\mathbb{R}^2$ be a set of $n$ points. Can there be $\\gg n$ many distinct distances each of which occurs for more than $n$ many pairs from $A$?", "tags": [ @@ -17368,7 +17443,7 @@ "formalized_url": "", "oeis_urls": [], "comments_problem_id": 756, - "comments_count": 0 + "comments_count": 1 }, { "problem_id": 757, @@ -17384,7 +17459,7 @@ "distances", "sidon sets" ], - "last_edited": "08 January 2026", + "last_edited": "10 April 2026", "latex_path": "/latex/757", "formalized": true, "formalized_url": "https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/ErdosProblems/757.lean", @@ -17439,21 +17514,21 @@ "problem_url": "/760", "status_bucket": "closed", "status_dom_id": "solved", - "status_label": "PROVED", - "status_detail": "This has been solved in the affirmative.", + "status_label": "PROVED (LEAN)", + "status_detail": "This has been solved in the affirmative and the proof verified in Lean.", "prize_amount": "", - "statement": "The cochromatic number of $G$, denoted by $\\zeta(G)$, is the minimum number of colours needed to colour the vertices of $G$ such that each colour class induces either a complete graph or empty graph. If $G$ is a graph with chromatic number $\\chi(G)=m$ then must $G$ contain a subgraph $H$ with\\[\\zeta(H) \\gg \\frac{m}{\\log m}?\\]", + "statement": "The cochromatic number of $G$, denoted by $\\zeta(G)$, is the minimum number of colours needed to colour the vertices of $G$ such that each colour class induces either a complete graph or independent set. If $G$ is a graph with chromatic number $\\chi(G)=m$ then must $G$ contain a subgraph $H$ with\\[\\zeta(H) \\gg \\frac{m}{\\log m}?\\]", "tags": [ "graph theory", "chromatic number" ], - "last_edited": "", + "last_edited": "24 April 2026", "latex_path": "/latex/760", "formalized": false, "formalized_url": "", "oeis_urls": [], "comments_problem_id": 760, - "comments_count": 0 + "comments_count": 2 }, { "problem_id": 761, @@ -17474,7 +17549,7 @@ "formalized_url": "", "oeis_urls": [], "comments_problem_id": 761, - "comments_count": 0 + "comments_count": 1 }, { "problem_id": 762, @@ -17544,8 +17619,8 @@ "problem_url": "/765", "status_bucket": "closed", "status_dom_id": "solved", - "status_label": "SOLVED", - "status_detail": "This has been resolved in some other way than a proof or disproof.", + "status_label": "SOLVED (LEAN)", + "status_detail": "This has been resolved in some other way than a proof or disproof, and that resolution verified in Lean.", "prize_amount": "", "statement": "Give an asymptotic formula for $\\mathrm{ex}(n;C_4)$.", "tags": [ @@ -17560,7 +17635,7 @@ "https://oeis.org/A006855" ], "comments_problem_id": 765, - "comments_count": 3 + "comments_count": 4 }, { "problem_id": 766, @@ -17644,7 +17719,9 @@ "latex_path": "/latex/769", "formalized": false, "formalized_url": "", - "oeis_urls": [], + "oeis_urls": [ + "https://oeis.org/A014544" + ], "comments_problem_id": 769, "comments_count": 1 }, @@ -17726,7 +17803,7 @@ "sidon sets", "squares" ], - "last_edited": "", + "last_edited": "24 April 2026", "latex_path": "/latex/773", "formalized": false, "formalized_url": "", @@ -17734,7 +17811,7 @@ "https://oeis.org/A390813" ], "comments_problem_id": 773, - "comments_count": 1 + "comments_count": 11 }, { "problem_id": 774, @@ -17750,8 +17827,8 @@ ], "last_edited": "28 December 2025", "latex_path": "/latex/774", - "formalized": false, - "formalized_url": "", + "formalized": true, + "formalized_url": "https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/ErdosProblems/774.lean", "oeis_urls": [], "comments_problem_id": 774, "comments_count": 2 @@ -17761,8 +17838,8 @@ "problem_url": "/775", "status_bucket": "closed", "status_dom_id": "solved", - "status_label": "DISPROVED", - "status_detail": "This has been solved in the negative.", + "status_label": "DISPROVED (LEAN)", + "status_detail": "This has been solved in the negative and the proof verified in Lean.", "prize_amount": "", "statement": "Is there a $3$-uniform hypergraph on $n$ vertices which contains at least $n-O(1)$ different sizes of cliques (maximal complete subgraphs)", "tags": [ @@ -17775,7 +17852,7 @@ "formalized_url": "", "oeis_urls": [], "comments_problem_id": 775, - "comments_count": 1 + "comments_count": 2 }, { "problem_id": 776, @@ -17789,13 +17866,13 @@ "tags": [ "combinatorics" ], - "last_edited": "", + "last_edited": "10 April 2026", "latex_path": "/latex/776", "formalized": false, "formalized_url": "", "oeis_urls": [], "comments_problem_id": 776, - "comments_count": 4 + "comments_count": 6 }, { "problem_id": 777, @@ -17926,16 +18003,16 @@ { "problem_id": 783, "problem_url": "/783", - "status_bucket": "open", - "status_dom_id": "open", - "status_label": "OPEN", - "status_detail": "This is open, and cannot be resolved with a finite computation.", + "status_bucket": "closed", + "status_dom_id": "solved", + "status_label": "SOLVED", + "status_detail": "This has been resolved in some other way than a proof or disproof.", "prize_amount": "", "statement": "Fix some constant $C>0$ and let $N$ be large. Let $A\\subseteq \\{2,\\ldots,N\\}$ be such that $(a,b)=1$ for all $a\\neq b\\in A$ and $\\sum_{n\\in A}\\frac{1}{n}\\leq C$. What choice of such an $A$ minimises the number of integers $m\\leq N$ not divisible by any $a\\in A$?", "tags": [ "number theory" ], - "last_edited": "08 February 2026", + "last_edited": "11 April 2026", "latex_path": "/latex/783", "formalized": false, "formalized_url": "", @@ -17955,7 +18032,7 @@ "tags": [ "number theory" ], - "last_edited": "20 December 2025", + "last_edited": "08 April 2026", "latex_path": "/latex/784", "formalized": false, "formalized_url": "", @@ -17968,20 +18045,20 @@ "problem_url": "/785", "status_bucket": "closed", "status_dom_id": "solved", - "status_label": "PROVED", - "status_detail": "This has been solved in the affirmative.", + "status_label": "PROVED (LEAN)", + "status_detail": "This has been solved in the affirmative and the proof verified in Lean.", "prize_amount": "", "statement": "Let $A,B\\subseteq \\mathbb{N}$ be infinite sets such that $A+B$ contains all large integers. Let $A(x)=\\lvert A\\cap [1,x]\\rvert$ and similarly for $B(x)$. Is it true that if $A(x)B(x)\\sim x$ then\\[A(x)B(x)-x\\to \\infty\\]as $x\\to \\infty$?", "tags": [ "additive combinatorics" ], - "last_edited": "", + "last_edited": "07 March 2026", "latex_path": "/latex/785", "formalized": false, "formalized_url": "", "oeis_urls": [], "comments_problem_id": 785, - "comments_count": 2 + "comments_count": 5 }, { "problem_id": 786, @@ -17995,7 +18072,7 @@ "tags": [ "number theory" ], - "last_edited": "02 February 2026", + "last_edited": "11 April 2026", "latex_path": "/latex/786", "formalized": true, "formalized_url": "https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/ErdosProblems/786.lean", @@ -18059,8 +18136,8 @@ ], "last_edited": "", "latex_path": "/latex/789", - "formalized": false, - "formalized_url": "", + "formalized": true, + "formalized_url": "https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/ErdosProblems/789.lean", "oeis_urls": [], "comments_problem_id": 789, "comments_count": 0 @@ -18181,13 +18258,13 @@ "tags": [ "number theory" ], - "last_edited": "", + "last_edited": "06 April 2026", "latex_path": "/latex/795", "formalized": false, "formalized_url": "", "oeis_urls": [], "comments_problem_id": 795, - "comments_count": 0 + "comments_count": 1 }, { "problem_id": 796, @@ -18235,8 +18312,8 @@ "problem_url": "/798", "status_bucket": "closed", "status_dom_id": "solved", - "status_label": "PROVED", - "status_detail": "This has been solved in the affirmative.", + "status_label": "PROVED (LEAN)", + "status_detail": "This has been solved in the affirmative and the proof verified in Lean.", "prize_amount": "", "statement": "Let $t(n)$ be the minimum number of points in $\\{1,\\ldots,n\\}^2$ such that the $\\binom{t}{2}$ lines determined by these points cover all points in $\\{1,\\ldots,n\\}^2$. Estimate $t(n)$. In particular, is it true that $t(n)=o(n)$?", "tags": [ @@ -18246,9 +18323,11 @@ "latex_path": "/latex/798", "formalized": false, "formalized_url": "", - "oeis_urls": [], + "oeis_urls": [ + "https://oeis.org/A116446" + ], "comments_problem_id": 798, - "comments_count": 0 + "comments_count": 1 }, { "problem_id": 799, @@ -18462,18 +18541,18 @@ "status_label": "OPEN", "status_detail": "This is open, and cannot be resolved with a finite computation.", "prize_amount": "", - "statement": "Let $k\\geq 3$ and define $F_k(n)$ to be the minimal $r$ such that there is a graph $G$ on $n$ vertices with $\\lfloor n^2/4\\rfloor+1$ many edges such that the edges can be $r$-coloured so that every subgraph isomorphic to $C_{2k+1}$ has no colour repeating on the edges. Is it true that\\[F_k(n)\\sim n^2/8?\\]", + "statement": "Define the anti-Ramsey number $\\chi_S(n,e,G)$ as the smallest $r$ such that there is a graph with $n$ vertices and $e$ edges with an $r$-colouring of its edges in which every copy of $G$ has entirely distinct edge colours. Is it true that, for all $k\\geq 3$,\\[\\chi_S(n, \\lfloor n^2/4\\rfloor+1,C_{2k+1})\\sim n^2/8?\\]", "tags": [ "graph theory", "ramsey theory" ], - "last_edited": "", + "last_edited": "01 April 2026", "latex_path": "/latex/809", "formalized": false, "formalized_url": "", "oeis_urls": [], "comments_problem_id": 809, - "comments_count": 0 + "comments_count": 1 }, { "problem_id": 810, @@ -18488,13 +18567,13 @@ "graph theory", "ramsey theory" ], - "last_edited": "", + "last_edited": "01 April 2026", "latex_path": "/latex/810", "formalized": false, "formalized_url": "", "oeis_urls": [], "comments_problem_id": 810, - "comments_count": 7 + "comments_count": 9 }, { "problem_id": 811, @@ -18532,9 +18611,11 @@ ], "last_edited": "", "latex_path": "/latex/812", - "formalized": false, - "formalized_url": "", - "oeis_urls": [], + "formalized": true, + "formalized_url": "https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/ErdosProblems/812.lean", + "oeis_urls": [ + "https://oeis.org/A059442" + ], "comments_problem_id": 812, "comments_count": 0 }, @@ -18643,8 +18724,8 @@ "problem_url": "/818", "status_bucket": "closed", "status_dom_id": "solved", - "status_label": "PROVED", - "status_detail": "This has been solved in the affirmative.", + "status_label": "PROVED (LEAN)", + "status_detail": "This has been solved in the affirmative and the proof verified in Lean.", "prize_amount": "", "statement": "Let $A$ be a finite set of integers such that $\\lvert A+A\\rvert \\ll \\lvert A\\rvert$. Is it true that\\[\\lvert AA\\rvert \\gg \\frac{\\lvert A\\rvert^2}{(\\log \\lvert A\\rvert)^C}\\]for some constant $C>0$?", "tags": [ @@ -18656,7 +18737,7 @@ "formalized_url": "", "oeis_urls": [], "comments_problem_id": 818, - "comments_count": 0 + "comments_count": 1 }, { "problem_id": 819, @@ -18676,7 +18757,7 @@ "formalized_url": "", "oeis_urls": [], "comments_problem_id": 819, - "comments_count": 0 + "comments_count": 2 }, { "problem_id": 820, @@ -18822,13 +18903,13 @@ "tags": [ "number theory" ], - "last_edited": "", + "last_edited": "17 April 2026", "latex_path": "/latex/826", "formalized": true, "formalized_url": "https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/ErdosProblems/826.lean", "oeis_urls": [], "comments_problem_id": 826, - "comments_count": 1 + "comments_count": 2 }, { "problem_id": 827, @@ -18842,13 +18923,13 @@ "tags": [ "geometry" ], - "last_edited": "24 October 2025", + "last_edited": "11 May 2026", "latex_path": "/latex/827", "formalized": false, "formalized_url": "", "oeis_urls": [], "comments_problem_id": 827, - "comments_count": 1 + "comments_count": 3 }, { "problem_id": 828, @@ -19016,7 +19097,7 @@ "formalized_url": "https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/ErdosProblems/835.lean", "oeis_urls": [], "comments_problem_id": 835, - "comments_count": 4 + "comments_count": 6 }, { "problem_id": 836, @@ -19038,7 +19119,7 @@ "formalized_url": "", "oeis_urls": [], "comments_problem_id": 836, - "comments_count": 0 + "comments_count": 3 }, { "problem_id": 837, @@ -19080,7 +19161,7 @@ "formalized_url": "", "oeis_urls": [], "comments_problem_id": 838, - "comments_count": 0 + "comments_count": 3 }, { "problem_id": 839, @@ -19100,7 +19181,7 @@ "formalized_url": "", "oeis_urls": [], "comments_problem_id": 839, - "comments_count": 1 + "comments_count": 2 }, { "problem_id": 840, @@ -19191,8 +19272,8 @@ "problem_url": "/844", "status_bucket": "closed", "status_dom_id": "solved", - "status_label": "PROVED", - "status_detail": "This has been solved in the affirmative.", + "status_label": "PROVED (LEAN)", + "status_detail": "This has been solved in the affirmative and the proof verified in Lean.", "prize_amount": "", "statement": "Let $A\\subseteq \\{1,\\ldots,N\\}$ be such that, for all $a,b\\in A$, the product $ab$ is not squarefree. Is the maximum size of such an $A$ achieved by taking $A$ to be the set of even numbers and odd non-squarefree numbers?", "tags": [ @@ -19205,7 +19286,7 @@ "formalized_url": "", "oeis_urls": [], "comments_problem_id": 844, - "comments_count": 0 + "comments_count": 1 }, { "problem_id": 845, @@ -19230,8 +19311,8 @@ { "problem_id": 846, "problem_url": "/846", - "status_bucket": "open", - "status_dom_id": "open", + "status_bucket": "closed", + "status_dom_id": "solved", "status_label": "DISPROVED (LEAN)", "status_detail": "This has been solved in the negative and the proof verified in Lean.", "prize_amount": "", @@ -19239,7 +19320,7 @@ "tags": [ "geometry" ], - "last_edited": "", + "last_edited": "10 April 2026", "latex_path": "/latex/846", "formalized": true, "formalized_url": "https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/ErdosProblems/846.lean", @@ -19285,7 +19366,7 @@ "formalized_url": "https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/ErdosProblems/848.lean", "oeis_urls": [], "comments_problem_id": 848, - "comments_count": 18 + "comments_count": 48 }, { "problem_id": 849, @@ -19342,8 +19423,8 @@ { "problem_id": 851, "problem_url": "/851", - "status_bucket": "open", - "status_dom_id": "open", + "status_bucket": "closed", + "status_dom_id": "solved", "status_label": "PROVED", "status_detail": "This has been solved in the affirmative.", "prize_amount": "", @@ -19351,7 +19432,7 @@ "tags": [ "number theory" ], - "last_edited": "", + "last_edited": "02 April 2026", "latex_path": "/latex/851", "formalized": true, "formalized_url": "https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/ErdosProblems/851.lean", @@ -19382,7 +19463,7 @@ "https://oeis.org/A078515" ], "comments_problem_id": 852, - "comments_count": 1 + "comments_count": 6 }, { "problem_id": 853, @@ -19406,7 +19487,7 @@ "https://oeis.org/A390769" ], "comments_problem_id": 853, - "comments_count": 2 + "comments_count": 3 }, { "problem_id": 854, @@ -19436,15 +19517,15 @@ "problem_url": "/855", "status_bucket": "open", "status_dom_id": "open", - "status_label": "FALSIFIABLE", - "status_detail": "Open, but could be disproved with a finite counterexample.", + "status_label": "OPEN", + "status_detail": "This is open, and cannot be resolved with a finite computation.", "prize_amount": "", "statement": "If $\\pi(x)$ counts the number of primes in $[1,x]$ then is it true that (for large $x$ and $y$)\\[\\pi(x+y) \\leq \\pi(x)+\\pi(y)?\\]", "tags": [ "number theory", "primes" ], - "last_edited": "12 January 2026", + "last_edited": "08 April 2026", "latex_path": "/latex/855", "formalized": true, "formalized_url": "https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/ErdosProblems/855.lean", @@ -19472,7 +19553,7 @@ "formalized_url": "", "oeis_urls": [], "comments_problem_id": 856, - "comments_count": 12 + "comments_count": 17 }, { "problem_id": 857, @@ -19488,8 +19569,8 @@ ], "last_edited": "", "latex_path": "/latex/857", - "formalized": false, - "formalized_url": "", + "formalized": true, + "formalized_url": "https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/ErdosProblems/857.lean", "oeis_urls": [], "comments_problem_id": 857, "comments_count": 1 @@ -19497,23 +19578,23 @@ { "problem_id": 858, "problem_url": "/858", - "status_bucket": "open", - "status_dom_id": "open", - "status_label": "OPEN", - "status_detail": "This is open, and cannot be resolved with a finite computation.", + "status_bucket": "closed", + "status_dom_id": "solved", + "status_label": "SOLVED", + "status_detail": "This has been resolved in some other way than a proof or disproof.", "prize_amount": "", "statement": "Let $A\\subseteq \\{1,\\ldots,N\\}$ be such that there is no solution to $at=b$ with $a,b\\in A$ and the smallest prime factor of $t$ is $>a$. Estimate the maximum of\\[\\frac{1}{\\log N}\\sum_{n\\in A}\\frac{1}{n}.\\]", "tags": [ "number theory", "primitive sets" ], - "last_edited": "", + "last_edited": "24 April 2026", "latex_path": "/latex/858", "formalized": false, "formalized_url": "", "oeis_urls": [], "comments_problem_id": 858, - "comments_count": 0 + "comments_count": 13 }, { "problem_id": 859, @@ -19612,24 +19693,24 @@ { "problem_id": 863, "problem_url": "/863", - "status_bucket": "open", - "status_dom_id": "open", - "status_label": "OPEN", - "status_detail": "This is open, and cannot be resolved with a finite computation.", + "status_bucket": "closed", + "status_dom_id": "solved", + "status_label": "PROVED", + "status_detail": "This has been solved in the affirmative.", "prize_amount": "", - "statement": "Let $r\\geq 2$ and let $A\\subseteq \\{1,\\ldots,N\\}$ be a set of maximal size such that there are at most $r$ solutions to $n=a+b$ with $a\\leq b$ for any $n$. (That is, $A$ is a $B_2[r]$ set.) Similarly, let $B\\subseteq \\{1,\\ldots,N\\}$ be a set of maximal size such that there are at most $r$ solutions to $n=a-b$ for any $n$. If $\\lvert A\\rvert\\sim c_rN^{1/2}$ as $N\\to \\infty$ and $\\lvert B\\rvert \\sim c_r'N^{1/2}$ as $N\\to \\infty$ then is it true that $c_r\\neq c_r'$ for $r\\geq 2$? Is it true that $c_r'\\epsilon \\log n$ (for all large $n$, for arbitrary fixed $\\epsilon>0$)?", "tags": [ "number theory", "additive basis" ], - "last_edited": "23 January 2026", + "last_edited": "03 April 2026", "latex_path": "/latex/868", "formalized": true, "formalized_url": "https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/ErdosProblems/868.lean", @@ -19741,23 +19822,23 @@ { "problem_id": 869, "problem_url": "/869", - "status_bucket": "open", - "status_dom_id": "open", - "status_label": "OPEN", - "status_detail": "This is open, and cannot be resolved with a finite computation.", + "status_bucket": "closed", + "status_dom_id": "solved", + "status_label": "DISPROVED", + "status_detail": "This has been solved in the negative.", "prize_amount": "", "statement": "If $A_1,A_2$ are disjoint additive bases of order $2$ (i.e. $A_i+A_i$ contains all large integers) then must $A=A_1\\cup A_2$ contain a minimal additive basis of order $2$ (one such that deleting any element creates infinitely many $n\\not\\in A+A$)?", "tags": [ "number theory", "additive basis" ], - "last_edited": "", + "last_edited": "02 May 2026", "latex_path": "/latex/869", "formalized": false, "formalized_url": "", "oeis_urls": [], "comments_problem_id": 869, - "comments_count": 1 + "comments_count": 2 }, { "problem_id": 870, @@ -19778,7 +19859,7 @@ "formalized_url": "", "oeis_urls": [], "comments_problem_id": 870, - "comments_count": 1 + "comments_count": 9 }, { "problem_id": 871, @@ -19814,13 +19895,13 @@ "number theory", "primitive sets" ], - "last_edited": "", + "last_edited": "24 April 2026", "latex_path": "/latex/872", "formalized": false, "formalized_url": "", "oeis_urls": [], "comments_problem_id": 872, - "comments_count": 14 + "comments_count": 33 }, { "problem_id": 873, @@ -19840,7 +19921,7 @@ "formalized_url": "https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/ErdosProblems/873.lean", "oeis_urls": [], "comments_problem_id": 873, - "comments_count": 1 + "comments_count": 8 }, { "problem_id": 874, @@ -19881,7 +19962,7 @@ "formalized_url": "", "oeis_urls": [], "comments_problem_id": 875, - "comments_count": 1 + "comments_count": 7 }, { "problem_id": 876, @@ -19931,11 +20012,11 @@ "status_label": "OPEN", "status_detail": "This is open, and cannot be resolved with a finite computation.", "prize_amount": "", - "statement": "If $n=\\prod_{1\\leq i\\leq t} p_i^{k_i}$ is the factorisation of $n$ into distinct primes then let\\[f(n)=\\sum p_i^{\\ell_i},\\]where $\\ell_i$ is chosen such that $n\\in [p_i^{\\ell_i},p_i^{\\ell_i+1})$. Furthermore, let\\[F(n)=\\max \\sum_{i=1}^t a_i\\]where the maximum is taken over all $a_1,\\ldots,a_t\\leq n$ such that $(a_i,a_j)=1$ for $i\\neq j$ and all prime factors of each $a_i$ are prime factors of $n$. Is it true that, for almost all $n$,\\[f(n)=o(n\\log\\log n)\\]and\\[F(n) \\gg n\\log\\log n?\\]Is it true that\\[\\max_{n\\leq x}f(n)\\sim \\frac{x\\log x}{\\log\\log x}?\\]Is it true that (for all $x$, or perhaps just for all large $x$)\\[\\max_{n\\leq x}f(n)=\\max_{n\\leq x}F(n)?\\]Find an asymptotic formula for the number of $nk$, then is it true that, for every $k\\geq 1$,\\[\\liminf_{n\\to \\infty}\\sum_{0\\leq in^2/4$ edges then it contains a triangle on $x,y,z$ such that\\[d(x)+d(y)+d(z) \\geq \\frac{3}{2}n.\\]", + "statement": "Let $r\\geq 2$ and let $t_r(n)$ be the Tur\u00e1n number (the maximal number of edges in a graph on $n$ vertices with no $K_{r+1}$). If $G$ is a graph with $n$ vertices and $m\\geq t_r(n)$ edges there exists a clique on $r$ vertices, say $x_1,\\ldots,x_r$, such that\\[d(x_1)+\\cdots+d(x_r)\\geq \\frac{2rm}{n}.\\]", "tags": [ "graph theory" ], - "last_edited": "06 October 2025", + "last_edited": "03 April 2026", "latex_path": "/latex/904", "formalized": false, "formalized_url": "", "oeis_urls": [], "comments_problem_id": 904, - "comments_count": 0 + "comments_count": 5 }, { "problem_id": 905, "problem_url": "/905", "status_bucket": "closed", "status_dom_id": "solved", - "status_label": "PROVED", - "status_detail": "This has been solved in the affirmative.", + "status_label": "PROVED (LEAN)", + "status_detail": "This has been solved in the affirmative and the proof verified in Lean.", "prize_amount": "", "statement": "Every graph with $n$ vertices and $>n^2/4$ edges contains an edge which is in at least $n/6$ triangles.", "tags": [ "graph theory" ], - "last_edited": "06 October 2025", + "last_edited": "07 April 2026", "latex_path": "/latex/905", "formalized": false, "formalized_url": "", "oeis_urls": [], "comments_problem_id": 905, - "comments_count": 2 + "comments_count": 3 }, { "problem_id": 906, @@ -20527,27 +20608,27 @@ "formalized_url": "https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/ErdosProblems/906.lean", "oeis_urls": [], "comments_problem_id": 906, - "comments_count": 8 + "comments_count": 11 }, { "problem_id": 907, "problem_url": "/907", "status_bucket": "closed", "status_dom_id": "solved", - "status_label": "PROVED", - "status_detail": "This has been solved in the affirmative.", + "status_label": "PROVED (LEAN)", + "status_detail": "This has been solved in the affirmative and the proof verified in Lean.", "prize_amount": "", - "statement": "Let $f:\\mathbb{R}\\to \\mathbb{R}$ be such that $f(x+h)-f(x)$ is continous for every $h>0$. Is it true that\\[f=g+h\\]for some continuous $g$ and additive $h$ (i.e. $h(x+y)=h(x)+h(y)$)?", + "statement": "Let $f:\\mathbb{R}\\to \\mathbb{R}$ be such that $f(x+h)-f(x)$ is continuous for every $h>0$. Is it true that\\[f=g+h\\]for some continuous $g$ and additive $h$ (i.e. $h(x+y)=h(x)+h(y)$)?", "tags": [ "analysis" ], - "last_edited": "30 December 2025", + "last_edited": "07 April 2026", "latex_path": "/latex/907", "formalized": false, "formalized_url": "", "oeis_urls": [], "comments_problem_id": 907, - "comments_count": 0 + "comments_count": 1 }, { "problem_id": 908, @@ -20681,20 +20762,20 @@ "problem_url": "/914", "status_bucket": "closed", "status_dom_id": "solved", - "status_label": "PROVED", - "status_detail": "This has been solved in the affirmative.", + "status_label": "PROVED (LEAN)", + "status_detail": "This has been solved in the affirmative and the proof verified in Lean.", "prize_amount": "", "statement": "Let $r\\geq 2$ and $m\\geq 1$. Every graph with $rm$ vertices and minimum degree at least $m(r-1)$ contains $m$ vertex disjoint copies of $K_r$.", "tags": [ "graph theory" ], - "last_edited": "", + "last_edited": "15 April 2026", "latex_path": "/latex/914", "formalized": false, "formalized_url": "", "oeis_urls": [], "comments_problem_id": 914, - "comments_count": 0 + "comments_count": 2 }, { "problem_id": 915, @@ -20869,8 +20950,8 @@ "problem_url": "/923", "status_bucket": "closed", "status_dom_id": "solved", - "status_label": "PROVED", - "status_detail": "This has been solved in the affirmative.", + "status_label": "PROVED (LEAN)", + "status_detail": "This has been solved in the affirmative and the proof verified in Lean.", "prize_amount": "", "statement": "Is it true that, for every $k$, there is some $f(k)$ such that if $G$ has chromatic number $\\geq f(k)$ then $G$ contains a triangle-free subgraph with chromatic number $\\geq k$?", "tags": [ @@ -20883,7 +20964,7 @@ "formalized_url": "", "oeis_urls": [], "comments_problem_id": 923, - "comments_count": 1 + "comments_count": 2 }, { "problem_id": 924, @@ -20979,7 +21060,7 @@ "tags": [ "number theory" ], - "last_edited": "", + "last_edited": "03 April 2026", "latex_path": "/latex/928", "formalized": false, "formalized_url": "", @@ -21047,7 +21128,7 @@ "formalized_url": "https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/ErdosProblems/931.lean", "oeis_urls": [], "comments_problem_id": 931, - "comments_count": 0 + "comments_count": 4 }, { "problem_id": 932, @@ -21109,7 +21190,7 @@ "formalized_url": "", "oeis_urls": [], "comments_problem_id": 934, - "comments_count": 0 + "comments_count": 3 }, { "problem_id": 935, @@ -21200,7 +21281,7 @@ "https://oeis.org/A001694" ], "comments_problem_id": 938, - "comments_count": 0 + "comments_count": 1 }, { "problem_id": 939, @@ -21221,7 +21302,7 @@ "formalized_url": "https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/ErdosProblems/939.lean", "oeis_urls": [], "comments_problem_id": 939, - "comments_count": 4 + "comments_count": 9 }, { "problem_id": 940, @@ -21307,7 +21388,7 @@ "formalized_url": "https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/ErdosProblems/943.lean", "oeis_urls": [], "comments_problem_id": 943, - "comments_count": 3 + "comments_count": 5 }, { "problem_id": 944, @@ -21375,7 +21456,7 @@ "https://oeis.org/A284783" ], "comments_problem_id": 946, - "comments_count": 3 + "comments_count": 5 }, { "problem_id": 947, @@ -21411,7 +21492,7 @@ "number theory", "ramsey theory" ], - "last_edited": "22 September 2025", + "last_edited": "10 April 2026", "latex_path": "/latex/948", "formalized": false, "formalized_url": "", @@ -21452,13 +21533,13 @@ "number theory", "primes" ], - "last_edited": "", + "last_edited": "15 April 2026", "latex_path": "/latex/950", "formalized": false, "formalized_url": "", "oeis_urls": [], "comments_problem_id": 950, - "comments_count": 2 + "comments_count": 3 }, { "problem_id": 951, @@ -21472,7 +21553,7 @@ "tags": [ "number theory" ], - "last_edited": "28 January 2026", + "last_edited": "06 April 2026", "latex_path": "/latex/951", "formalized": true, "formalized_url": "https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/ErdosProblems/951.lean", @@ -21492,7 +21573,7 @@ "tags": [ "number theory" ], - "last_edited": "", + "last_edited": "08 April 2026", "latex_path": "/latex/952", "formalized": true, "formalized_url": "https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/ErdosProblems/952.lean", @@ -21519,7 +21600,7 @@ "formalized_url": "", "oeis_urls": [], "comments_problem_id": 953, - "comments_count": 3 + "comments_count": 17 }, { "problem_id": 954, @@ -21583,7 +21664,7 @@ "formalized_url": "", "oeis_urls": [], "comments_problem_id": 956, - "comments_count": 0 + "comments_count": 7 }, { "problem_id": 957, @@ -21651,16 +21732,16 @@ { "problem_id": 960, "problem_url": "/960", - "status_bucket": "open", - "status_dom_id": "open", - "status_label": "OPEN", - "status_detail": "This is open, and cannot be resolved with a finite computation.", + "status_bucket": "closed", + "status_dom_id": "solved", + "status_label": "DISPROVED", + "status_detail": "This has been solved in the negative.", "prize_amount": "", "statement": "Let $r,k\\geq 2$ be fixed. Let $A\\subset \\mathbb{R}^2$ be a set of $n$ points with no $k$ points on a line. Determine the threshold $f_{r,k}(n)$ such that if there are at least $f_{r,k}(n)$ many ordinary lines (lines containing exactly two points) then there is a set $A'\\subseteq A$ of $r$ points such that all $\\binom{r}{2}$ many lines determined by $A'$ are ordinary. Is it true that $f_{r,k}(n)=o(n^2)$, or perhaps even $\\ll n$?", "tags": [ "geometry" ], - "last_edited": "", + "last_edited": "09 April 2026", "latex_path": "/latex/960", "formalized": false, "formalized_url": "", @@ -21680,7 +21761,7 @@ "tags": [ "number theory" ], - "last_edited": "", + "last_edited": "03 April 2026", "latex_path": "/latex/961", "formalized": true, "formalized_url": "https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/ErdosProblems/961.lean", @@ -21698,19 +21779,19 @@ "status_label": "OPEN", "status_detail": "This is open, and cannot be resolved with a finite computation.", "prize_amount": "", - "statement": "Let $k(n)$ be the maximal $k$ such that there exists $m\\leq n$ such that each of the integers\\[m+1,\\ldots,m+k\\]are divisible by at least one prime $>k$. Estimate $k(n)$.", + "statement": "Let $k(n)$ be the maximal $k$ such that there exists $m\\leq n$ such that each of the integers\\[m+1,\\ldots,m+k\\]are divisible by at least one prime $>k$. Estimate $k(n)$ - in particular, is it true that\\[\\log k(n) \\leq (\\log n)^{1/2+o(1)}?\\]", "tags": [ "number theory" ], - "last_edited": "30 December 2025", + "last_edited": "03 April 2026", "latex_path": "/latex/962", - "formalized": false, - "formalized_url": "", + "formalized": true, + "formalized_url": "https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/ErdosProblems/962.lean", "oeis_urls": [ "https://oeis.org/A327909" ], "comments_problem_id": 962, - "comments_count": 3 + "comments_count": 5 }, { "problem_id": 963, @@ -21828,7 +21909,7 @@ "tags": [ "number theory" ], - "last_edited": "", + "last_edited": "31 March 2026", "latex_path": "/latex/968", "formalized": true, "formalized_url": "https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/ErdosProblems/968.lean", @@ -21942,15 +22023,15 @@ "formalized_url": "", "oeis_urls": [], "comments_problem_id": 973, - "comments_count": 7 + "comments_count": 6 }, { "problem_id": 974, "problem_url": "/974", "status_bucket": "closed", "status_dom_id": "solved", - "status_label": "PROVED", - "status_detail": "This has been solved in the affirmative.", + "status_label": "PROVED (LEAN)", + "status_detail": "This has been solved in the affirmative and the proof verified in Lean.", "prize_amount": "", "statement": "Let $z_1,\\ldots,z_n\\in \\mathbb{C}$ be a sequence such that $z_1=1$. Suppose that the sequence of\\[s_k=\\sum_{1\\leq i\\leq n}z_i^k\\]contains infinitely many $(n-1)$-tuples of consecutive values of $s_k$ which are all $0$. Then (essentially)\\[z_j=e(j/n),\\]where $e(x)=e^{2\\pi ix}$.", "tags": [ @@ -21962,7 +22043,7 @@ "formalized_url": "", "oeis_urls": [], "comments_problem_id": 974, - "comments_count": 13 + "comments_count": 14 }, { "problem_id": 975, @@ -22006,7 +22087,7 @@ "formalized_url": "", "oeis_urls": [], "comments_problem_id": 976, - "comments_count": 1 + "comments_count": 3 }, { "problem_id": 977, @@ -22039,17 +22120,17 @@ "status_label": "OPEN", "status_detail": "This is open, and cannot be resolved with a finite computation.", "prize_amount": "", - "statement": "Let $f\\in \\mathbb{Z}[x]$ be an irreducible polynomial of degree $k>2$ (and suppose that $k\\neq 2^l$ for any $l\\geq 1$) such that the leading coefficient of $f$ is positive. Does the set of integers $n\\geq 1$ for which $f(n)$ is $(k-1)$-power-free have positive density? If $k>3$ then are there infinitely many $n$ for which $f(n)$ is $(k-2)$-power-free? In particular, does\\[n^4+2\\]represent infinitely many squarefree numbers?", + "statement": "Let $f\\in \\mathbb{Z}[x]$ be an irreducible polynomial of degree $k>2$ (and suppose that $k\\neq 2^l$ for any $l\\geq 1$) such that the leading coefficient of $f$ is positive. Does the set of integers $n\\geq 1$ for which $f(n)$ is $(k-1)$-power-free have positive density? If $k>3$, and for all primes $p$ there exists $n$ such that $p^{k-2}\\nmid f(n)$, then are there infinitely many $n$ for which $f(n)$ is $(k-2)$-power-free? In particular, does\\[n^4+2\\]represent infinitely many squarefree numbers?", "tags": [ "number theory" ], - "last_edited": "05 March 2026", + "last_edited": "31 March 2026", "latex_path": "/latex/978", "formalized": true, "formalized_url": "https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/ErdosProblems/978.lean", "oeis_urls": [], "comments_problem_id": 978, - "comments_count": 8 + "comments_count": 16 }, { "problem_id": 979, @@ -22071,7 +22152,7 @@ "https://oeis.org/A385316" ], "comments_problem_id": 979, - "comments_count": 12 + "comments_count": 11 }, { "problem_id": 980, @@ -22158,7 +22239,7 @@ "formalized_url": "", "oeis_urls": [], "comments_problem_id": 983, - "comments_count": 4 + "comments_count": 7 }, { "problem_id": 984, @@ -22173,7 +22254,7 @@ "arithmetic progressions", "additive combinatorics" ], - "last_edited": "", + "last_edited": "04 April 2026", "latex_path": "/latex/984", "formalized": false, "formalized_url": "", @@ -22203,7 +22284,7 @@ "https://oeis.org/A219429" ], "comments_problem_id": 985, - "comments_count": 1 + "comments_count": 2 }, { "problem_id": 986, @@ -22232,23 +22313,23 @@ { "problem_id": 987, "problem_url": "/987", - "status_bucket": "open", - "status_dom_id": "open", - "status_label": "OPEN", - "status_detail": "This is open, and cannot be resolved with a finite computation.", + "status_bucket": "closed", + "status_dom_id": "solved", + "status_label": "PROVED", + "status_detail": "This has been solved in the affirmative.", "prize_amount": "", "statement": "Let $x_1,x_2,\\ldots \\in (0,1)$ be an infinite sequence and let\\[A_k=\\limsup_{n\\to \\infty}\\left\\lvert \\sum_{j\\leq n} e(kx_j)\\right\\rvert,\\]where $e(x)=e^{2\\pi ix}$. Is it true that\\[\\limsup_{k\\to \\infty} A_k=\\infty?\\]Is it possible for $A_k=o(k)$?", "tags": [ "analysis", "discrepancy" ], - "last_edited": "29 December 2025", + "last_edited": "09 April 2026", "latex_path": "/latex/987", "formalized": false, "formalized_url": "", "oeis_urls": [], "comments_problem_id": 987, - "comments_count": 7 + "comments_count": 9 }, { "problem_id": 988, @@ -22293,22 +22374,22 @@ { "problem_id": 990, "problem_url": "/990", - "status_bucket": "open", - "status_dom_id": "open", - "status_label": "OPEN", - "status_detail": "This is open, and cannot be resolved with a finite computation.", + "status_bucket": "closed", + "status_dom_id": "solved", + "status_label": "DISPROVED (LEAN)", + "status_detail": "This has been solved in the negative and the proof verified in Lean.", "prize_amount": "", "statement": "Let $f=a_0+\\cdots+a_dx^d\\in \\mathbb{C}[x]$ be a polynomial. Is it true that, if $f$ has roots $z_1,\\ldots,z_d$ with corresponding arguments $\\theta_1,\\ldots,\\theta_d\\in [0,2\\pi]$, then for all intervals $I\\subseteq [0,2\\pi]$\\[\\left\\lvert (\\# \\theta_i \\in I) - \\frac{\\lvert I\\rvert}{2\\pi}d\\right\\rvert \\ll \\left(n\\log M\\right)^{1/2},\\]where $n$ is the number of non-zero coefficients of $f$ and\\[M=\\frac{\\lvert a_0\\rvert+\\cdots +\\lvert a_d\\rvert}{(\\lvert a_0\\rvert\\lvert a_d\\rvert)^{1/2}}.\\]", "tags": [ "analysis" ], - "last_edited": "", + "last_edited": "10 April 2026", "latex_path": "/latex/990", "formalized": false, "formalized_url": "", "oeis_urls": [], "comments_problem_id": 990, - "comments_count": 1 + "comments_count": 3 }, { "problem_id": 991, @@ -22368,7 +22449,7 @@ "formalized_url": "", "oeis_urls": [], "comments_problem_id": 993, - "comments_count": 1 + "comments_count": 3 }, { "problem_id": 994, @@ -22410,7 +22491,7 @@ "formalized_url": "", "oeis_urls": [], "comments_problem_id": 995, - "comments_count": 0 + "comments_count": 7 }, { "problem_id": 996, @@ -22430,15 +22511,15 @@ "formalized_url": "https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/ErdosProblems/996.lean", "oeis_urls": [], "comments_problem_id": 996, - "comments_count": 0 + "comments_count": 1 }, { "problem_id": 997, "problem_url": "/997", - "status_bucket": "open", - "status_dom_id": "open", - "status_label": "OPEN", - "status_detail": "This is open, and cannot be resolved with a finite computation.", + "status_bucket": "closed", + "status_dom_id": "solved", + "status_label": "PROVED (LEAN)", + "status_detail": "This has been solved in the affirmative and the proof verified in Lean.", "prize_amount": "", "statement": "Call $x_1,x_2,\\ldots \\in (0,1)$ well-distributed if, for every $\\epsilon>0$, if $k$ is sufficiently large then, for all $n>0$ and intervals $I\\subseteq [0,1]$,\\[\\lvert \\# \\{ n0$ such that\\[\\lim_k \\frac{R(k+1,k)}{R(k,k)}> 1+c.\\]", + "statement": "Let $R(k,l)$ be the usual Ramsey number: the smallest $n$ such that if the edges of $K_n$ are coloured red and blue then there exists either a red $K_k$ or a blue $K_l$. Prove the existence of some $c>0$ such that\\[\\lim_{k\\to \\infty}\\frac{R(k+1,k)}{R(k,k)}> 1+c.\\]", "tags": [ "graph theory", "ramsey theory" ], - "last_edited": "03 December 2025", + "last_edited": "23 March 2026", "latex_path": "/latex/1030", "formalized": false, "formalized_url": "", @@ -23145,7 +23228,7 @@ "https://oeis.org/A059442" ], "comments_problem_id": 1030, - "comments_count": 1 + "comments_count": 2 }, { "problem_id": 1031, @@ -23186,7 +23269,7 @@ "formalized_url": "", "oeis_urls": [], "comments_problem_id": 1032, - "comments_count": 1 + "comments_count": 3 }, { "problem_id": 1033, @@ -23200,7 +23283,7 @@ "tags": [ "graph theory" ], - "last_edited": "", + "last_edited": "03 April 2026", "latex_path": "/latex/1033", "formalized": false, "formalized_url": "", @@ -23306,7 +23389,7 @@ "formalized_url": "https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/ErdosProblems/1038.lean", "oeis_urls": [], "comments_problem_id": 1038, - "comments_count": 127 + "comments_count": 137 }, { "problem_id": 1039, @@ -23327,7 +23410,7 @@ "formalized_url": "", "oeis_urls": [], "comments_problem_id": 1039, - "comments_count": 1 + "comments_count": 15 }, { "problem_id": 1040, @@ -23347,7 +23430,7 @@ "formalized_url": "", "oeis_urls": [], "comments_problem_id": 1040, - "comments_count": 0 + "comments_count": 1 }, { "problem_id": 1041, @@ -23368,7 +23451,7 @@ "formalized_url": "https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/ErdosProblems/1041.lean", "oeis_urls": [], "comments_problem_id": 1041, - "comments_count": 2 + "comments_count": 46 }, { "problem_id": 1042, @@ -23382,13 +23465,13 @@ "tags": [ "analysis" ], - "last_edited": "15 September 2025", + "last_edited": "12 April 2026", "latex_path": "/latex/1042", "formalized": false, "formalized_url": "", "oeis_urls": [], "comments_problem_id": 1042, - "comments_count": 0 + "comments_count": 1 }, { "problem_id": 1043, @@ -23415,8 +23498,8 @@ "problem_url": "/1044", "status_bucket": "closed", "status_dom_id": "solved", - "status_label": "SOLVED", - "status_detail": "This has been resolved in some other way than a proof or disproof.", + "status_label": "SOLVED (LEAN)", + "status_detail": "This has been resolved in some other way than a proof or disproof, and that resolution verified in Lean.", "prize_amount": "", "statement": "Let $f(z)=\\prod_{i=1}^n(z-z_i)\\in\\mathbb{C}[x]$ where $\\lvert z_i\\rvert\\leq 1$ for all $i$. If $\\Lambda(f)$ is the maximum of the lengths of the boundaries of the connected components of\\[\\{ z: \\lvert f(z)\\rvert<1\\}\\]then determine the infimum of $\\Lambda(f)$.", "tags": [ @@ -23428,27 +23511,27 @@ "formalized_url": "", "oeis_urls": [], "comments_problem_id": 1044, - "comments_count": 7 + "comments_count": 8 }, { "problem_id": 1045, "problem_url": "/1045", "status_bucket": "open", "status_dom_id": "open", - "status_label": "FALSIFIABLE", - "status_detail": "Open, but could be disproved with a finite counterexample.", + "status_label": "OPEN", + "status_detail": "This is open, and cannot be resolved with a finite computation.", "prize_amount": "", "statement": "Let $z_1,\\ldots,z_n\\in \\mathbb{C}$ with $\\lvert z_i-z_j\\rvert\\leq 2$ for all $i,j$, and\\[\\Delta(z_1,\\ldots,z_n)=\\prod_{i\\neq j}\\lvert z_i-z_j\\rvert.\\]What is the maximum possible value of $\\Delta$? Is it maximised by taking the $z_i$ to be the vertices of a regular polygon?", "tags": [ "analysis" ], - "last_edited": "30 December 2025", + "last_edited": "02 April 2026", "latex_path": "/latex/1045", "formalized": false, "formalized_url": "", "oeis_urls": [], "comments_problem_id": 1045, - "comments_count": 43 + "comments_count": 47 }, { "problem_id": 1046, @@ -23528,7 +23611,7 @@ "formalized_url": "https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/ErdosProblems/1049.lean", "oeis_urls": [], "comments_problem_id": 1049, - "comments_count": 1 + "comments_count": 3 }, { "problem_id": 1050, @@ -23637,7 +23720,7 @@ "https://oeis.org/A167485" ], "comments_problem_id": 1054, - "comments_count": 5 + "comments_count": 22 }, { "problem_id": 1055, @@ -23813,7 +23896,7 @@ "https://oeis.org/A038372" ], "comments_problem_id": 1062, - "comments_count": 4 + "comments_count": 13 }, { "problem_id": 1063, @@ -23902,7 +23985,7 @@ "formalized_url": "", "oeis_urls": [], "comments_problem_id": 1066, - "comments_count": 1 + "comments_count": 2 }, { "problem_id": 1067, @@ -24074,7 +24157,7 @@ "https://oeis.org/A064164" ], "comments_problem_id": 1074, - "comments_count": 3 + "comments_count": 5 }, { "problem_id": 1075, @@ -24234,7 +24317,7 @@ "geometry", "distances" ], - "last_edited": "20 December 2025", + "last_edited": "11 April 2026", "latex_path": "/latex/1082", "formalized": true, "formalized_url": "https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/ErdosProblems/1082.lean", @@ -24259,7 +24342,9 @@ "latex_path": "/latex/1083", "formalized": false, "formalized_url": "", - "oeis_urls": [], + "oeis_urls": [ + "https://oeis.org/A186704" + ], "comments_problem_id": 1083, "comments_count": 0 }, @@ -24299,13 +24384,15 @@ "geometry", "distances" ], - "last_edited": "17 October 2025", + "last_edited": "23 May 2026", "latex_path": "/latex/1085", "formalized": true, "formalized_url": "https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/ErdosProblems/1085.lean", - "oeis_urls": [], + "oeis_urls": [ + "https://oeis.org/A186705" + ], "comments_problem_id": 1085, - "comments_count": 0 + "comments_count": 1 }, { "problem_id": 1086, @@ -24361,7 +24448,7 @@ "tags": [ "geometry" ], - "last_edited": "16 October 2025", + "last_edited": "08 April 2026", "latex_path": "/latex/1088", "formalized": false, "formalized_url": "", @@ -24414,17 +24501,17 @@ { "problem_id": 1091, "problem_url": "/1091", - "status_bucket": "open", - "status_dom_id": "open", - "status_label": "OPEN", - "status_detail": "This is open, and cannot be resolved with a finite computation.", + "status_bucket": "closed", + "status_dom_id": "solved", + "status_label": "SOLVED", + "status_detail": "This has been resolved in some other way than a proof or disproof.", "prize_amount": "", "statement": "Let $G$ be a $K_4$-free graph with chromatic number $4$. Must $G$ contain an odd cycle with at least two diagonals? More generally, is there some $f(r)\\to \\infty$ such that every graph with chromatic number $4$, in which every subgraph on $\\leq r$ vertices has chromatic number $\\leq 3$, contains an odd cycle with at least $f(r)$ diagonals?", "tags": [ "graph theory", "chromatic number" ], - "last_edited": "06 December 2025", + "last_edited": "09 April 2026", "latex_path": "/latex/1091", "formalized": false, "formalized_url": "", @@ -24435,23 +24522,23 @@ { "problem_id": 1092, "problem_url": "/1092", - "status_bucket": "open", - "status_dom_id": "open", - "status_label": "OPEN", - "status_detail": "This is open, and cannot be resolved with a finite computation.", + "status_bucket": "closed", + "status_dom_id": "solved", + "status_label": "DISPROVED", + "status_detail": "This has been solved in the negative.", "prize_amount": "", - "statement": "Let $f_r(n)$ be maximal such that, if a graph $G$ has the property that every subgraph $H$ on $m$ vertices is the union of a graph with chromatic number $r$ and a graph with $\\leq f_r(m)$ edges, then $G$ has chromatic number $\\leq r+1$. Is it true that $f_2(n) \\gg n$? More generally, is $f_r(n)\\gg_r n$?", + "statement": "Let $f_r(n)$ be maximal such that, if a graph $G$ has the property that every subgraph $H$ on $m$ vertices is the union of a graph with chromatic number $\\leq r$ and a graph with $\\leq f_r(m)$ edges, then $G$ has chromatic number $\\leq r+1$. Is it true that $f_2(n) \\gg n$? More generally, is $f_r(n)\\gg_r n$?", "tags": [ "graph theory", "chromatic number" ], - "last_edited": "06 December 2025", + "last_edited": "12 May 2026", "latex_path": "/latex/1092", "formalized": true, "formalized_url": "https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/ErdosProblems/1092.lean", "oeis_urls": [], "comments_problem_id": 1092, - "comments_count": 2 + "comments_count": 6 }, { "problem_id": 1093, @@ -24516,27 +24603,27 @@ "https://oeis.org/A003458" ], "comments_problem_id": 1095, - "comments_count": 6 + "comments_count": 8 }, { "problem_id": 1096, "problem_url": "/1096", - "status_bucket": "open", - "status_dom_id": "open", - "status_label": "OPEN", - "status_detail": "This is open, and cannot be resolved with a finite computation.", + "status_bucket": "closed", + "status_dom_id": "solved", + "status_label": "PROVED", + "status_detail": "This has been solved in the affirmative.", "prize_amount": "", "statement": "Let $10$ is sufficiently small, $x_{k+1}-x_k \\to 0$?", "tags": [ "number theory" ], - "last_edited": "19 October 2025", + "last_edited": "16 April 2026", "latex_path": "/latex/1096", - "formalized": false, - "formalized_url": "", + "formalized": true, + "formalized_url": "https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/ErdosProblems/1096.lean", "oeis_urls": [], "comments_problem_id": 1096, - "comments_count": 0 + "comments_count": 3 }, { "problem_id": 1097, @@ -24546,26 +24633,26 @@ "status_label": "OPEN", "status_detail": "This is open, and cannot be resolved with a finite computation.", "prize_amount": "", - "statement": "Let $A$ be a set of $n$ integers. How many distinct $d$ can occur as the common difference of a three-term arithmetic progression in $A$? Are there always $O(n^{3/2})$ many such $d$?", + "statement": "Let $A$ be a set of $n$ integers. How many distinct $d$ can occur as the common difference of a three-term arithmetic progression in $A$? In particular, are there always $O(n^{3/2})$ many such $d$?", "tags": [ "number theory", "additive combinatorics" ], - "last_edited": "03 December 2025", + "last_edited": "01 April 2026", "latex_path": "/latex/1097", "formalized": true, "formalized_url": "https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/ErdosProblems/1097.lean", "oeis_urls": [], "comments_problem_id": 1097, - "comments_count": 11 + "comments_count": 17 }, { "problem_id": 1098, "problem_url": "/1098", "status_bucket": "closed", "status_dom_id": "solved", - "status_label": "PROVED", - "status_detail": "This has been solved in the affirmative.", + "status_label": "PROVED (LEAN)", + "status_detail": "This has been solved in the affirmative and the proof verified in Lean.", "prize_amount": "", "statement": "Let $G$ be a group and $\\Gamma=\\Gamma(G)$ be the non-commuting graph, with vertices the elements of $G$ and an edge between $g$ and $h$ if and only if $g$ and $h$ do not commute, $gh\\neq hg$. If $\\Gamma$ contains no infinite complete subgraph, then is there a finite bound on the size of complete subgraphs of $\\Gamma$?", "tags": [ @@ -24577,7 +24664,7 @@ "formalized_url": "", "oeis_urls": [], "comments_problem_id": 1098, - "comments_count": 0 + "comments_count": 1 }, { "problem_id": 1099, @@ -24621,7 +24708,7 @@ "https://oeis.org/A325864" ], "comments_problem_id": 1100, - "comments_count": 2 + "comments_count": 5 }, { "problem_id": 1101, @@ -24641,7 +24728,7 @@ "formalized_url": "https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/ErdosProblems/1101.lean", "oeis_urls": [], "comments_problem_id": 1101, - "comments_count": 1 + "comments_count": 3 }, { "problem_id": 1102, @@ -24683,7 +24770,7 @@ "https://oeis.org/A392164" ], "comments_problem_id": 1103, - "comments_count": 4 + "comments_count": 5 }, { "problem_id": 1104, @@ -24702,7 +24789,9 @@ "latex_path": "/latex/1104", "formalized": true, "formalized_url": "https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/ErdosProblems/1104.lean", - "oeis_urls": [], + "oeis_urls": [ + "https://oeis.org/A292528" + ], "comments_problem_id": 1104, "comments_count": 2 }, @@ -24835,13 +24924,13 @@ "tags": [ "number theory" ], - "last_edited": "22 January 2026", + "last_edited": "01 April 2026", "latex_path": "/latex/1110", "formalized": false, "formalized_url": "", "oeis_urls": [], "comments_problem_id": 1110, - "comments_count": 3 + "comments_count": 4 }, { "problem_id": 1111, @@ -24898,8 +24987,8 @@ ], "last_edited": "29 December 2025", "latex_path": "/latex/1113", - "formalized": false, - "formalized_url": "", + "formalized": true, + "formalized_url": "https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/ErdosProblems/1113.lean", "oeis_urls": [ "https://oeis.org/A076336" ], @@ -25053,20 +25142,20 @@ "problem_url": "/1121", "status_bucket": "closed", "status_dom_id": "solved", - "status_label": "PROVED", - "status_detail": "This has been solved in the affirmative.", + "status_label": "PROVED (LEAN)", + "status_detail": "This has been solved in the affirmative and the proof verified in Lean.", "prize_amount": "", "statement": "If $C_1,\\ldots,C_n$ are circles in $\\mathbb{R}^2$ with radii $r_1,\\ldots,r_n$ such that no line disjoint from all the circles divides them into two non-empty sets then the circles can be covered by a circle of radius $r=\\sum r_i$.", "tags": [ "geometry" ], - "last_edited": "30 December 2025", + "last_edited": "17 April 2026", "latex_path": "/latex/1121", "formalized": false, "formalized_url": "", "oeis_urls": [], "comments_problem_id": 1121, - "comments_count": 0 + "comments_count": 5 }, { "problem_id": 1122, @@ -25080,13 +25169,13 @@ "tags": [ "number theory" ], - "last_edited": "30 December 2025", + "last_edited": "01 April 2026", "latex_path": "/latex/1122", "formalized": false, "formalized_url": "", "oeis_urls": [], "comments_problem_id": 1122, - "comments_count": 0 + "comments_count": 1 }, { "problem_id": 1123, @@ -25100,7 +25189,7 @@ "tags": [ "algebra" ], - "last_edited": "31 December 2025", + "last_edited": "05 March 2026", "latex_path": "/latex/1123", "formalized": false, "formalized_url": "", @@ -25133,8 +25222,8 @@ "problem_url": "/1125", "status_bucket": "closed", "status_dom_id": "solved", - "status_label": "PROVED", - "status_detail": "This has been solved in the affirmative.", + "status_label": "PROVED (LEAN)", + "status_detail": "This has been solved in the affirmative and the proof verified in Lean.", "prize_amount": "", "statement": "Let $f:\\mathbb{R}\\to \\mathbb{R}$ be such that\\[2f(x) \\leq f(x+h)+f(x+2h)\\]for every $x\\in \\mathbb{R}$ and $h>0$. Must $f$ be monotonic?", "tags": [ @@ -25146,7 +25235,7 @@ "formalized_url": "", "oeis_urls": [], "comments_problem_id": 1125, - "comments_count": 0 + "comments_count": 1 }, { "problem_id": 1126, @@ -25273,7 +25362,7 @@ "formalized_url": "", "oeis_urls": [], "comments_problem_id": 1131, - "comments_count": 2 + "comments_count": 5 }, { "problem_id": 1132, @@ -25288,13 +25377,13 @@ "analysis", "polynomials" ], - "last_edited": "23 January 2026", + "last_edited": "01 April 2026", "latex_path": "/latex/1132", "formalized": false, "formalized_url": "", "oeis_urls": [], "comments_problem_id": 1132, - "comments_count": 1 + "comments_count": 2 }, { "problem_id": 1133, @@ -25315,7 +25404,7 @@ "formalized_url": "", "oeis_urls": [], "comments_problem_id": 1133, - "comments_count": 0 + "comments_count": 2 }, { "problem_id": 1134, @@ -25368,8 +25457,8 @@ "problem_url": "/1136", "status_bucket": "closed", "status_dom_id": "solved", - "status_label": "PROVED", - "status_detail": "This has been solved in the affirmative.", + "status_label": "PROVED (LEAN)", + "status_detail": "This has been solved in the affirmative and the proof verified in Lean.", "prize_amount": "", "statement": "Does there exist $A\\subset \\mathbb{N}$ with lower density $>1/3$ such that $a+b\\neq 2^k$ for any $a,b\\in A$ and $k\\geq 0$?", "tags": [ @@ -25381,7 +25470,7 @@ "formalized_url": "", "oeis_urls": [], "comments_problem_id": 1136, - "comments_count": 0 + "comments_count": 3 }, { "problem_id": 1137, @@ -25412,8 +25501,8 @@ "problem_url": "/1138", "status_bucket": "open", "status_dom_id": "open", - "status_label": "OPEN", - "status_detail": "This is open, and cannot be resolved with a finite computation.", + "status_label": "DISPROVED (LEAN)", + "status_detail": "This has been solved in the negative and the proof verified in Lean.", "prize_amount": "", "statement": "Let $x/21$. If $d=\\max_{p_n0$. Does there exist a constant $C_\\epsilon$ such that, for all primes $p$, every residue modulo $p$ is the sum of at most $C_\\epsilon$ many elements of\\[\\{ n^{-1} : 1\\leq n\\leq p^\\epsilon\\}\\]where $n^{-1}$ denotes the inverse of $n$ modulo $p$?", + "tags": [ + "number theory" + ], + "last_edited": "06 March 2026", + "latex_path": "/latex/1180", + "formalized": false, + "formalized_url": "", + "oeis_urls": [], + "comments_problem_id": 1180, + "comments_count": 0 + }, + { + "problem_id": 1181, + "problem_url": "/1181", + "status_bucket": "open", + "status_dom_id": "open", + "status_label": "OPEN", + "status_detail": "This is open, and cannot be resolved with a finite computation.", + "prize_amount": "", + "statement": "Let $q(n,k)$ denote the least prime which does not divide $\\prod_{1\\leq i\\leq k}(n+i)$. Is it true that there exists some $c>0$ such that, for all large $n$,\\[q(n,\\log n)<(1-c)(\\log n)^2?\\]", + "tags": [ + "number theory" + ], + "last_edited": "07 March 2026", + "latex_path": "/latex/1181", + "formalized": false, + "formalized_url": "", + "oeis_urls": [], + "comments_problem_id": 1181, + "comments_count": 0 + }, + { + "problem_id": 1182, + "problem_url": "/1182", + "status_bucket": "open", + "status_dom_id": "open", + "status_label": "OPEN", + "status_detail": "This is open, and cannot be resolved with a finite computation.", + "prize_amount": "", + "statement": "Let $f(n)$ be maximal such that there is a connected graph $G$ with $n$ vertices and $f(n)$ edges such that\\[R(K_3,G)= 2n-1.\\]Let $F(n)$ be maximal such that every connected graph $G$ with $n$ vertices and $\\leq F(n)$ edges has\\[R(K_3,G)= 2n-1.\\]Estimate $f(n)$ and $F(n)$. In particular, is it true that $F(n)/n\\to \\infty$?", + "tags": [ + "graph theory", + "ramsey theory" + ], + "last_edited": "11 April 2026", + "latex_path": "/latex/1182", + "formalized": false, + "formalized_url": "", + "oeis_urls": [], + "comments_problem_id": 1182, + "comments_count": 3 + }, + { + "problem_id": 1183, + "problem_url": "/1183", + "status_bucket": "open", + "status_dom_id": "open", + "status_label": "OPEN", + "status_detail": "This is open, and cannot be resolved with a finite computation.", + "prize_amount": "", + "statement": "Let $f(n)$ be maximal such that in any $2$-colouring of the subsets of $\\{1,\\ldots,n\\}$ there is always a monochromatic family of at least $f(n)$ sets which is closed under taking unions and intersections. Estimate $f(n)$. Let $F(n)$ be defined similarly, except that we only require the family be closed under taking unions. Estimate $F(n)$. In particular, is it true that $F(n)\\geq n^{\\omega(n)}$ for some $\\omega(n)\\to \\infty$ as $n\\to \\infty$, and $F(n)<(1+o(1))^n$?", + "tags": [ + "combinatorics", + "ramsey theory" + ], + "last_edited": "", + "latex_path": "/latex/1183", + "formalized": false, + "formalized_url": "", + "oeis_urls": [], + "comments_problem_id": 1183, + "comments_count": 10 + }, + { + "problem_id": 1184, + "problem_url": "/1184", + "status_bucket": "open", + "status_dom_id": "open", + "status_label": "OPEN", + "status_detail": "This is open, and cannot be resolved with a finite computation.", + "prize_amount": "", + "statement": "Let $f(n,k)$ count the number of $1\\leq i\\leq k$ such that $P(n+i)>k$ (where $P(m)$ is the largest prime divisor of $m$). Is it true that, if $\\alpha>1$ is such that $n=k^{\\alpha+o(1)}$, then\\[f(n,k)=(1-\\rho(\\alpha)+o(1))k,\\]where $\\rho$ is the Dickman function ?", + "tags": [ + "number theory", + "primes" + ], + "last_edited": "06 April 2026", + "latex_path": "/latex/1184", + "formalized": false, + "formalized_url": "", + "oeis_urls": [], + "comments_problem_id": 1184, + "comments_count": 1 + }, + { + "problem_id": 1185, + "problem_url": "/1185", + "status_bucket": "closed", + "status_dom_id": "solved", + "status_label": "SOLVED", + "status_detail": "This has been resolved in some other way than a proof or disproof.", + "prize_amount": "", + "statement": "Let $\\delta>0$ and $k\\geq 3$. Is it true that there exists $m\\geq 1$ (depending only on $\\delta$ and $k$) such that, for all large $N$, if $A,B\\subseteq \\{1,\\ldots,N\\}$ with $\\lvert A\\rvert \\geq \\delta N$ and $\\lvert B\\rvert \\geq m$ then there is a non-trivial $k$-term arithmetic progression in $A$ whose common difference is in $B-B$?", + "tags": [ + "additive combinatorics", + "arithmetic progressions" + ], + "last_edited": "05 April 2026", + "latex_path": "/latex/1185", + "formalized": false, + "formalized_url": "", + "oeis_urls": [], + "comments_problem_id": 1185, + "comments_count": 0 + }, + { + "problem_id": 1186, + "problem_url": "/1186", + "status_bucket": "open", + "status_dom_id": "open", + "status_label": "OPEN", + "status_detail": "This is open, and cannot be resolved with a finite computation.", + "prize_amount": "", + "statement": "Let $\\delta_k$ be such that in any $2$-colouring of $\\{1,\\ldots,n\\}$ there exist at least $(\\delta_k+o(1))n^2$ many monochromatic $k$-term arithmetic progressions. Give reasonable bounds (or even an asymptotic formula) for $\\delta_k$.", + "tags": [ + "additive combinatorics", + "arithmetic progressions" + ], + "last_edited": "08 April 2026", + "latex_path": "/latex/1186", + "formalized": false, + "formalized_url": "", + "oeis_urls": [], + "comments_problem_id": 1186, + "comments_count": 0 + }, + { + "problem_id": 1187, + "problem_url": "/1187", + "status_bucket": "closed", + "status_dom_id": "solved", + "status_label": "SOLVED", + "status_detail": "This has been resolved in some other way than a proof or disproof.", + "prize_amount": "", + "statement": "Let $k\\geq 3$. Is it true that, in any finite colouring of the integers, there are monochromatic arithmetic progressions of primes of length $k$? Are there monochromatic arithmetic progressions of length $k$ whose common difference is a prime?", + "tags": [ + "number theory", + "additive combinatorics", + "arithmetic progressions", + "primes" + ], + "last_edited": "08 April 2026", + "latex_path": "/latex/1187", + "formalized": false, + "formalized_url": "", + "oeis_urls": [], + "comments_problem_id": 1187, + "comments_count": 1 + }, + { + "problem_id": 1188, + "problem_url": "/1188", + "status_bucket": "open", + "status_dom_id": "open", + "status_label": "OPEN", + "status_detail": "This is open, and cannot be resolved with a finite computation.", + "prize_amount": "", + "statement": "Call a set of distinct integers $11$) form an irreducible covering set?", + "tags": [ + "number theory", + "covering systems" + ], + "last_edited": "08 April 2026", + "latex_path": "/latex/1189", + "formalized": false, + "formalized_url": "", + "oeis_urls": [], + "comments_problem_id": 1189, + "comments_count": 6 + }, + { + "problem_id": 1190, + "problem_url": "/1190", + "status_bucket": "open", + "status_dom_id": "open", + "status_label": "OPEN", + "status_detail": "This is open, and cannot be resolved with a finite computation.", + "prize_amount": "", + "statement": "Let\\[\\epsilon_m=\\max \\sum \\frac{1}{n_i}\\]where the maximum is taken over all finite sequences $m0\\]for some $c>0$?", + "tags": [ + "additive combinatorics", + "sidon sets" + ], + "last_edited": "06 April 2026", + "latex_path": "/latex/1191", + "formalized": false, + "formalized_url": "", + "oeis_urls": [], + "comments_problem_id": 1191, + "comments_count": 0 + }, + { + "problem_id": 1192, + "problem_url": "/1192", + "status_bucket": "open", + "status_dom_id": "open", + "status_label": "OPEN", + "status_detail": "This is open, and cannot be resolved with a finite computation.", + "prize_amount": "", + "statement": "For $A\\subset \\mathbb{N}$ let $f_r(n)$ count the number of solutions to $n=a_1+\\cdots+a_r$ with $a_i\\in A$. Does there exist, for all $r\\geq 2$, a basis $A$ of order $r$ (so that $f_r(n)>0$ for all large $n$) such that\\[\\sum_{n\\leq x}f_r(n)^2 \\ll x\\]for all $x$?", + "tags": [ + "additive combinatorics", + "additive basis" + ], + "last_edited": "06 April 2026", + "latex_path": "/latex/1192", + "formalized": false, + "formalized_url": "", + "oeis_urls": [], + "comments_problem_id": 1192, + "comments_count": 0 + }, + { + "problem_id": 1193, + "problem_url": "/1193", + "status_bucket": "closed", + "status_dom_id": "solved", + "status_label": "SOLVED (LEAN)", + "status_detail": "This has been resolved in some other way than a proof or disproof, and that resolution verified in Lean.", + "prize_amount": "", + "statement": "Let $A\\subset \\mathbb{N}$ and let $g(n)$ be a non-decreasing function of $n$ which is always $>0$. Is the lower density of\\[\\{ n : 1_A\\ast 1_A(n)=g(n)\\}\\]always $0$? Is the upper density always $0$, for all sufficiently large (depending on $x$) integers $n$ there exists an integer $r\\geq 1$ such that $nx\\in r\\cdot E$?", + "tags": [ + "analysis" + ], + "last_edited": "15 April 2026", + "latex_path": "/latex/1197", + "formalized": false, + "formalized_url": "", + "oeis_urls": [], + "comments_problem_id": 1197, + "comments_count": 4 + }, + { + "problem_id": 1198, + "problem_url": "/1198", + "status_bucket": "closed", + "status_dom_id": "solved", + "status_label": "DISPROVED", + "status_detail": "This has been solved in the negative.", + "prize_amount": "", + "statement": "If $\\mathbb{N}$ is $2$-coloured then must there exist an infinite set $A=\\{a_1<\\cdots\\}$ such that all expressions of the shape\\[\\prod_{i\\in S_1}a_i+\\cdots+\\prod_{i\\in S_k}a_i,\\]for disjoint $S_1,\\ldots,S_k$ (excluding the trivial expressions $a_i$) are the same colour?", + "tags": [ + "additive combinatorics", + "ramsey theory" + ], + "last_edited": "17 April 2026", + "latex_path": "/latex/1198", + "formalized": false, + "formalized_url": "", + "oeis_urls": [], + "comments_problem_id": 1198, + "comments_count": 4 + }, + { + "problem_id": 1199, + "problem_url": "/1199", + "status_bucket": "open", + "status_dom_id": "open", + "status_label": "OPEN", + "status_detail": "This is open, and cannot be resolved with a finite computation.", + "prize_amount": "", + "statement": "Is it true that in any $2$-colouring of $\\mathbb{N}$ there exists an infinite set $A$ such that all elements of $A+A$ are the same colour?", + "tags": [ + "additive combinatorics", + "ramsey theory" + ], + "last_edited": "", + "latex_path": "/latex/1199", + "formalized": true, + "formalized_url": "https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/ErdosProblems/1199.lean", + "oeis_urls": [], + "comments_problem_id": 1199, + "comments_count": 3 + }, + { + "problem_id": 1200, + "problem_url": "/1200", + "status_bucket": "open", + "status_dom_id": "open", + "status_label": "OPEN", + "status_detail": "This is open, and cannot be resolved with a finite computation.", + "prize_amount": "", + "statement": "There exists a constant $C$ such that for all large $x$ there is a collection of primes $p_1<\\ldots0$ there exists a $k$ such that the density of $n$ for which\\[P(n(n+1)\\cdots(n+k))>n^{1-\\epsilon}\\]is at least $1-\\eta$ (where $P(m)$ is the greatest prime divisor of $m$)?", + "tags": [ + "number theory", + "primes" + ], + "last_edited": "", + "latex_path": "/latex/1201", + "formalized": false, + "formalized_url": "", + "oeis_urls": [], + "comments_problem_id": 1201, + "comments_count": 10 + }, + { + "problem_id": 1202, + "problem_url": "/1202", + "status_bucket": "closed", + "status_dom_id": "solved", + "status_label": "SOLVED", + "status_detail": "This has been resolved in some other way than a proof or disproof.", + "prize_amount": "", + "statement": "Let $\\epsilon,\\eta>0$. Does there exist a $k$ such that, given any set of $k$ primes $p_1<\\cdots0$?", + "tags": [ + "geometry", + "distances" + ], + "last_edited": "08 April 2026", + "latex_path": "/latex/1207", + "formalized": false, + "formalized_url": "", + "oeis_urls": [], + "comments_problem_id": 1207, + "comments_count": 0 + }, + { + "problem_id": 1208, + "problem_url": "/1208", + "status_bucket": "open", + "status_dom_id": "open", + "status_label": "OPEN", + "status_detail": "This is open, and cannot be resolved with a finite computation.", + "prize_amount": "", + "statement": "For $d\\geq 2$ let $F_d(n)$ be minimal such that every set of $n$ points in $\\mathbb{R}^d$ contains a set of $F_d(n)$ points with distinct distances. Estimate $F_d(n)$ for fixed $d$ as $n\\to \\infty$.", + "tags": [ + "geometry", + "distances" + ], + "last_edited": "08 April 2026", + "latex_path": "/latex/1208", + "formalized": false, + "formalized_url": "", + "oeis_urls": [], + "comments_problem_id": 1208, + "comments_count": 0 + }, + { + "problem_id": 1209, + "problem_url": "/1209", + "status_bucket": "open", + "status_dom_id": "open", + "status_label": "OPEN", + "status_detail": "This is open, and cannot be resolved with a finite computation.", + "prize_amount": "", + "statement": "Let $A=\\{a_11$ and at least one of $x$ or $y$ is composite?", + "tags": [ + "number theory", + "primes" + ], + "last_edited": "08 April 2026", + "latex_path": "/latex/1212", + "formalized": false, + "formalized_url": "", + "oeis_urls": [], + "comments_problem_id": 1212, + "comments_count": 0 + }, + { + "problem_id": 1213, + "problem_url": "/1213", + "status_bucket": "closed", + "status_dom_id": "solved", + "status_label": "PROVED", + "status_detail": "This has been solved in the affirmative.", + "prize_amount": "", + "statement": "Let $a,K\\geq 1$. Does there exist $f(a,K)$ such that if\\[a=a_1<\\cdots f(a,K)$ and with bounded gaps $a_{i+1}-a_i\\leq K$ then there are two distinct intervals $I$ and $J$ such that\\[\\sum_{i\\in I}a_i=\\sum_{j\\in J}a_j?\\]", + "tags": [ + "additive combinatorics" + ], + "last_edited": "10 April 2026", + "latex_path": "/latex/1213", + "formalized": false, + "formalized_url": "", + "oeis_urls": [], + "comments_problem_id": 1213, + "comments_count": 0 + }, + { + "problem_id": 1214, + "problem_url": "/1214", + "status_bucket": "closed", + "status_dom_id": "solved", + "status_label": "PROVED", + "status_detail": "This has been solved in the affirmative.", + "prize_amount": "", + "statement": "Let $x,y\\geq 1$ be integers such that, for all $n\\geq 1$, the set of primes dividing $x^{n}-1$ is equal to set of primes dividing $y^n-1$. Must $x=y$?", + "tags": [ + "number theory" + ], + "last_edited": "12 April 2026", + "latex_path": "/latex/1214", + "formalized": true, + "formalized_url": "https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/ErdosProblems/1214.lean", + "oeis_urls": [], + "comments_problem_id": 1214, + "comments_count": 0 + }, + { + "problem_id": 1215, + "problem_url": "/1215", + "status_bucket": "closed", + "status_dom_id": "solved", + "status_label": "DISPROVED", + "status_detail": "This has been solved in the negative.", + "prize_amount": "", + "statement": "Does there exist a constant $C$ such that for every polynomial $P$ with $P(0)=1$, all of whose roots are on the unit circle, there exists a path in\\[\\{ z: \\lvert P(z)\\rvert < 1\\}\\]which connects $0$ to the unit circle of length at most $C$?", + "tags": [ + "analysis" + ], + "last_edited": "12 April 2026", + "latex_path": "/latex/1215", + "formalized": false, + "formalized_url": "", + "oeis_urls": [], + "comments_problem_id": 1215, + "comments_count": 3 + }, + { + "problem_id": 1216, + "problem_url": "/1216", + "status_bucket": "closed", + "status_dom_id": "solved", + "status_label": "DISPROVED", + "status_detail": "This has been solved in the negative.", + "prize_amount": "", + "statement": "A tournament is a complete directed graph. Let $f(n)$ be such that every tournament on $n$ vertices contains a transitive tournament on $f(n)$ vertices (i.e. one such that if $i\\to j\\to k$ then $i\\to k$). Is it true that $f(n)=\\lfloor \\log_2 n\\rfloor +1$?", + "tags": [ + "graph theory", + "ramsey theory" + ], + "last_edited": "12 April 2026", + "latex_path": "/latex/1216", + "formalized": false, + "formalized_url": "", + "oeis_urls": [], + "comments_problem_id": 1216, + "comments_count": 3 + }, + { + "problem_id": 1217, + "problem_url": "/1217", + "status_bucket": "closed", + "status_dom_id": "solved", + "status_label": "PROVED", + "status_detail": "This has been solved in the affirmative.", + "prize_amount": "", + "statement": "Let $A=\\{a_10$ for $0<\\alpha <1 $? The Schnirelmann density is defined by\\[d_s(A) = \\inf_{N\\geq 1}\\frac{\\lvert A\\cap\\{1,\\ldots,N\\}\\rvert}{N}.\\]", + "tags": [ + "number theory" + ], + "last_edited": "02 May 2026", + "latex_path": "/latex/38", + "formalized": true, + "formalized_url": "https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/ErdosProblems/38.lean", + "oeis_urls": [], + "comments_problem_id": 38, + "comments_count": 6 + }, + { + "problem_id": 42, + "problem_url": "/42", + "status_bucket": "closed", + "status_dom_id": "solved", + "status_label": "SOLVED (LEAN)", + "status_detail": "This has been resolved in some other way than a proof or disproof, and that resolution verified in Lean.", + "prize_amount": "", + "statement": "Let $M\\geq 1$ and $N$ be sufficiently large in terms of $M$. Is it true that for every Sidon set $A\\subset \\{1,\\ldots,N\\}$ there is another Sidon set $B\\subset \\{1,\\ldots,N\\}$ of size $M$ such that $(A-A)\\cap(B-B)=\\{0\\}$?", + "tags": [ + "number theory", + "sidon sets", + "additive combinatorics" + ], + "last_edited": "10 May 2026", + "latex_path": "/latex/42", + "formalized": true, + "formalized_url": "https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/ErdosProblems/42.lean", + "oeis_urls": [], + "comments_problem_id": 42, + "comments_count": 40 + }, + { + "problem_id": 43, + "problem_url": "/43", + "status_bucket": "closed", + "status_dom_id": "solved", + "status_label": "DISPROVED", + "status_detail": "This has been solved in the negative.", + "prize_amount": "$100", + "statement": "If $A,B\\subset \\{1,\\ldots,N\\}$ are two Sidon sets such that $(A-A)\\cap(B-B)=\\{0\\}$ then is it true that\\[ \\binom{\\lvert A\\rvert}{2}+\\binom{\\lvert B\\rvert}{2}\\leq\\binom{f(N)}{2}+O(1),\\]where $f(N)$ is the maximum possible size of a Sidon set in $\\{1,\\ldots,N\\}$? If $\\lvert A\\rvert=\\lvert B\\rvert$ then can this bound be improved to\\[\\binom{\\lvert A\\rvert}{2}+\\binom{\\lvert B\\rvert}{2}\\leq (1-c+o(1))\\binom{f(N)}{2}\\]for some constant $c>0$?", + "tags": [ + "number theory", + "sidon sets", + "additive combinatorics" + ], + "last_edited": "10 May 2026", + "latex_path": "/latex/43", + "formalized": true, + "formalized_url": "https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/ErdosProblems/43.lean", + "oeis_urls": [ + "https://oeis.org/A003022", + "https://oeis.org/A143824", + "https://oeis.org/A227590" + ], + "comments_problem_id": 43, + "comments_count": 11 + }, { "problem_id": 45, "problem_url": "/45", "status_bucket": "closed", "status_dom_id": "solved", - "status_label": "PROVED", - "status_detail": "This has been solved in the affirmative.", + "status_label": "PROVED (LEAN)", + "status_detail": "This has been solved in the affirmative and the proof verified in Lean.", "prize_amount": "", "statement": "Let $k\\geq 2$. Is there an integer $n_k$ such that, if $D=\\{ 11$ there is an associated $a_d$ such that every integer is congruent to some $a_d\\pmod{d}$, and if there is some integer $x$ with\\[x\\equiv a_d\\pmod{d}\\textrm{ and }x\\equiv a_{d'}\\pmod{d'}\\]then $(d,d')=1$.", "tags": [ @@ -2225,7 +2437,7 @@ "formalized_url": "https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/ErdosProblems/204.lean", "oeis_urls": [], "comments_problem_id": 204, - "comments_count": 0 + "comments_count": 1 }, { "problem_id": 205, @@ -2239,21 +2451,21 @@ "tags": [ "number theory" ], - "last_edited": "23 January 2026", + "last_edited": "05 April 2026", "latex_path": "/latex/205", "formalized": false, "formalized_url": "", "oeis_urls": [], "comments_problem_id": 205, - "comments_count": 21 + "comments_count": 22 }, { "problem_id": 206, "problem_url": "/206", "status_bucket": "closed", "status_dom_id": "solved", - "status_label": "DISPROVED", - "status_detail": "This has been solved in the negative.", + "status_label": "DISPROVED (LEAN)", + "status_detail": "This has been solved in the negative and the proof verified in Lean.", "prize_amount": "", "statement": "Let $x>0$ be a real number. For any $n\\geq 1$ let\\[R_n(x) = \\sum_{i=1}^n\\frac{1}{m_i}cn$ but there is no covering system whose moduli all divide $n$?", + "statement": "Is it true that, for every $c$, there exists an $n$ such that $\\sigma(n)>cn$ but there is no covering system whose moduli all distinct divisors of $n$ (which are $>1$)?", "tags": [ "number theory", "covering systems" ], - "last_edited": "06 October 2025", + "last_edited": "10 April 2026", "latex_path": "/latex/277", "formalized": true, "formalized_url": "https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/ErdosProblems/277.lean", @@ -3046,8 +3279,8 @@ "problem_url": "/280", "status_bucket": "closed", "status_dom_id": "solved", - "status_label": "DISPROVED", - "status_detail": "This has been solved in the negative.", + "status_label": "DISPROVED (LEAN)", + "status_detail": "This has been solved in the negative and the proof verified in Lean.", "prize_amount": "", "statement": "Let $n_10$ we have $n_k>(1+\\epsilon)k\\log k$ for all $k$. Then\\[\\#\\{ m0$ and $k\\geq 2$. Is it true that, for all sufficiently large $n$, there is a sequence of $k$ consecutive integers in $\\{1,\\ldots,n\\}$ all of which are $n^\\epsilon$-smooth?", + "tags": [ + "number theory" + ], + "last_edited": "28 April 2026", + "latex_path": "/latex/369", + "formalized": false, + "formalized_url": "", + "oeis_urls": [], + "comments_problem_id": 369, + "comments_count": 11 }, { "problem_id": 370, @@ -3890,6 +4253,31 @@ "comments_problem_id": 379, "comments_count": 8 }, + { + "problem_id": 380, + "problem_url": "/380", + "status_bucket": "closed", + "status_dom_id": "solved", + "status_label": "PROVED", + "status_detail": "This has been solved in the affirmative.", + "prize_amount": "", + "statement": "We call an interval $[u,v]$ 'bad' if the greatest prime factor of $\\prod_{u\\leq m\\leq v}m$ occurs with an exponent greater than $1$. Let $B(x)$ count the number of $n\\leq x$ which are contained in at least one bad interval. Is it true that\\[B(x)\\sim \\#\\{ n\\leq x: P(n)^2\\mid n\\},\\]where $P(n)$ is the largest prime factor of $n$?", + "tags": [ + "number theory" + ], + "last_edited": "10 April 2026", + "latex_path": "/latex/380", + "formalized": false, + "formalized_url": "", + "oeis_urls": [ + "https://oeis.org/A070003", + "https://oeis.org/A387054", + "https://oeis.org/A388654", + "https://oeis.org/A389100" + ], + "comments_problem_id": 380, + "comments_count": 10 + }, { "problem_id": 381, "problem_url": "/381", @@ -4060,7 +4448,7 @@ "formalized_url": "", "oeis_urls": [], "comments_problem_id": 401, - "comments_count": 18 + "comments_count": 21 }, { "problem_id": 402, @@ -4074,7 +4462,7 @@ "tags": [ "number theory" ], - "last_edited": "", + "last_edited": "08 April 2026", "latex_path": "/latex/402", "formalized": true, "formalized_url": "https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/ErdosProblems/402.lean", @@ -4195,8 +4583,8 @@ "problem_url": "/426", "status_bucket": "closed", "status_dom_id": "solved", - "status_label": "DISPROVED", - "status_detail": "This has been solved in the negative.", + "status_label": "DISPROVED (LEAN)", + "status_detail": "This has been solved in the negative and the proof verified in Lean.", "prize_amount": "$25", "statement": "We say $H$ is a unique subgraph of $G$ if there is exactly one way to find $H$ as a subgraph (not necessarily induced) of $G$. Is there a graph on $n$ vertices with\\[\\gg \\frac{2^{\\binom{n}{2}}}{n!}\\]many distinct unique subgraphs?", "tags": [ @@ -4208,15 +4596,15 @@ "formalized_url": "", "oeis_urls": [], "comments_problem_id": 426, - "comments_count": 0 + "comments_count": 2 }, { "problem_id": 427, "problem_url": "/427", "status_bucket": "closed", "status_dom_id": "solved", - "status_label": "PROVED", - "status_detail": "This has been solved in the affirmative.", + "status_label": "PROVED (LEAN)", + "status_detail": "This has been solved in the affirmative and the proof verified in Lean.", "prize_amount": "", "statement": "Is it true that, for every $n$ and $d$, there exists $k$ such that\\[d \\mid p_{n+1}+\\cdots+p_{n+k},\\]where $p_r$ denotes the $r$th prime?", "tags": [ @@ -4229,7 +4617,7 @@ "formalized_url": "https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/ErdosProblems/427.lean", "oeis_urls": [], "comments_problem_id": 427, - "comments_count": 1 + "comments_count": 2 }, { "problem_id": 429, @@ -4243,7 +4631,7 @@ "tags": [ "number theory" ], - "last_edited": "", + "last_edited": "08 April 2026", "latex_path": "/latex/429", "formalized": false, "formalized_url": "", @@ -4345,7 +4733,7 @@ "tags": [ "number theory" ], - "last_edited": "18 November 2025", + "last_edited": "07 April 2026", "latex_path": "/latex/438", "formalized": false, "formalized_url": "", @@ -4368,7 +4756,7 @@ "number theory", "ramsey theory" ], - "last_edited": "", + "last_edited": "07 April 2026", "latex_path": "/latex/439", "formalized": false, "formalized_url": "", @@ -4576,7 +4964,7 @@ "tags": [ "number theory" ], - "last_edited": "26 October 2025", + "last_edited": "08 April 2026", "latex_path": "/latex/453", "formalized": false, "formalized_url": "", @@ -4584,13 +4972,35 @@ "comments_problem_id": 453, "comments_count": 1 }, + { + "problem_id": 457, + "problem_url": "/457", + "status_bucket": "closed", + "status_dom_id": "solved", + "status_label": "PROVED (LEAN)", + "status_detail": "This has been solved in the affirmative and the proof verified in Lean.", + "prize_amount": "", + "statement": "Is there some $\\epsilon>0$ such that there are infinitely many $n$ where all primes $p\\leq (2+\\epsilon)\\log n$ divide\\[\\prod_{1\\leq i\\leq \\log n}(n+i)?\\]", + "tags": [ + "number theory" + ], + "last_edited": "07 March 2026", + "latex_path": "/latex/457", + "formalized": true, + "formalized_url": "https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/ErdosProblems/457.lean", + "oeis_urls": [ + "https://oeis.org/A391668" + ], + "comments_problem_id": 457, + "comments_count": 10 + }, { "problem_id": 459, "problem_url": "/459", "status_bucket": "closed", "status_dom_id": "solved", - "status_label": "SOLVED", - "status_detail": "This has been resolved in some other way than a proof or disproof.", + "status_label": "SOLVED (LEAN)", + "status_detail": "This has been resolved in some other way than a proof or disproof, and that resolution verified in Lean.", "prize_amount": "", "statement": "Let $f(u)$ be the largest $v$ such that no $m\\in (u,v)$ is composed entirely of primes dividing $uv$. Estimate $f(u)$.", "tags": [ @@ -4605,7 +5015,7 @@ "https://oeis.org/A289280" ], "comments_problem_id": 459, - "comments_count": 2 + "comments_count": 3 }, { "problem_id": 464, @@ -4665,7 +5075,7 @@ "formalized_url": "", "oeis_urls": [], "comments_problem_id": 466, - "comments_count": 0 + "comments_count": 1 }, { "problem_id": 471, @@ -4797,8 +5207,8 @@ "problem_url": "/484", "status_bucket": "closed", "status_dom_id": "solved", - "status_label": "PROVED", - "status_detail": "This has been solved in the affirmative.", + "status_label": "PROVED (LEAN)", + "status_detail": "This has been solved in the affirmative and the proof verified in Lean.", "prize_amount": "", "statement": "Prove that there exists an absolute constant $c>0$ such that, whenever $\\{1,\\ldots,N\\}$ is $k$-coloured (and $N$ is large enough depending on $k$) then there are at least $cN$ many integers in $\\{1,\\ldots,N\\}$ which are representable as a monochromatic sum (that is, $a+b$ where $a,b\\in \\{1,\\ldots,N\\}$ are in the same colour class and $a\\neq b$).", "tags": [ @@ -4806,13 +5216,13 @@ "additive combinatorics", "ramsey theory" ], - "last_edited": "", + "last_edited": "08 April 2026", "latex_path": "/latex/484", "formalized": false, "formalized_url": "", "oeis_urls": [], "comments_problem_id": 484, - "comments_count": 0 + "comments_count": 1 }, { "problem_id": 485, @@ -4827,7 +5237,7 @@ "analysis", "polynomials" ], - "last_edited": "29 December 2025", + "last_edited": "08 April 2026", "latex_path": "/latex/485", "formalized": false, "formalized_url": "", @@ -4873,7 +5283,7 @@ "formalized_url": "", "oeis_urls": [], "comments_problem_id": 490, - "comments_count": 0 + "comments_count": 3 }, { "problem_id": 491, @@ -4887,13 +5297,13 @@ "tags": [ "number theory" ], - "last_edited": "30 December 2025", + "last_edited": "01 April 2026", "latex_path": "/latex/491", "formalized": false, "formalized_url": "", "oeis_urls": [], "comments_problem_id": 491, - "comments_count": 1 + "comments_count": 0 }, { "problem_id": 492, @@ -4954,7 +5364,7 @@ "formalized_url": "https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/ErdosProblems/494.lean", "oeis_urls": [], "comments_problem_id": 494, - "comments_count": 2 + "comments_count": 3 }, { "problem_id": 496, @@ -5097,8 +5507,8 @@ ], "last_edited": "30 December 2025", "latex_path": "/latex/505", - "formalized": false, - "formalized_url": "", + "formalized": true, + "formalized_url": "https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/ErdosProblems/505.lean", "oeis_urls": [], "comments_problem_id": 505, "comments_count": 1 @@ -5229,8 +5639,8 @@ "problem_url": "/519", "status_bucket": "closed", "status_dom_id": "solved", - "status_label": "PROVED", - "status_detail": "This has been solved in the affirmative.", + "status_label": "PROVED (LEAN)", + "status_detail": "This has been solved in the affirmative and the proof verified in Lean.", "prize_amount": "", "statement": "Let $z_1,\\ldots,z_n\\in \\mathbb{C}$ with $z_1=1$. Must there exist an absolute constant $c>0$ such that\\[\\max_{1\\leq k\\leq n}\\left\\lvert \\sum_{i}z_i^k\\right\\rvert>c?\\]", "tags": [ @@ -5242,7 +5652,7 @@ "formalized_url": "", "oeis_urls": [], "comments_problem_id": 519, - "comments_count": 2 + "comments_count": 3 }, { "problem_id": 523, @@ -5384,7 +5794,7 @@ "number theory", "intersecting family" ], - "last_edited": "", + "last_edited": "08 April 2026", "latex_path": "/latex/534", "formalized": false, "formalized_url": "", @@ -5420,14 +5830,14 @@ "problem_url": "/540", "status_bucket": "closed", "status_dom_id": "solved", - "status_label": "PROVED", - "status_detail": "This has been solved in the affirmative.", + "status_label": "PROVED (LEAN)", + "status_detail": "This has been solved in the affirmative and the proof verified in Lean.", "prize_amount": "", "statement": "Is it true that if $A\\subseteq \\mathbb{Z}/N\\mathbb{Z}$ has size $\\gg N^{1/2}$ then there exists some non-empty $S\\subseteq A$ such that $\\sum_{n\\in S}n\\equiv 0\\pmod{N}$?", "tags": [ "number theory" ], - "last_edited": "28 December 2025", + "last_edited": "06 March 2026", "latex_path": "/latex/540", "formalized": false, "formalized_url": "", @@ -5435,7 +5845,7 @@ "https://oeis.org/A034463" ], "comments_problem_id": 540, - "comments_count": 1 + "comments_count": 2 }, { "problem_id": 541, @@ -5449,13 +5859,13 @@ "tags": [ "number theory" ], - "last_edited": "", + "last_edited": "08 April 2026", "latex_path": "/latex/541", "formalized": true, "formalized_url": "https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/ErdosProblems/541.lean", "oeis_urls": [], "comments_problem_id": 541, - "comments_count": 4 + "comments_count": 5 }, { "problem_id": 542, @@ -5469,7 +5879,7 @@ "tags": [ "number theory" ], - "last_edited": "17 October 2025", + "last_edited": "08 April 2026", "latex_path": "/latex/542", "formalized": false, "formalized_url": "", @@ -5538,7 +5948,7 @@ "formalized_url": "", "oeis_urls": [], "comments_problem_id": 547, - "comments_count": 2 + "comments_count": 3 }, { "problem_id": 549, @@ -5691,6 +6101,27 @@ "comments_problem_id": 570, "comments_count": 3 }, + { + "problem_id": 574, + "problem_url": "/574", + "status_bucket": "closed", + "status_dom_id": "solved", + "status_label": "DISPROVED", + "status_detail": "This has been solved in the negative.", + "prize_amount": "", + "statement": "Is it true that, for $k\\geq 2$,\\[\\mathrm{ex}(n;\\{C_{2k-1},C_{2k}\\})=(1+o(1))(n/2)^{1+\\frac{1}{k}}.\\]", + "tags": [ + "graph theory", + "turan number" + ], + "last_edited": "01 April 2026", + "latex_path": "/latex/574", + "formalized": false, + "formalized_url": "", + "oeis_urls": [], + "comments_problem_id": 574, + "comments_count": 1 + }, { "problem_id": 577, "problem_url": "/577", @@ -5995,8 +6426,8 @@ ], "last_edited": "01 December 2025", "latex_path": "/latex/613", - "formalized": false, - "formalized_url": "", + "formalized": true, + "formalized_url": "https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/ErdosProblems/613.lean", "oeis_urls": [], "comments_problem_id": 613, "comments_count": 4 @@ -6047,8 +6478,8 @@ "problem_url": "/621", "status_bucket": "closed", "status_dom_id": "solved", - "status_label": "PROVED", - "status_detail": "This has been solved in the affirmative.", + "status_label": "PROVED (LEAN)", + "status_detail": "This has been solved in the affirmative and the proof verified in Lean.", "prize_amount": "", "statement": "Let $G$ be a graph on $n$ vertices, $\\alpha_1(G)$ be the maximum number of edges that contain at most one edge from every triangle, and $\\tau_1(G)$ be the minimum number of edges that contain at least one edge from every triangle. Is it true that\\[\\alpha_1(G)+\\tau_1(G) \\leq \\frac{n^2}{4}?\\]", "tags": [ @@ -6060,7 +6491,7 @@ "formalized_url": "", "oeis_urls": [], "comments_problem_id": 621, - "comments_count": 1 + "comments_count": 2 }, { "problem_id": 622, @@ -6145,6 +6576,26 @@ "comments_problem_id": 632, "comments_count": 0 }, + { + "problem_id": 633, + "problem_url": "/633", + "status_bucket": "closed", + "status_dom_id": "solved", + "status_label": "SOLVED", + "status_detail": "This has been resolved in some other way than a proof or disproof.", + "prize_amount": "$25", + "statement": "Classify those triangles which can only be cut into a square number of congruent triangles.", + "tags": [ + "geometry" + ], + "last_edited": "08 April 2026", + "latex_path": "/latex/633", + "formalized": true, + "formalized_url": "https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/ErdosProblems/633.lean", + "oeis_urls": [], + "comments_problem_id": 633, + "comments_count": 7 + }, { "problem_id": 636, "problem_url": "/636", @@ -6192,8 +6643,8 @@ "problem_url": "/639", "status_bucket": "closed", "status_dom_id": "solved", - "status_label": "PROVED", - "status_detail": "This has been solved in the affirmative.", + "status_label": "PROVED (LEAN)", + "status_detail": "This has been solved in the affirmative and the proof verified in Lean.", "prize_amount": "", "statement": "Is it true that if the edges of $K_n$ are 2-coloured then there are at most $n^2/4$ many edges which do not occur in a monochromatic triangle?", "tags": [ @@ -6206,7 +6657,7 @@ "formalized_url": "", "oeis_urls": [], "comments_problem_id": 639, - "comments_count": 0 + "comments_count": 1 }, { "problem_id": 641, @@ -6240,9 +6691,10 @@ "tags": [ "number theory", "additive combinatorics", - "ramsey theory" + "ramsey theory", + "arithmetic progressions" ], - "last_edited": "24 October 2025", + "last_edited": "04 April 2026", "latex_path": "/latex/645", "formalized": true, "formalized_url": "https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/ErdosProblems/645.lean", @@ -6291,7 +6743,7 @@ "https://oeis.org/A391750" ], "comments_problem_id": 648, - "comments_count": 2 + "comments_count": 3 }, { "problem_id": 649, @@ -6313,6 +6765,28 @@ "comments_problem_id": 649, "comments_count": 2 }, + { + "problem_id": 650, + "problem_url": "/650", + "status_bucket": "closed", + "status_dom_id": "solved", + "status_label": "SOLVED (LEAN)", + "status_detail": "This has been resolved in some other way than a proof or disproof, and that resolution verified in Lean.", + "prize_amount": "", + "statement": "Let $f(m)$ be such that if $A\\subseteq \\{1,\\ldots,N\\}$ has $\\lvert A\\rvert=m$ then every interval in $[1,\\infty)$ of length $2N$ contains $\\geq f(m)$ many distinct integers $b_1,\\ldots,b_r$ where each $b_i$ is divisible by some $a_i\\in A$, where $a_1,\\ldots,a_r$ are distinct. Estimate $f(m)$. In particular is it true that $f(m)\\leq \\sqrt{m}$?", + "tags": [ + "number theory" + ], + "last_edited": "02 April 2026", + "latex_path": "/latex/650", + "formalized": false, + "formalized_url": "", + "oeis_urls": [ + "https://oeis.org/A027434" + ], + "comments_problem_id": 650, + "comments_count": 28 + }, { "problem_id": 651, "problem_url": "/651", @@ -6363,12 +6837,12 @@ "status_label": "PROVED", "status_detail": "This has been solved in the affirmative.", "prize_amount": "", - "statement": "Let $A\\subset \\mathbb{N}$ be a set with positive upper density. Must there exist an infinite set $B$ and integer $t$ such that\\[B+B+t\\subseteq A?\\]", + "statement": "Let $A\\subset \\mathbb{N}$ be a set with positive upper density. Must there exist an infinite set $B\\subseteq A$ and integer $t$ such that\\[\\{b_1+b_2: b_1\\neq b_2\\in B\\}+t\\subseteq A?\\]", "tags": [ "number theory", "additive combinatorics" ], - "last_edited": "", + "last_edited": "07 April 2026", "latex_path": "/latex/656", "formalized": false, "formalized_url": "", @@ -6381,8 +6855,8 @@ "problem_url": "/658", "status_bucket": "closed", "status_dom_id": "solved", - "status_label": "PROVED", - "status_detail": "This has been solved in the affirmative.", + "status_label": "PROVED (LEAN)", + "status_detail": "This has been solved in the affirmative and the proof verified in Lean.", "prize_amount": "", "statement": "Let $\\delta>0$ and $N$ be sufficiently large depending on $\\delta$. Is it true that if $A\\subseteq \\{1,\\ldots,N\\}^2$ has $\\lvert A\\rvert \\geq \\delta N^2$ then $A$ must contain the vertices of a square?", "tags": [ @@ -6394,7 +6868,7 @@ "formalized_url": "", "oeis_urls": [], "comments_problem_id": 658, - "comments_count": 0 + "comments_count": 2 }, { "problem_id": 659, @@ -6541,13 +7015,33 @@ "comments_problem_id": 682, "comments_count": 0 }, + { + "problem_id": 690, + "problem_url": "/690", + "status_bucket": "closed", + "status_dom_id": "solved", + "status_label": "SOLVED", + "status_detail": "This has been resolved in some other way than a proof or disproof.", + "prize_amount": "", + "statement": "Let $d_k(p)$ be the density of those integers whose $k$th smallest prime factor is $p$ (i.e. if $p_1n+1$ (i.e. increases until some $m$ then decreases thereafter)? For fixed $n$, where does $\\delta_1(n,m)$ achieve its maximum?", "tags": [ @@ -6560,7 +7054,48 @@ "formalized_url": "", "oeis_urls": [], "comments_problem_id": 692, - "comments_count": 0 + "comments_count": 1 + }, + { + "problem_id": 694, + "problem_url": "/694", + "status_bucket": "closed", + "status_dom_id": "solved", + "status_label": "SOLVED (LEAN)", + "status_detail": "This has been resolved in some other way than a proof or disproof, and that resolution verified in Lean.", + "prize_amount": "", + "statement": "Let $f_{\\max}(n)$ be the largest $m$ such that $\\phi(m)=n$, and $f_{\\min}(n)$ be the smallest such $m$, where $\\phi$ is Euler's totient function. Investigate\\[\\max_{n\\leq x}\\frac{f_{\\max}(n)}{f_{\\min}(n)}\\](where the maximum is restricted to those $n$ of the form $n=\\phi(m)$ for some $m$.)", + "tags": [ + "number theory" + ], + "last_edited": "02 May 2026", + "latex_path": "/latex/694", + "formalized": true, + "formalized_url": "https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/ErdosProblems/694.lean", + "oeis_urls": [], + "comments_problem_id": 694, + "comments_count": 13 + }, + { + "problem_id": 696, + "problem_url": "/696", + "status_bucket": "closed", + "status_dom_id": "solved", + "status_label": "SOLVED (LEAN)", + "status_detail": "This has been resolved in some other way than a proof or disproof, and that resolution verified in Lean.", + "prize_amount": "", + "statement": "Let $h(n)$ be the largest $\\ell$ such that there is a sequence of primes $p_1<\\cdots < p_\\ell$ all dividing $n$ with $p_{i+1}\\equiv 1\\pmod{p_i}$. Let $H(n)$ be the largest $u$ such that there is a sequence of integers $d_1<\\cdots < d_u$ all dividing $n$ with $d_{i+1}\\equiv 1\\pmod{d_i}$. Estimate $h(n)$ and $H(n)$. Is it true that $H(n)/h(n)\\to \\infty$ for almost all $n$?", + "tags": [ + "number theory", + "divisors" + ], + "last_edited": "11 May 2026", + "latex_path": "/latex/696", + "formalized": false, + "formalized_url": "", + "oeis_urls": [], + "comments_problem_id": 696, + "comments_count": 11 }, { "problem_id": 697, @@ -6782,7 +7317,7 @@ "graph theory", "ramsey theory" ], - "last_edited": "26 October 2025", + "last_edited": "07 March 2026", "latex_path": "/latex/720", "formalized": false, "formalized_url": "", @@ -6804,7 +7339,7 @@ "additive combinatorics", "ramsey theory" ], - "last_edited": "", + "last_edited": "04 April 2026", "latex_path": "/latex/721", "formalized": false, "formalized_url": "", @@ -6832,7 +7367,7 @@ "formalized_url": "", "oeis_urls": [], "comments_problem_id": 722, - "comments_count": 0 + "comments_count": 1 }, { "problem_id": 728, @@ -6853,7 +7388,7 @@ "formalized_url": "https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/ErdosProblems/728.lean", "oeis_urls": [], "comments_problem_id": 728, - "comments_count": 84 + "comments_count": 86 }, { "problem_id": 729, @@ -6958,6 +7493,26 @@ "comments_problem_id": 737, "comments_count": 1 }, + { + "problem_id": 741, + "problem_url": "/741", + "status_bucket": "closed", + "status_dom_id": "solved", + "status_label": "SOLVED (LEAN)", + "status_detail": "This has been resolved in some other way than a proof or disproof, and that resolution verified in Lean.", + "prize_amount": "", + "statement": "Let $A\\subseteq \\mathbb{N}$ be such that $A+A$ has positive (upper) density. Can one always decompose $A=A_1\\sqcup A_2$ such that $A_1+A_1$ and $A_2+A_2$ both have positive (upper) density? Is there a basis $A$ of order $2$ such that if $A=A_1\\sqcup A_2$ then $A_1+A_1$ and $A_2+A_2$ cannot both have bounded gaps?", + "tags": [ + "additive combinatorics" + ], + "last_edited": "02 May 2026", + "latex_path": "/latex/741", + "formalized": true, + "formalized_url": "https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/ErdosProblems/741.lean", + "oeis_urls": [], + "comments_problem_id": 741, + "comments_count": 6 + }, { "problem_id": 742, "problem_url": "/742", @@ -7129,8 +7684,8 @@ "problem_url": "/753", "status_bucket": "closed", "status_dom_id": "solved", - "status_label": "DISPROVED", - "status_detail": "This has been solved in the negative.", + "status_label": "DISPROVED (LEAN)", + "status_detail": "This has been solved in the negative and the proof verified in Lean.", "prize_amount": "", "statement": "The list chromatic number $\\chi_L(G)$ is defined to be the minimal $k$ such that for any assignment of a list of $k$ colours to each vertex of $G$ (perhaps different lists for different vertices) a colouring of each vertex by a colour on its list can be chosen such that adjacent vertices receive distinct colours. Does there exist some constant $c>0$ such that\\[\\chi_L(G)+\\chi_L(G^c)> n^{1/2+c}\\]for every graph $G$ on $n$ vertices (where $G^c$ is the complement of $G$)?", "tags": [ @@ -7143,7 +7698,7 @@ "formalized_url": "", "oeis_urls": [], "comments_problem_id": 753, - "comments_count": 0 + "comments_count": 1 }, { "problem_id": 754, @@ -7191,8 +7746,8 @@ "problem_url": "/756", "status_bucket": "closed", "status_dom_id": "solved", - "status_label": "PROVED", - "status_detail": "This has been solved in the affirmative.", + "status_label": "PROVED (LEAN)", + "status_detail": "This has been solved in the affirmative and the proof verified in Lean.", "prize_amount": "", "statement": "Let $A\\subset \\mathbb{R}^2$ be a set of $n$ points. Can there be $\\gg n$ many distinct distances each of which occurs for more than $n$ many pairs from $A$?", "tags": [ @@ -7205,7 +7760,7 @@ "formalized_url": "", "oeis_urls": [], "comments_problem_id": 756, - "comments_count": 0 + "comments_count": 1 }, { "problem_id": 758, @@ -7254,21 +7809,21 @@ "problem_url": "/760", "status_bucket": "closed", "status_dom_id": "solved", - "status_label": "PROVED", - "status_detail": "This has been solved in the affirmative.", + "status_label": "PROVED (LEAN)", + "status_detail": "This has been solved in the affirmative and the proof verified in Lean.", "prize_amount": "", - "statement": "The cochromatic number of $G$, denoted by $\\zeta(G)$, is the minimum number of colours needed to colour the vertices of $G$ such that each colour class induces either a complete graph or empty graph. If $G$ is a graph with chromatic number $\\chi(G)=m$ then must $G$ contain a subgraph $H$ with\\[\\zeta(H) \\gg \\frac{m}{\\log m}?\\]", + "statement": "The cochromatic number of $G$, denoted by $\\zeta(G)$, is the minimum number of colours needed to colour the vertices of $G$ such that each colour class induces either a complete graph or independent set. If $G$ is a graph with chromatic number $\\chi(G)=m$ then must $G$ contain a subgraph $H$ with\\[\\zeta(H) \\gg \\frac{m}{\\log m}?\\]", "tags": [ "graph theory", "chromatic number" ], - "last_edited": "", + "last_edited": "24 April 2026", "latex_path": "/latex/760", "formalized": false, "formalized_url": "", "oeis_urls": [], "comments_problem_id": 760, - "comments_count": 0 + "comments_count": 2 }, { "problem_id": 762, @@ -7338,8 +7893,8 @@ "problem_url": "/765", "status_bucket": "closed", "status_dom_id": "solved", - "status_label": "SOLVED", - "status_detail": "This has been resolved in some other way than a proof or disproof.", + "status_label": "SOLVED (LEAN)", + "status_detail": "This has been resolved in some other way than a proof or disproof, and that resolution verified in Lean.", "prize_amount": "", "statement": "Give an asymptotic formula for $\\mathrm{ex}(n;C_4)$.", "tags": [ @@ -7354,7 +7909,7 @@ "https://oeis.org/A006855" ], "comments_problem_id": 765, - "comments_count": 3 + "comments_count": 4 }, { "problem_id": 767, @@ -7424,8 +7979,8 @@ "problem_url": "/775", "status_bucket": "closed", "status_dom_id": "solved", - "status_label": "DISPROVED", - "status_detail": "This has been solved in the negative.", + "status_label": "DISPROVED (LEAN)", + "status_detail": "This has been solved in the negative and the proof verified in Lean.", "prize_amount": "", "statement": "Is there a $3$-uniform hypergraph on $n$ vertices which contains at least $n-O(1)$ different sizes of cliques (maximal complete subgraphs)", "tags": [ @@ -7438,7 +7993,7 @@ "formalized_url": "", "oeis_urls": [], "comments_problem_id": 775, - "comments_count": 1 + "comments_count": 2 }, { "problem_id": 777, @@ -7503,6 +8058,26 @@ "comments_problem_id": 781, "comments_count": 0 }, + { + "problem_id": 783, + "problem_url": "/783", + "status_bucket": "closed", + "status_dom_id": "solved", + "status_label": "SOLVED", + "status_detail": "This has been resolved in some other way than a proof or disproof.", + "prize_amount": "", + "statement": "Fix some constant $C>0$ and let $N$ be large. Let $A\\subseteq \\{2,\\ldots,N\\}$ be such that $(a,b)=1$ for all $a\\neq b\\in A$ and $\\sum_{n\\in A}\\frac{1}{n}\\leq C$. What choice of such an $A$ minimises the number of integers $m\\leq N$ not divisible by any $a\\in A$?", + "tags": [ + "number theory" + ], + "last_edited": "11 April 2026", + "latex_path": "/latex/783", + "formalized": false, + "formalized_url": "", + "oeis_urls": [], + "comments_problem_id": 783, + "comments_count": 28 + }, { "problem_id": 784, "problem_url": "/784", @@ -7515,7 +8090,7 @@ "tags": [ "number theory" ], - "last_edited": "20 December 2025", + "last_edited": "08 April 2026", "latex_path": "/latex/784", "formalized": false, "formalized_url": "", @@ -7528,20 +8103,20 @@ "problem_url": "/785", "status_bucket": "closed", "status_dom_id": "solved", - "status_label": "PROVED", - "status_detail": "This has been solved in the affirmative.", + "status_label": "PROVED (LEAN)", + "status_detail": "This has been solved in the affirmative and the proof verified in Lean.", "prize_amount": "", "statement": "Let $A,B\\subseteq \\mathbb{N}$ be infinite sets such that $A+B$ contains all large integers. Let $A(x)=\\lvert A\\cap [1,x]\\rvert$ and similarly for $B(x)$. Is it true that if $A(x)B(x)\\sim x$ then\\[A(x)B(x)-x\\to \\infty\\]as $x\\to \\infty$?", "tags": [ "additive combinatorics" ], - "last_edited": "", + "last_edited": "07 March 2026", "latex_path": "/latex/785", "formalized": false, "formalized_url": "", "oeis_urls": [], "comments_problem_id": 785, - "comments_count": 2 + "comments_count": 5 }, { "problem_id": 794, @@ -7577,13 +8152,13 @@ "tags": [ "number theory" ], - "last_edited": "", + "last_edited": "06 April 2026", "latex_path": "/latex/795", "formalized": false, "formalized_url": "", "oeis_urls": [], "comments_problem_id": 795, - "comments_count": 0 + "comments_count": 1 }, { "problem_id": 797, @@ -7611,8 +8186,8 @@ "problem_url": "/798", "status_bucket": "closed", "status_dom_id": "solved", - "status_label": "PROVED", - "status_detail": "This has been solved in the affirmative.", + "status_label": "PROVED (LEAN)", + "status_detail": "This has been solved in the affirmative and the proof verified in Lean.", "prize_amount": "", "statement": "Let $t(n)$ be the minimum number of points in $\\{1,\\ldots,n\\}^2$ such that the $\\binom{t}{2}$ lines determined by these points cover all points in $\\{1,\\ldots,n\\}^2$. Estimate $t(n)$. In particular, is it true that $t(n)=o(n)$?", "tags": [ @@ -7622,9 +8197,11 @@ "latex_path": "/latex/798", "formalized": false, "formalized_url": "", - "oeis_urls": [], + "oeis_urls": [ + "https://oeis.org/A116446" + ], "comments_problem_id": 798, - "comments_count": 0 + "comments_count": 1 }, { "problem_id": 799, @@ -7855,8 +8432,8 @@ "problem_url": "/818", "status_bucket": "closed", "status_dom_id": "solved", - "status_label": "PROVED", - "status_detail": "This has been solved in the affirmative.", + "status_label": "PROVED (LEAN)", + "status_detail": "This has been solved in the affirmative and the proof verified in Lean.", "prize_amount": "", "statement": "Let $A$ be a finite set of integers such that $\\lvert A+A\\rvert \\ll \\lvert A\\rvert$. Is it true that\\[\\lvert AA\\rvert \\gg \\frac{\\lvert A\\rvert^2}{(\\log \\lvert A\\rvert)^C}\\]for some constant $C>0$?", "tags": [ @@ -7868,7 +8445,7 @@ "formalized_url": "", "oeis_urls": [], "comments_problem_id": 818, - "comments_count": 0 + "comments_count": 1 }, { "problem_id": 822, @@ -8071,8 +8648,8 @@ "problem_url": "/844", "status_bucket": "closed", "status_dom_id": "solved", - "status_label": "PROVED", - "status_detail": "This has been solved in the affirmative.", + "status_label": "PROVED (LEAN)", + "status_detail": "This has been solved in the affirmative and the proof verified in Lean.", "prize_amount": "", "statement": "Let $A\\subseteq \\{1,\\ldots,N\\}$ be such that, for all $a,b\\in A$, the product $ab$ is not squarefree. Is the maximum size of such an $A$ achieved by taking $A$ to be the set of even numbers and odd non-squarefree numbers?", "tags": [ @@ -8085,7 +8662,7 @@ "formalized_url": "", "oeis_urls": [], "comments_problem_id": 844, - "comments_count": 0 + "comments_count": 1 }, { "problem_id": 845, @@ -8107,6 +8684,26 @@ "comments_problem_id": 845, "comments_count": 13 }, + { + "problem_id": 846, + "problem_url": "/846", + "status_bucket": "closed", + "status_dom_id": "solved", + "status_label": "DISPROVED (LEAN)", + "status_detail": "This has been solved in the negative and the proof verified in Lean.", + "prize_amount": "", + "statement": "Let $A\\subset \\mathbb{R}^2$ be an infinite set for which there exists some $\\epsilon>0$ such that in any subset of $A$ of size $n$ there are always at least $\\epsilon n$ with no three on a line. Is it true that $A$ is the union of a finite number of sets where no three are on a line?", + "tags": [ + "geometry" + ], + "last_edited": "10 April 2026", + "latex_path": "/latex/846", + "formalized": true, + "formalized_url": "https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/ErdosProblems/846.lean", + "oeis_urls": [], + "comments_problem_id": 846, + "comments_count": 8 + }, { "problem_id": 847, "problem_url": "/847", @@ -8145,7 +8742,48 @@ "formalized_url": "https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/ErdosProblems/848.lean", "oeis_urls": [], "comments_problem_id": 848, - "comments_count": 18 + "comments_count": 48 + }, + { + "problem_id": 851, + "problem_url": "/851", + "status_bucket": "closed", + "status_dom_id": "solved", + "status_label": "PROVED", + "status_detail": "This has been solved in the affirmative.", + "prize_amount": "", + "statement": "Let $\\epsilon>0$. Is there some $r\\ll_\\epsilon 1$ such that the density of integers of the form $2^k+n$, where $k\\geq 0$ and $n$ has at most $r$ prime divisors, is at least $1-\\epsilon$?", + "tags": [ + "number theory" + ], + "last_edited": "02 April 2026", + "latex_path": "/latex/851", + "formalized": true, + "formalized_url": "https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/ErdosProblems/851.lean", + "oeis_urls": [], + "comments_problem_id": 851, + "comments_count": 5 + }, + { + "problem_id": 858, + "problem_url": "/858", + "status_bucket": "closed", + "status_dom_id": "solved", + "status_label": "SOLVED", + "status_detail": "This has been resolved in some other way than a proof or disproof.", + "prize_amount": "", + "statement": "Let $A\\subseteq \\{1,\\ldots,N\\}$ be such that there is no solution to $at=b$ with $a,b\\in A$ and the smallest prime factor of $t$ is $>a$. Estimate the maximum of\\[\\frac{1}{\\log N}\\sum_{n\\in A}\\frac{1}{n}.\\]", + "tags": [ + "number theory", + "primitive sets" + ], + "last_edited": "24 April 2026", + "latex_path": "/latex/858", + "formalized": false, + "formalized_url": "", + "oeis_urls": [], + "comments_problem_id": 858, + "comments_count": 13 }, { "problem_id": 861, @@ -8196,13 +8834,35 @@ "comments_problem_id": 862, "comments_count": 3 }, + { + "problem_id": 863, + "problem_url": "/863", + "status_bucket": "closed", + "status_dom_id": "solved", + "status_label": "PROVED", + "status_detail": "This has been solved in the affirmative.", + "prize_amount": "", + "statement": "Let $r\\geq 2$ and let $A\\subseteq \\{1,\\ldots,N\\}$ be a set of maximal size such that there are at most $r$ solutions to $n=a+b$ with $a\\leq b$ for any $n$. (That is, $A$ is a $B_2[r]$ set.) Similarly, let $B\\subseteq \\{1,\\ldots,N\\}$ be a set of maximal size such that there are at most $r$ solutions to $n=a-b$ for any $n\\geq 1$. If $\\lvert A\\rvert\\sim c_rN^{1/2}$ as $N\\to \\infty$ and $\\lvert B\\rvert \\sim c_r'N^{1/2}$ as $N\\to \\infty$ then is it true that $c_r\\neq c_r'$ for $r\\geq 2$? Is it true that $c_r'\\epsilon \\log n$ (for all large $n$, for arbitrary fixed $\\epsilon>0$)?", "tags": [ "number theory", "additive basis" ], - "last_edited": "23 January 2026", + "last_edited": "03 April 2026", "latex_path": "/latex/868", "formalized": true, "formalized_url": "https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/ErdosProblems/868.lean", @@ -8237,6 +8897,27 @@ "comments_problem_id": 868, "comments_count": 8 }, + { + "problem_id": 869, + "problem_url": "/869", + "status_bucket": "closed", + "status_dom_id": "solved", + "status_label": "DISPROVED", + "status_detail": "This has been solved in the negative.", + "prize_amount": "", + "statement": "If $A_1,A_2$ are disjoint additive bases of order $2$ (i.e. $A_i+A_i$ contains all large integers) then must $A=A_1\\cup A_2$ contain a minimal additive basis of order $2$ (one such that deleting any element creates infinitely many $n\\not\\in A+A$)?", + "tags": [ + "number theory", + "additive basis" + ], + "last_edited": "02 May 2026", + "latex_path": "/latex/869", + "formalized": false, + "formalized_url": "", + "oeis_urls": [], + "comments_problem_id": 869, + "comments_count": 2 + }, { "problem_id": 871, "problem_url": "/871", @@ -8342,12 +9023,33 @@ "comments_count": 0 }, { - "problem_id": 894, - "problem_url": "/894", + "problem_id": 884, + "problem_url": "/884", "status_bucket": "closed", "status_dom_id": "solved", - "status_label": "PROVED", - "status_detail": "This has been solved in the affirmative.", + "status_label": "DISPROVED", + "status_detail": "This has been solved in the negative.", + "prize_amount": "", + "statement": "Is it true that, for any $n$, if $d_1<\\cdots 0$ with $n_{k+1}\\geq (1+\\epsilon)n_k$ for all $k$). Is it true that there must exist a finite colouring of $\\mathbb{N}$ with no monochromatic solutions to $a-b\\in A$?", "tags": [ @@ -8383,6 +9085,26 @@ "comments_problem_id": 895, "comments_count": 0 }, + { + "problem_id": 896, + "problem_url": "/896", + "status_bucket": "closed", + "status_dom_id": "solved", + "status_label": "SOLVED", + "status_detail": "This has been resolved in some other way than a proof or disproof.", + "prize_amount": "", + "statement": "Estimate the maximum of $F(A,B)$ as $A,B$ range over all subsets of $\\{1,\\ldots,N\\}$, where $F(A,B)$ counts the number of $m$ such that $m=ab$ has exactly one solution (with $a\\in A$ and $b\\in B$).", + "tags": [ + "number theory" + ], + "last_edited": "02 May 2026", + "latex_path": "/latex/896", + "formalized": false, + "formalized_url": "", + "oeis_urls": [], + "comments_problem_id": 896, + "comments_count": 5 + }, { "problem_id": 897, "problem_url": "/897", @@ -8395,7 +9117,7 @@ "tags": [ "number theory" ], - "last_edited": "27 December 2025", + "last_edited": "01 April 2026", "latex_path": "/latex/897", "formalized": true, "formalized_url": "https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/ErdosProblems/897.lean", @@ -8455,7 +9177,7 @@ "tags": [ "graph theory" ], - "last_edited": "", + "last_edited": "07 March 2026", "latex_path": "/latex/900", "formalized": false, "formalized_url": "", @@ -8488,60 +9210,60 @@ "problem_url": "/904", "status_bucket": "closed", "status_dom_id": "solved", - "status_label": "PROVED", - "status_detail": "This has been solved in the affirmative.", + "status_label": "PROVED (LEAN)", + "status_detail": "This has been solved in the affirmative and the proof verified in Lean.", "prize_amount": "", - "statement": "If $G$ is a graph with $n$ vertices and $>n^2/4$ edges then it contains a triangle on $x,y,z$ such that\\[d(x)+d(y)+d(z) \\geq \\frac{3}{2}n.\\]", + "statement": "Let $r\\geq 2$ and let $t_r(n)$ be the Tur\u00e1n number (the maximal number of edges in a graph on $n$ vertices with no $K_{r+1}$). If $G$ is a graph with $n$ vertices and $m\\geq t_r(n)$ edges there exists a clique on $r$ vertices, say $x_1,\\ldots,x_r$, such that\\[d(x_1)+\\cdots+d(x_r)\\geq \\frac{2rm}{n}.\\]", "tags": [ "graph theory" ], - "last_edited": "06 October 2025", + "last_edited": "03 April 2026", "latex_path": "/latex/904", "formalized": false, "formalized_url": "", "oeis_urls": [], "comments_problem_id": 904, - "comments_count": 0 + "comments_count": 5 }, { "problem_id": 905, "problem_url": "/905", "status_bucket": "closed", "status_dom_id": "solved", - "status_label": "PROVED", - "status_detail": "This has been solved in the affirmative.", + "status_label": "PROVED (LEAN)", + "status_detail": "This has been solved in the affirmative and the proof verified in Lean.", "prize_amount": "", "statement": "Every graph with $n$ vertices and $>n^2/4$ edges contains an edge which is in at least $n/6$ triangles.", "tags": [ "graph theory" ], - "last_edited": "06 October 2025", + "last_edited": "07 April 2026", "latex_path": "/latex/905", "formalized": false, "formalized_url": "", "oeis_urls": [], "comments_problem_id": 905, - "comments_count": 2 + "comments_count": 3 }, { "problem_id": 907, "problem_url": "/907", "status_bucket": "closed", "status_dom_id": "solved", - "status_label": "PROVED", - "status_detail": "This has been solved in the affirmative.", + "status_label": "PROVED (LEAN)", + "status_detail": "This has been solved in the affirmative and the proof verified in Lean.", "prize_amount": "", - "statement": "Let $f:\\mathbb{R}\\to \\mathbb{R}$ be such that $f(x+h)-f(x)$ is continous for every $h>0$. Is it true that\\[f=g+h\\]for some continuous $g$ and additive $h$ (i.e. $h(x+y)=h(x)+h(y)$)?", + "statement": "Let $f:\\mathbb{R}\\to \\mathbb{R}$ be such that $f(x+h)-f(x)$ is continuous for every $h>0$. Is it true that\\[f=g+h\\]for some continuous $g$ and additive $h$ (i.e. $h(x+y)=h(x)+h(y)$)?", "tags": [ "analysis" ], - "last_edited": "30 December 2025", + "last_edited": "07 April 2026", "latex_path": "/latex/907", "formalized": false, "formalized_url": "", "oeis_urls": [], "comments_problem_id": 907, - "comments_count": 0 + "comments_count": 1 }, { "problem_id": 908, @@ -8609,20 +9331,20 @@ "problem_url": "/914", "status_bucket": "closed", "status_dom_id": "solved", - "status_label": "PROVED", - "status_detail": "This has been solved in the affirmative.", + "status_label": "PROVED (LEAN)", + "status_detail": "This has been solved in the affirmative and the proof verified in Lean.", "prize_amount": "", "statement": "Let $r\\geq 2$ and $m\\geq 1$. Every graph with $rm$ vertices and minimum degree at least $m(r-1)$ contains $m$ vertex disjoint copies of $K_r$.", "tags": [ "graph theory" ], - "last_edited": "", + "last_edited": "15 April 2026", "latex_path": "/latex/914", "formalized": false, "formalized_url": "", "oeis_urls": [], "comments_problem_id": 914, - "comments_count": 0 + "comments_count": 2 }, { "problem_id": 915, @@ -8713,8 +9435,8 @@ "problem_url": "/923", "status_bucket": "closed", "status_dom_id": "solved", - "status_label": "PROVED", - "status_detail": "This has been solved in the affirmative.", + "status_label": "PROVED (LEAN)", + "status_detail": "This has been solved in the affirmative and the proof verified in Lean.", "prize_amount": "", "statement": "Is it true that, for every $k$, there is some $f(k)$ such that if $G$ has chromatic number $\\geq f(k)$ then $G$ contains a triangle-free subgraph with chromatic number $\\geq k$?", "tags": [ @@ -8727,7 +9449,7 @@ "formalized_url": "", "oeis_urls": [], "comments_problem_id": 923, - "comments_count": 1 + "comments_count": 2 }, { "problem_id": 924, @@ -8877,7 +9599,7 @@ "https://oeis.org/A284783" ], "comments_problem_id": 946, - "comments_count": 3 + "comments_count": 5 }, { "problem_id": 947, @@ -8942,6 +9664,26 @@ "comments_problem_id": 958, "comments_count": 2 }, + { + "problem_id": 960, + "problem_url": "/960", + "status_bucket": "closed", + "status_dom_id": "solved", + "status_label": "DISPROVED", + "status_detail": "This has been solved in the negative.", + "prize_amount": "", + "statement": "Let $r,k\\geq 2$ be fixed. Let $A\\subset \\mathbb{R}^2$ be a set of $n$ points with no $k$ points on a line. Determine the threshold $f_{r,k}(n)$ such that if there are at least $f_{r,k}(n)$ many ordinary lines (lines containing exactly two points) then there is a set $A'\\subseteq A$ of $r$ points such that all $\\binom{r}{2}$ many lines determined by $A'$ are ordinary. Is it true that $f_{r,k}(n)=o(n^2)$, or perhaps even $\\ll n$?", + "tags": [ + "geometry" + ], + "last_edited": "09 April 2026", + "latex_path": "/latex/960", + "formalized": false, + "formalized_url": "", + "oeis_urls": [], + "comments_problem_id": 960, + "comments_count": 1 + }, { "problem_id": 964, "problem_url": "/964", @@ -9031,8 +9773,8 @@ "problem_url": "/974", "status_bucket": "closed", "status_dom_id": "solved", - "status_label": "PROVED", - "status_detail": "This has been solved in the affirmative.", + "status_label": "PROVED (LEAN)", + "status_detail": "This has been solved in the affirmative and the proof verified in Lean.", "prize_amount": "", "statement": "Let $z_1,\\ldots,z_n\\in \\mathbb{C}$ be a sequence such that $z_1=1$. Suppose that the sequence of\\[s_k=\\sum_{1\\leq i\\leq n}z_i^k\\]contains infinitely many $(n-1)$-tuples of consecutive values of $s_k$ which are all $0$. Then (essentially)\\[z_j=e(j/n),\\]where $e(x)=e^{2\\pi ix}$.", "tags": [ @@ -9044,7 +9786,7 @@ "formalized_url": "", "oeis_urls": [], "comments_problem_id": 974, - "comments_count": 13 + "comments_count": 14 }, { "problem_id": 977, @@ -9125,7 +9867,7 @@ "arithmetic progressions", "additive combinatorics" ], - "last_edited": "", + "last_edited": "04 April 2026", "latex_path": "/latex/984", "formalized": false, "formalized_url": "", @@ -9133,6 +9875,27 @@ "comments_problem_id": 984, "comments_count": 3 }, + { + "problem_id": 987, + "problem_url": "/987", + "status_bucket": "closed", + "status_dom_id": "solved", + "status_label": "PROVED", + "status_detail": "This has been solved in the affirmative.", + "prize_amount": "", + "statement": "Let $x_1,x_2,\\ldots \\in (0,1)$ be an infinite sequence and let\\[A_k=\\limsup_{n\\to \\infty}\\left\\lvert \\sum_{j\\leq n} e(kx_j)\\right\\rvert,\\]where $e(x)=e^{2\\pi ix}$. Is it true that\\[\\limsup_{k\\to \\infty} A_k=\\infty?\\]Is it possible for $A_k=o(k)$?", + "tags": [ + "analysis", + "discrepancy" + ], + "last_edited": "09 April 2026", + "latex_path": "/latex/987", + "formalized": false, + "formalized_url": "", + "oeis_urls": [], + "comments_problem_id": 987, + "comments_count": 9 + }, { "problem_id": 988, "problem_url": "/988", @@ -9173,6 +9936,26 @@ "comments_problem_id": 989, "comments_count": 0 }, + { + "problem_id": 990, + "problem_url": "/990", + "status_bucket": "closed", + "status_dom_id": "solved", + "status_label": "DISPROVED (LEAN)", + "status_detail": "This has been solved in the negative and the proof verified in Lean.", + "prize_amount": "", + "statement": "Let $f=a_0+\\cdots+a_dx^d\\in \\mathbb{C}[x]$ be a polynomial. Is it true that, if $f$ has roots $z_1,\\ldots,z_d$ with corresponding arguments $\\theta_1,\\ldots,\\theta_d\\in [0,2\\pi]$, then for all intervals $I\\subseteq [0,2\\pi]$\\[\\left\\lvert (\\# \\theta_i \\in I) - \\frac{\\lvert I\\rvert}{2\\pi}d\\right\\rvert \\ll \\left(n\\log M\\right)^{1/2},\\]where $n$ is the number of non-zero coefficients of $f$ and\\[M=\\frac{\\lvert a_0\\rvert+\\cdots +\\lvert a_d\\rvert}{(\\lvert a_0\\rvert\\lvert a_d\\rvert)^{1/2}}.\\]", + "tags": [ + "analysis" + ], + "last_edited": "10 April 2026", + "latex_path": "/latex/990", + "formalized": false, + "formalized_url": "", + "oeis_urls": [], + "comments_problem_id": 990, + "comments_count": 3 + }, { "problem_id": 991, "problem_url": "/991", @@ -9234,6 +10017,28 @@ "comments_problem_id": 994, "comments_count": 0 }, + { + "problem_id": 997, + "problem_url": "/997", + "status_bucket": "closed", + "status_dom_id": "solved", + "status_label": "PROVED (LEAN)", + "status_detail": "This has been solved in the affirmative and the proof verified in Lean.", + "prize_amount": "", + "statement": "Call $x_1,x_2,\\ldots \\in (0,1)$ well-distributed if, for every $\\epsilon>0$, if $k$ is sufficiently large then, for all $n>0$ and intervals $I\\subseteq [0,1]$,\\[\\lvert \\# \\{ n0$ is sufficiently small, $x_{k+1}-x_k \\to 0$?", + "tags": [ + "number theory" + ], + "last_edited": "16 April 2026", + "latex_path": "/latex/1096", + "formalized": true, + "formalized_url": "https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/ErdosProblems/1096.lean", + "oeis_urls": [], + "comments_problem_id": 1096, + "comments_count": 3 + }, + { + "problem_id": 1098, + "problem_url": "/1098", + "status_bucket": "closed", + "status_dom_id": "solved", + "status_label": "PROVED (LEAN)", + "status_detail": "This has been solved in the affirmative and the proof verified in Lean.", + "prize_amount": "", "statement": "Let $G$ be a group and $\\Gamma=\\Gamma(G)$ be the non-commuting graph, with vertices the elements of $G$ and an edge between $g$ and $h$ if and only if $g$ and $h$ do not commute, $gh\\neq hg$. If $\\Gamma$ contains no infinite complete subgraph, then is there a finite bound on the size of complete subgraphs of $\\Gamma$?", "tags": [ "group theory" @@ -10200,7 +11088,7 @@ "formalized_url": "", "oeis_urls": [], "comments_problem_id": 1098, - "comments_count": 0 + "comments_count": 1 }, { "problem_id": 1099, @@ -10371,20 +11259,20 @@ "problem_url": "/1121", "status_bucket": "closed", "status_dom_id": "solved", - "status_label": "PROVED", - "status_detail": "This has been solved in the affirmative.", + "status_label": "PROVED (LEAN)", + "status_detail": "This has been solved in the affirmative and the proof verified in Lean.", "prize_amount": "", "statement": "If $C_1,\\ldots,C_n$ are circles in $\\mathbb{R}^2$ with radii $r_1,\\ldots,r_n$ such that no line disjoint from all the circles divides them into two non-empty sets then the circles can be covered by a circle of radius $r=\\sum r_i$.", "tags": [ "geometry" ], - "last_edited": "30 December 2025", + "last_edited": "17 April 2026", "latex_path": "/latex/1121", "formalized": false, "formalized_url": "", "oeis_urls": [], "comments_problem_id": 1121, - "comments_count": 0 + "comments_count": 5 }, { "problem_id": 1123, @@ -10398,7 +11286,7 @@ "tags": [ "algebra" ], - "last_edited": "31 December 2025", + "last_edited": "05 March 2026", "latex_path": "/latex/1123", "formalized": false, "formalized_url": "", @@ -10431,8 +11319,8 @@ "problem_url": "/1125", "status_bucket": "closed", "status_dom_id": "solved", - "status_label": "PROVED", - "status_detail": "This has been solved in the affirmative.", + "status_label": "PROVED (LEAN)", + "status_detail": "This has been solved in the affirmative and the proof verified in Lean.", "prize_amount": "", "statement": "Let $f:\\mathbb{R}\\to \\mathbb{R}$ be such that\\[2f(x) \\leq f(x+h)+f(x+2h)\\]for every $x\\in \\mathbb{R}$ and $h>0$. Must $f$ be monotonic?", "tags": [ @@ -10444,7 +11332,7 @@ "formalized_url": "", "oeis_urls": [], "comments_problem_id": 1125, - "comments_count": 0 + "comments_count": 1 }, { "problem_id": 1126, @@ -10579,8 +11467,8 @@ "problem_url": "/1136", "status_bucket": "closed", "status_dom_id": "solved", - "status_label": "PROVED", - "status_detail": "This has been solved in the affirmative.", + "status_label": "PROVED (LEAN)", + "status_detail": "This has been solved in the affirmative and the proof verified in Lean.", "prize_amount": "", "statement": "Does there exist $A\\subset \\mathbb{N}$ with lower density $>1/3$ such that $a+b\\neq 2^k$ for any $a,b\\in A$ and $k\\geq 0$?", "tags": [ @@ -10592,7 +11480,7 @@ "formalized_url": "", "oeis_urls": [], "comments_problem_id": 1136, - "comments_count": 0 + "comments_count": 3 }, { "problem_id": 1140, @@ -10614,6 +11502,29 @@ "comments_problem_id": 1140, "comments_count": 4 }, + { + "problem_id": 1141, + "problem_url": "/1141", + "status_bucket": "closed", + "status_dom_id": "solved", + "status_label": "DISPROVED (LEAN)", + "status_detail": "This has been solved in the negative and the proof verified in Lean.", + "prize_amount": "", + "statement": "Are there infinitely many $n$ such that $n-k^2$ is prime for all $k$ with $(n,k)=1$ and $k^2 \\left(\\frac{2}{\\pi}-o(1)\\right)\\log n?\\]", + "tags": [ + "analysis", + "polynomials" + ], + "last_edited": "01 April 2026", + "latex_path": "/latex/1153", + "formalized": false, + "formalized_url": "", + "oeis_urls": [], + "comments_problem_id": 1153, + "comments_count": 10 + }, { "problem_id": 1161, "problem_url": "/1161", @@ -10755,6 +11710,338 @@ "oeis_urls": [], "comments_problem_id": 1179, "comments_count": 0 + }, + { + "problem_id": 1180, + "problem_url": "/1180", + "status_bucket": "closed", + "status_dom_id": "solved", + "status_label": "PROVED", + "status_detail": "This has been solved in the affirmative.", + "prize_amount": "", + "statement": "Let $\\epsilon>0$. Does there exist a constant $C_\\epsilon$ such that, for all primes $p$, every residue modulo $p$ is the sum of at most $C_\\epsilon$ many elements of\\[\\{ n^{-1} : 1\\leq n\\leq p^\\epsilon\\}\\]where $n^{-1}$ denotes the inverse of $n$ modulo $p$?", + "tags": [ + "number theory" + ], + "last_edited": "06 March 2026", + "latex_path": "/latex/1180", + "formalized": false, + "formalized_url": "", + "oeis_urls": [], + "comments_problem_id": 1180, + "comments_count": 0 + }, + { + "problem_id": 1185, + "problem_url": "/1185", + "status_bucket": "closed", + "status_dom_id": "solved", + "status_label": "SOLVED", + "status_detail": "This has been resolved in some other way than a proof or disproof.", + "prize_amount": "", + "statement": "Let $\\delta>0$ and $k\\geq 3$. Is it true that there exists $m\\geq 1$ (depending only on $\\delta$ and $k$) such that, for all large $N$, if $A,B\\subseteq \\{1,\\ldots,N\\}$ with $\\lvert A\\rvert \\geq \\delta N$ and $\\lvert B\\rvert \\geq m$ then there is a non-trivial $k$-term arithmetic progression in $A$ whose common difference is in $B-B$?", + "tags": [ + "additive combinatorics", + "arithmetic progressions" + ], + "last_edited": "05 April 2026", + "latex_path": "/latex/1185", + "formalized": false, + "formalized_url": "", + "oeis_urls": [], + "comments_problem_id": 1185, + "comments_count": 0 + }, + { + "problem_id": 1187, + "problem_url": "/1187", + "status_bucket": "closed", + "status_dom_id": "solved", + "status_label": "SOLVED", + "status_detail": "This has been resolved in some other way than a proof or disproof.", + "prize_amount": "", + "statement": "Let $k\\geq 3$. Is it true that, in any finite colouring of the integers, there are monochromatic arithmetic progressions of primes of length $k$? Are there monochromatic arithmetic progressions of length $k$ whose common difference is a prime?", + "tags": [ + "number theory", + "additive combinatorics", + "arithmetic progressions", + "primes" + ], + "last_edited": "08 April 2026", + "latex_path": "/latex/1187", + "formalized": false, + "formalized_url": "", + "oeis_urls": [], + "comments_problem_id": 1187, + "comments_count": 1 + }, + { + "problem_id": 1193, + "problem_url": "/1193", + "status_bucket": "closed", + "status_dom_id": "solved", + "status_label": "SOLVED (LEAN)", + "status_detail": "This has been resolved in some other way than a proof or disproof, and that resolution verified in Lean.", + "prize_amount": "", + "statement": "Let $A\\subset \\mathbb{N}$ and let $g(n)$ be a non-decreasing function of $n$ which is always $>0$. Is the lower density of\\[\\{ n : 1_A\\ast 1_A(n)=g(n)\\}\\]always $0$? Is the upper density always $0$, for all sufficiently large (depending on $x$) integers $n$ there exists an integer $r\\geq 1$ such that $nx\\in r\\cdot E$?", + "tags": [ + "analysis" + ], + "last_edited": "15 April 2026", + "latex_path": "/latex/1197", + "formalized": false, + "formalized_url": "", + "oeis_urls": [], + "comments_problem_id": 1197, + "comments_count": 4 + }, + { + "problem_id": 1198, + "problem_url": "/1198", + "status_bucket": "closed", + "status_dom_id": "solved", + "status_label": "DISPROVED", + "status_detail": "This has been solved in the negative.", + "prize_amount": "", + "statement": "If $\\mathbb{N}$ is $2$-coloured then must there exist an infinite set $A=\\{a_1<\\cdots\\}$ such that all expressions of the shape\\[\\prod_{i\\in S_1}a_i+\\cdots+\\prod_{i\\in S_k}a_i,\\]for disjoint $S_1,\\ldots,S_k$ (excluding the trivial expressions $a_i$) are the same colour?", + "tags": [ + "additive combinatorics", + "ramsey theory" + ], + "last_edited": "17 April 2026", + "latex_path": "/latex/1198", + "formalized": false, + "formalized_url": "", + "oeis_urls": [], + "comments_problem_id": 1198, + "comments_count": 4 + }, + { + "problem_id": 1202, + "problem_url": "/1202", + "status_bucket": "closed", + "status_dom_id": "solved", + "status_label": "SOLVED", + "status_detail": "This has been resolved in some other way than a proof or disproof.", + "prize_amount": "", + "statement": "Let $\\epsilon,\\eta>0$. Does there exist a $k$ such that, given any set of $k$ primes $p_1<\\cdots f(a,K)$ and with bounded gaps $a_{i+1}-a_i\\leq K$ then there are two distinct intervals $I$ and $J$ such that\\[\\sum_{i\\in I}a_i=\\sum_{j\\in J}a_j?\\]", + "tags": [ + "additive combinatorics" + ], + "last_edited": "10 April 2026", + "latex_path": "/latex/1213", + "formalized": false, + "formalized_url": "", + "oeis_urls": [], + "comments_problem_id": 1213, + "comments_count": 0 + }, + { + "problem_id": 1214, + "problem_url": "/1214", + "status_bucket": "closed", + "status_dom_id": "solved", + "status_label": "PROVED", + "status_detail": "This has been solved in the affirmative.", + "prize_amount": "", + "statement": "Let $x,y\\geq 1$ be integers such that, for all $n\\geq 1$, the set of primes dividing $x^{n}-1$ is equal to set of primes dividing $y^n-1$. Must $x=y$?", + "tags": [ + "number theory" + ], + "last_edited": "12 April 2026", + "latex_path": "/latex/1214", + "formalized": true, + "formalized_url": "https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/ErdosProblems/1214.lean", + "oeis_urls": [], + "comments_problem_id": 1214, + "comments_count": 0 + }, + { + "problem_id": 1215, + "problem_url": "/1215", + "status_bucket": "closed", + "status_dom_id": "solved", + "status_label": "DISPROVED", + "status_detail": "This has been solved in the negative.", + "prize_amount": "", + "statement": "Does there exist a constant $C$ such that for every polynomial $P$ with $P(0)=1$, all of whose roots are on the unit circle, there exists a path in\\[\\{ z: \\lvert P(z)\\rvert < 1\\}\\]which connects $0$ to the unit circle of length at most $C$?", + "tags": [ + "analysis" + ], + "last_edited": "12 April 2026", + "latex_path": "/latex/1215", + "formalized": false, + "formalized_url": "", + "oeis_urls": [], + "comments_problem_id": 1215, + "comments_count": 3 + }, + { + "problem_id": 1216, + "problem_url": "/1216", + "status_bucket": "closed", + "status_dom_id": "solved", + "status_label": "DISPROVED", + "status_detail": "This has been solved in the negative.", + "prize_amount": "", + "statement": "A tournament is a complete directed graph. Let $f(n)$ be such that every tournament on $n$ vertices contains a transitive tournament on $f(n)$ vertices (i.e. one such that if $i\\to j\\to k$ then $i\\to k$). Is it true that $f(n)=\\lfloor \\log_2 n\\rfloor +1$?", + "tags": [ + "graph theory", + "ramsey theory" + ], + "last_edited": "12 April 2026", + "latex_path": "/latex/1216", + "formalized": false, + "formalized_url": "", + "oeis_urls": [], + "comments_problem_id": 1216, + "comments_count": 3 + }, + { + "problem_id": 1217, + "problem_url": "/1217", + "status_bucket": "closed", + "status_dom_id": "solved", + "status_label": "PROVED", + "status_detail": "This has been solved in the affirmative.", + "prize_amount": "", + "statement": "Let $A=\\{a_10$ for $0<\\alpha <1 $? The Schnirelmann density is defined by\\[d_s(A) = \\inf_{N\\geq 1}\\frac{\\lvert A\\cap\\{1,\\ldots,N\\}\\rvert}{N}.\\]", - "tags": [ - "number theory" - ], - "last_edited": "16 September 2025", - "latex_path": "/latex/38", - "formalized": true, - "formalized_url": "https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/ErdosProblems/38.lean", - "oeis_urls": [], - "comments_problem_id": 38, - "comments_count": 1 + "comments_count": 5 }, { "problem_id": 39, @@ -1211,7 +1170,7 @@ "sidon sets", "additive combinatorics" ], - "last_edited": "23 January 2026", + "last_edited": "06 April 2026", "latex_path": "/latex/39", "formalized": true, "formalized_url": "https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/ErdosProblems/39.lean", @@ -1254,7 +1213,7 @@ "sidon sets", "additive combinatorics" ], - "last_edited": "23 January 2026", + "last_edited": "06 April 2026", "latex_path": "/latex/41", "formalized": true, "formalized_url": "https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/ErdosProblems/41.lean", @@ -1262,54 +1221,6 @@ "comments_problem_id": 41, "comments_count": 0 }, - { - "problem_id": 42, - "problem_url": "/42", - "status_bucket": "open", - "status_dom_id": "open", - "status_label": "OPEN", - "status_detail": "This is open, and cannot be resolved with a finite computation.", - "prize_amount": "", - "statement": "Let $M\\geq 1$ and $N$ be sufficiently large in terms of $M$. Is it true that for every Sidon set $A\\subset \\{1,\\ldots,N\\}$ there is another Sidon set $B\\subset \\{1,\\ldots,N\\}$ of size $M$ such that $(A-A)\\cap(B-B)=\\{0\\}$?", - "tags": [ - "number theory", - "sidon sets", - "additive combinatorics" - ], - "last_edited": "23 January 2026", - "latex_path": "/latex/42", - "formalized": true, - "formalized_url": "https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/ErdosProblems/42.lean", - "oeis_urls": [], - "comments_problem_id": 42, - "comments_count": 6 - }, - { - "problem_id": 43, - "problem_url": "/43", - "status_bucket": "open", - "status_dom_id": "open", - "status_label": "OPEN", - "status_detail": "This is open, and cannot be resolved with a finite computation.", - "prize_amount": "$100", - "statement": "If $A,B\\subset \\{1,\\ldots,N\\}$ are two Sidon sets such that $(A-A)\\cap(B-B)=\\{0\\}$ then is it true that\\[ \\binom{\\lvert A\\rvert}{2}+\\binom{\\lvert B\\rvert}{2}\\leq\\binom{f(N)}{2}+O(1),\\]where $f(N)$ is the maximum possible size of a Sidon set in $\\{1,\\ldots,N\\}$? If $\\lvert A\\rvert=\\lvert B\\rvert$ then can this bound be improved to\\[\\binom{\\lvert A\\rvert}{2}+\\binom{\\lvert B\\rvert}{2}\\leq (1-c+o(1))\\binom{f(N)}{2}\\]for some constant $c>0$?", - "tags": [ - "number theory", - "sidon sets", - "additive combinatorics" - ], - "last_edited": "20 December 2025", - "latex_path": "/latex/43", - "formalized": false, - "formalized_url": "", - "oeis_urls": [ - "https://oeis.org/A003022", - "https://oeis.org/A143824", - "https://oeis.org/A227590" - ], - "comments_problem_id": 43, - "comments_count": 9 - }, { "problem_id": 44, "problem_url": "/44", @@ -1346,8 +1257,8 @@ ], "last_edited": "", "latex_path": "/latex/50", - "formalized": false, - "formalized_url": "", + "formalized": true, + "formalized_url": "https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/ErdosProblems/50.lean", "oeis_urls": [], "comments_problem_id": 50, "comments_count": 0 @@ -1388,10 +1299,10 @@ "number theory", "additive combinatorics" ], - "last_edited": "23 January 2026", + "last_edited": "08 April 2026", "latex_path": "/latex/52", - "formalized": false, - "formalized_url": "", + "formalized": true, + "formalized_url": "https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/ErdosProblems/52.lean", "oeis_urls": [ "https://oeis.org/A263996" ], @@ -1415,7 +1326,9 @@ "latex_path": "/latex/60", "formalized": false, "formalized_url": "", - "oeis_urls": [], + "oeis_urls": [ + "https://oeis.org/A006855" + ], "comments_problem_id": 60, "comments_count": 0 }, @@ -1431,13 +1344,13 @@ "tags": [ "graph theory" ], - "last_edited": "23 January 2026", + "last_edited": "10 April 2026", "latex_path": "/latex/61", "formalized": true, "formalized_url": "https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/ErdosProblems/61.lean", "oeis_urls": [], "comments_problem_id": 61, - "comments_count": 3 + "comments_count": 6 }, { "problem_id": 62, @@ -1472,7 +1385,7 @@ "graph theory", "cycles" ], - "last_edited": "18 January 2026", + "last_edited": "10 April 2026", "latex_path": "/latex/64", "formalized": true, "formalized_url": "https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/ErdosProblems/64.lean", @@ -1514,7 +1427,7 @@ "number theory", "additive basis" ], - "last_edited": "23 January 2026", + "last_edited": "06 April 2026", "latex_path": "/latex/66", "formalized": true, "formalized_url": "https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/ErdosProblems/66.lean", @@ -1604,11 +1517,11 @@ ], "last_edited": "29 January 2026", "latex_path": "/latex/75", - "formalized": false, - "formalized_url": "", + "formalized": true, + "formalized_url": "https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/ErdosProblems/75.lean", "oeis_urls": [], "comments_problem_id": 75, - "comments_count": 0 + "comments_count": 3 }, { "problem_id": 77, @@ -1631,7 +1544,7 @@ "https://oeis.org/A059442" ], "comments_problem_id": 77, - "comments_count": 2 + "comments_count": 3 }, { "problem_id": 78, @@ -1654,7 +1567,7 @@ "https://oeis.org/A059442" ], "comments_problem_id": 78, - "comments_count": 3 + "comments_count": 5 }, { "problem_id": 80, @@ -1669,13 +1582,13 @@ "graph theory", "ramsey theory" ], - "last_edited": "", + "last_edited": "07 April 2026", "latex_path": "/latex/80", "formalized": false, "formalized_url": "", "oeis_urls": [], "comments_problem_id": 80, - "comments_count": 0 + "comments_count": 1 }, { "problem_id": 81, @@ -1709,17 +1622,28 @@ "tags": [ "graph theory" ], - "last_edited": "06 October 2025", + "last_edited": "10 April 2026", "latex_path": "/latex/82", "formalized": true, "formalized_url": "https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/ErdosProblems/82.lean", "oeis_urls": [ "https://oeis.org/A120414", "https://oeis.org/A390256", - "https://oeis.org/A390257" + "https://oeis.org/A390257", + "https://oeis.org/A390919", + "https://oeis.org/A392636", + "https://oeis.org/A394400", + "https://oeis.org/A394462", + "https://oeis.org/A394539", + "https://oeis.org/A394563", + "https://oeis.org/A394564", + "https://oeis.org/A394573", + "https://oeis.org/A394574", + "https://oeis.org/A394930", + "https://oeis.org/A394933" ], "comments_problem_id": 82, - "comments_count": 16 + "comments_count": 19 }, { "problem_id": 84, @@ -1747,8 +1671,8 @@ "problem_url": "/85", "status_bucket": "open", "status_dom_id": "open", - "status_label": "FALSIFIABLE", - "status_detail": "Open, but could be disproved with a finite counterexample.", + "status_label": "OPEN", + "status_detail": "This is open, and cannot be resolved with a finite computation.", "prize_amount": "", "statement": "Let $n\\geq 4$ and $f(n)$ be minimal such that every graph on $n$ vertices with minimal degree $\\geq f(n)$ contains a $C_4$. Is it true that, for all large $n$, $f(n+1)\\geq f(n)$?", "tags": [ @@ -1785,7 +1709,7 @@ "https://oeis.org/A245762" ], "comments_problem_id": 86, - "comments_count": 1 + "comments_count": 2 }, { "problem_id": 87, @@ -1834,29 +1758,6 @@ "comments_problem_id": 89, "comments_count": 0 }, - { - "problem_id": 90, - "problem_url": "/90", - "status_bucket": "open", - "status_dom_id": "open", - "status_label": "OPEN", - "status_detail": "This is open, and cannot be resolved with a finite computation.", - "prize_amount": "$500", - "statement": "Does every set of $n$ distinct points in $\\mathbb{R}^2$ contain at most $n^{1+O(1/\\log\\log n)}$ many pairs which are distance 1 apart?", - "tags": [ - "geometry", - "distances" - ], - "last_edited": "23 January 2026", - "latex_path": "/latex/90", - "formalized": true, - "formalized_url": "https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/ErdosProblems/90.lean", - "oeis_urls": [ - "https://oeis.org/A186705" - ], - "comments_problem_id": 90, - "comments_count": 0 - }, { "problem_id": 91, "problem_url": "/91", @@ -1865,41 +1766,20 @@ "status_label": "OPEN", "status_detail": "This is open, and cannot be resolved with a finite computation.", "prize_amount": "", - "statement": "Let $n$ be a sufficently large integer. Suppose $A\\subset \\mathbb{R}^2$ has $\\lvert A\\rvert=n$ and minimises the number of distinct distances between points in $A$. Prove that there are at least two (and probably many) such $A$ which are non-similar.", + "statement": "Let $n$ be a sufficiently large integer. Suppose $A\\subset \\mathbb{R}^2$ has $\\lvert A\\rvert=n$ and minimises the number of distinct distances between points in $A$. Prove that there are at least two (and probably many) such $A$ which are non-similar.", "tags": [ "geometry", "distances" ], - "last_edited": "16 January 2026", + "last_edited": "13 April 2026", "latex_path": "/latex/91", - "formalized": false, - "formalized_url": "", + "formalized": true, + "formalized_url": "https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/ErdosProblems/91.lean", "oeis_urls": [ "https://oeis.org/A186704" ], "comments_problem_id": 91, - "comments_count": 5 - }, - { - "problem_id": 92, - "problem_url": "/92", - "status_bucket": "open", - "status_dom_id": "open", - "status_label": "OPEN", - "status_detail": "This is open, and cannot be resolved with a finite computation.", - "prize_amount": "$500", - "statement": "Let $f(n)$ be maximal such that there exists a set $A$ of $n$ points in $\\mathbb{R}^2$ in which every $x\\in A$ has at least $f(n)$ points in $A$ equidistant from $x$. Is it true that $f(n)\\leq n^{o(1)}$? Or even $f(n) < n^{O(1/\\log\\log n)}$?", - "tags": [ - "geometry", - "distances" - ], - "last_edited": "28 December 2025", - "latex_path": "/latex/92", - "formalized": true, - "formalized_url": "https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/ErdosProblems/92.lean", - "oeis_urls": [], - "comments_problem_id": 92, - "comments_count": 1 + "comments_count": 7 }, { "problem_id": 96, @@ -1917,8 +1797,8 @@ ], "last_edited": "23 January 2026", "latex_path": "/latex/96", - "formalized": false, - "formalized_url": "", + "formalized": true, + "formalized_url": "https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/ErdosProblems/96.lean", "oeis_urls": [], "comments_problem_id": 96, "comments_count": 5 @@ -1943,7 +1823,7 @@ "formalized_url": "https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/ErdosProblems/97.lean", "oeis_urls": [], "comments_problem_id": 97, - "comments_count": 0 + "comments_count": 2 }, { "problem_id": 98, @@ -1960,8 +1840,8 @@ ], "last_edited": "15 October 2025", "latex_path": "/latex/98", - "formalized": false, - "formalized_url": "", + "formalized": true, + "formalized_url": "https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/ErdosProblems/98.lean", "oeis_urls": [], "comments_problem_id": 98, "comments_count": 0 @@ -2022,8 +1902,8 @@ ], "last_edited": "27 December 2025", "latex_path": "/latex/101", - "formalized": false, - "formalized_url": "", + "formalized": true, + "formalized_url": "https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/ErdosProblems/101.lean", "oeis_urls": [ "https://oeis.org/A006065" ], @@ -2105,7 +1985,7 @@ "tags": [ "geometry" ], - "last_edited": "", + "last_edited": "06 March 2026", "latex_path": "/latex/106", "formalized": false, "formalized_url": "", @@ -2126,7 +2006,7 @@ "geometry", "convex" ], - "last_edited": "23 January 2026", + "last_edited": "11 April 2026", "latex_path": "/latex/107", "formalized": true, "formalized_url": "https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/ErdosProblems/107.lean", @@ -2134,7 +2014,7 @@ "https://oeis.org/A000051" ], "comments_problem_id": 107, - "comments_count": 3 + "comments_count": 2 }, { "problem_id": 108, @@ -2206,8 +2086,8 @@ "problem_url": "/114", "status_bucket": "open", "status_dom_id": "open", - "status_label": "OPEN", - "status_detail": "This is open, and cannot be resolved with a finite computation.", + "status_label": "FALSIFIABLE", + "status_detail": "Open, but could be disproved with a finite counterexample.", "prize_amount": "$250", "statement": "If $p(z)\\in\\mathbb{C}[z]$ is a monic polynomial of degree $n$ then is the length of the curve $\\{ z\\in \\mathbb{C} : \\lvert p(z)\\rvert=1\\}$ maximised when $p(z)=z^n-1$?", "tags": [ @@ -2220,7 +2100,7 @@ "formalized_url": "", "oeis_urls": [], "comments_problem_id": 114, - "comments_count": 3 + "comments_count": 6 }, { "problem_id": 117, @@ -2281,7 +2161,7 @@ "formalized_url": "https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/ErdosProblems/120.lean", "oeis_urls": [], "comments_problem_id": 120, - "comments_count": 1 + "comments_count": 3 }, { "problem_id": 122, @@ -2291,17 +2171,17 @@ "status_label": "OPEN", "status_detail": "This is open, and cannot be resolved with a finite computation.", "prize_amount": "", - "statement": "For which number theoretic functions $f$ is it true that, for any $F(n)$ such that $f(n)/F(n)\\to 0$ for almost all $n$, there are infinitely many $x$ such that\\[\\frac{\\#\\{ n\\in \\mathbb{N} : n+f(n)\\in (x,x+F(x))\\}}{F(x)}\\to \\infty?\\]", + "statement": "For which number theoretic functions $f$ is it true that, for any $F(n)$ such that $F(n)/f(n)\\to 0$ for almost all $n$, there are infinitely many $x$ such that\\[\\frac{\\#\\{ n\\in \\mathbb{N} : n+f(n)\\in (x,x+F(x))\\}}{F(x)}\\to \\infty?\\]", "tags": [ "number theory" ], - "last_edited": "01 February 2026", + "last_edited": "01 April 2026", "latex_path": "/latex/122", "formalized": false, "formalized_url": "", "oeis_urls": [], "comments_problem_id": 122, - "comments_count": 0 + "comments_count": 3 }, { "problem_id": 123, @@ -2321,7 +2201,7 @@ "formalized_url": "https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/ErdosProblems/123.lean", "oeis_urls": [], "comments_problem_id": 123, - "comments_count": 1 + "comments_count": 4 }, { "problem_id": 124, @@ -2343,30 +2223,7 @@ "formalized_url": "https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/ErdosProblems/124.lean", "oeis_urls": [], "comments_problem_id": 124, - "comments_count": 13 - }, - { - "problem_id": 125, - "problem_url": "/125", - "status_bucket": "open", - "status_dom_id": "open", - "status_label": "OPEN", - "status_detail": "This is open, and cannot be resolved with a finite computation.", - "prize_amount": "", - "statement": "Let $A = \\{ \\sum\\epsilon_k3^k : \\epsilon_k\\in \\{0,1\\}\\}$ be the set of integers which have only the digits $0,1$ when written base $3$, and $B=\\{ \\sum\\epsilon_k4^k : \\epsilon_k\\in \\{0,1\\}\\}$ be the set of integers which have only the digits $0,1$ when written base $4$. Does $A+B$ have positive density?", - "tags": [ - "number theory", - "base representations" - ], - "last_edited": "", - "latex_path": "/latex/125", - "formalized": true, - "formalized_url": "https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/ErdosProblems/125.lean", - "oeis_urls": [ - "https://oeis.org/A367090" - ], - "comments_problem_id": 125, - "comments_count": 13 + "comments_count": 14 }, { "problem_id": 126, @@ -2525,7 +2382,7 @@ "tags": [ "additive combinatorics" ], - "last_edited": "28 December 2025", + "last_edited": "10 April 2026", "latex_path": "/latex/138", "formalized": true, "formalized_url": "https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/ErdosProblems/138.lean", @@ -2533,7 +2390,7 @@ "https://oeis.org/A005346" ], "comments_problem_id": 138, - "comments_count": 4 + "comments_count": 7 }, { "problem_id": 141, @@ -2572,7 +2429,7 @@ "additive combinatorics", "arithmetic progressions" ], - "last_edited": "23 January 2026", + "last_edited": "04 April 2026", "latex_path": "/latex/142", "formalized": true, "formalized_url": "https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/ErdosProblems/142.lean", @@ -2597,13 +2454,13 @@ "tags": [ "primitive sets" ], - "last_edited": "", + "last_edited": "24 April 2026", "latex_path": "/latex/143", "formalized": true, "formalized_url": "https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/ErdosProblems/143.lean", "oeis_urls": [], "comments_problem_id": 143, - "comments_count": 0 + "comments_count": 1 }, { "problem_id": 145, @@ -2680,11 +2537,11 @@ "status_label": "OPEN", "status_detail": "This is open, and cannot be resolved with a finite computation.", "prize_amount": "", - "statement": "Let $G$ be a graph with maximum degree $\\Delta$. Is $G$ the union of at most $\\tfrac{5}{4}\\Delta^2$ sets of strongly independent edges (sets such that the induced subgraph is the union of vertex-disjoint edges)?", + "statement": "The strong chromatic index of a graph $G$, denoted by $\\mathrm{sq}(G)$, is the minimum $k$ such that the edges of $G$ can be partitioned into $k$ sets of 'strongly independent' edges, that is, such that the subgraph of $G$ induced by each set is the union of vertex-disjoint edges. Is it true that, for any graph $G$ with maximum degree $\\Delta$,\\[\\mathrm{sq}(G)\\leq\\frac{5}{4}\\Delta^2?\\]", "tags": [ "graph theory" ], - "last_edited": "01 February 2026", + "last_edited": "10 April 2026", "latex_path": "/latex/149", "formalized": false, "formalized_url": "", @@ -2712,26 +2569,6 @@ "comments_problem_id": 151, "comments_count": 1 }, - { - "problem_id": 152, - "problem_url": "/152", - "status_bucket": "open", - "status_dom_id": "open", - "status_label": "OPEN", - "status_detail": "This is open, and cannot be resolved with a finite computation.", - "prize_amount": "", - "statement": "For any $M\\geq 1$, if $A\\subset \\mathbb{N}$ is a sufficiently large finite Sidon set then there are at least $M$ many $a\\in A+A$ such that $a+1,a-1\\not\\in A+A$.", - "tags": [ - "sidon sets" - ], - "last_edited": "", - "latex_path": "/latex/152", - "formalized": true, - "formalized_url": "https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/ErdosProblems/152.lean", - "oeis_urls": [], - "comments_problem_id": 152, - "comments_count": 0 - }, { "problem_id": 153, "problem_url": "/153", @@ -2750,7 +2587,7 @@ "formalized_url": "https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/ErdosProblems/153.lean", "oeis_urls": [], "comments_problem_id": 153, - "comments_count": 2 + "comments_count": 7 }, { "problem_id": 155, @@ -2826,13 +2663,13 @@ "status_dom_id": "open", "status_label": "OPEN", "status_detail": "This is open, and cannot be resolved with a finite computation.", - "prize_amount": "", + "prize_amount": "$100", "statement": "There exists some constant $c>0$ such that $$R(C_4,K_n) \\ll n^{2-c}.$$", "tags": [ "graph theory", "ramsey theory" ], - "last_edited": "", + "last_edited": "07 March 2026", "latex_path": "/latex/159", "formalized": false, "formalized_url": "", @@ -2904,7 +2741,7 @@ "formalized_url": "", "oeis_urls": [], "comments_problem_id": 162, - "comments_count": 3 + "comments_count": 4 }, { "problem_id": 165, @@ -2919,7 +2756,7 @@ "graph theory", "ramsey theory" ], - "last_edited": "28 December 2025", + "last_edited": "07 March 2026", "latex_path": "/latex/165", "formalized": false, "formalized_url": "", @@ -2961,7 +2798,7 @@ "tags": [ "additive combinatorics" ], - "last_edited": "24 October 2025", + "last_edited": "23 March 2026", "latex_path": "/latex/168", "formalized": true, "formalized_url": "https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/ErdosProblems/168.lean", @@ -2987,7 +2824,7 @@ "additive combinatorics", "arithmetic progressions" ], - "last_edited": "", + "last_edited": "04 April 2026", "latex_path": "/latex/169", "formalized": false, "formalized_url": "", @@ -3032,7 +2869,7 @@ "additive combinatorics", "ramsey theory" ], - "last_edited": "01 February 2026", + "last_edited": "06 April 2026", "latex_path": "/latex/172", "formalized": true, "formalized_url": "https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/ErdosProblems/172.lean", @@ -3096,13 +2933,13 @@ "arithmetic progressions", "discrepancy" ], - "last_edited": "28 December 2025", + "last_edited": "04 April 2026", "latex_path": "/latex/176", "formalized": false, "formalized_url": "", "oeis_urls": [], "comments_problem_id": 176, - "comments_count": 1 + "comments_count": 8 }, { "problem_id": 177, @@ -3144,7 +2981,7 @@ "formalized_url": "", "oeis_urls": [], "comments_problem_id": 180, - "comments_count": 1 + "comments_count": 4 }, { "problem_id": 181, @@ -3180,7 +3017,7 @@ "graph theory", "ramsey theory" ], - "last_edited": "", + "last_edited": "10 April 2026", "latex_path": "/latex/183", "formalized": false, "formalized_url": "", @@ -3203,13 +3040,13 @@ "graph theory", "cycles" ], - "last_edited": "", + "last_edited": "01 April 2026", "latex_path": "/latex/184", - "formalized": false, - "formalized_url": "", + "formalized": true, + "formalized_url": "https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/ErdosProblems/184.lean", "oeis_urls": [], "comments_problem_id": 184, - "comments_count": 0 + "comments_count": 1 }, { "problem_id": 187, @@ -3225,7 +3062,7 @@ "ramsey theory", "arithmetic progressions" ], - "last_edited": "01 January 2026", + "last_edited": "04 April 2026", "latex_path": "/latex/187", "formalized": false, "formalized_url": "", @@ -3273,7 +3110,7 @@ "formalized_url": "", "oeis_urls": [], "comments_problem_id": 190, - "comments_count": 0 + "comments_count": 4 }, { "problem_id": 193, @@ -3289,8 +3126,8 @@ ], "last_edited": "", "latex_path": "/latex/193", - "formalized": false, - "formalized_url": "", + "formalized": true, + "formalized_url": "https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/ErdosProblems/193.lean", "oeis_urls": [ "https://oeis.org/A231255" ], @@ -3393,7 +3230,7 @@ "additive combinatorics", "arithmetic progressions" ], - "last_edited": "", + "last_edited": "08 April 2026", "latex_path": "/latex/201", "formalized": false, "formalized_url": "", @@ -3418,7 +3255,7 @@ "tags": [ "covering systems" ], - "last_edited": "23 January 2026", + "last_edited": "06 April 2026", "latex_path": "/latex/202", "formalized": false, "formalized_url": "", @@ -3426,7 +3263,7 @@ "https://oeis.org/A389975" ], "comments_problem_id": 202, - "comments_count": 0 + "comments_count": 5 }, { "problem_id": 203, @@ -3527,13 +3364,13 @@ "geometry", "distances" ], - "last_edited": "02 October 2025", + "last_edited": "13 April 2026", "latex_path": "/latex/217", "formalized": false, "formalized_url": "", "oeis_urls": [], "comments_problem_id": 217, - "comments_count": 0 + "comments_count": 1 }, { "problem_id": 218, @@ -3685,7 +3522,7 @@ "additive combinatorics", "sidon sets" ], - "last_edited": "30 September 2025", + "last_edited": "06 April 2026", "latex_path": "/latex/241", "formalized": true, "formalized_url": "https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/ErdosProblems/241.lean", @@ -3693,7 +3530,7 @@ "https://oeis.org/A387704" ], "comments_problem_id": 241, - "comments_count": 2 + "comments_count": 3 }, { "problem_id": 242, @@ -3708,7 +3545,7 @@ "number theory", "unit fractions" ], - "last_edited": "28 January 2026", + "last_edited": "07 May 2026", "latex_path": "/latex/242", "formalized": true, "formalized_url": "https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/ErdosProblems/242.lean", @@ -3744,7 +3581,7 @@ "https://oeis.org/A000058" ], "comments_problem_id": 243, - "comments_count": 6 + "comments_count": 7 }, { "problem_id": 244, @@ -3809,7 +3646,7 @@ "https://oeis.org/A256936" ], "comments_problem_id": 249, - "comments_count": 0 + "comments_count": 2 }, { "problem_id": 251, @@ -3832,7 +3669,7 @@ "https://oeis.org/A098990" ], "comments_problem_id": 251, - "comments_count": 1 + "comments_count": 4 }, { "problem_id": 252, @@ -3856,7 +3693,7 @@ "https://oeis.org/A227989" ], "comments_problem_id": 252, - "comments_count": 2 + "comments_count": 3 }, { "problem_id": 254, @@ -3872,8 +3709,8 @@ ], "last_edited": "07 December 2025", "latex_path": "/latex/254", - "formalized": false, - "formalized_url": "", + "formalized": true, + "formalized_url": "https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/ErdosProblems/254.lean", "oeis_urls": [], "comments_problem_id": 254, "comments_count": 3 @@ -3910,7 +3747,7 @@ "tags": [ "irrationality" ], - "last_edited": "02 December 2025", + "last_edited": "15 April 2026", "latex_path": "/latex/257", "formalized": true, "formalized_url": "https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/ErdosProblems/257.lean", @@ -3918,26 +3755,6 @@ "comments_problem_id": 257, "comments_count": 6 }, - { - "problem_id": 258, - "problem_url": "/258", - "status_bucket": "open", - "status_dom_id": "open", - "status_label": "OPEN", - "status_detail": "This is open, and cannot be resolved with a finite computation.", - "prize_amount": "", - "statement": "Let $a_1,a_2,\\ldots$ be a sequence of positive integers with $a_n\\to \\infty$. Is\\[\\sum_{n} \\frac{\\tau(n)}{a_1\\cdots a_n}\\]irrational, where $\\tau(n)$ is the number of divisors of $n$?", - "tags": [ - "irrationality" - ], - "last_edited": "20 January 2026", - "latex_path": "/latex/258", - "formalized": true, - "formalized_url": "https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/ErdosProblems/258.lean", - "oeis_urls": [], - "comments_problem_id": 258, - "comments_count": 0 - }, { "problem_id": 260, "problem_url": "/260", @@ -3952,8 +3769,8 @@ ], "last_edited": "01 February 2026", "latex_path": "/latex/260", - "formalized": false, - "formalized_url": "", + "formalized": true, + "formalized_url": "https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/ErdosProblems/260.lean", "oeis_urls": [], "comments_problem_id": 260, "comments_count": 2 @@ -3976,7 +3793,7 @@ "formalized_url": "", "oeis_urls": [], "comments_problem_id": 261, - "comments_count": 1 + "comments_count": 3 }, { "problem_id": 263, @@ -3986,17 +3803,17 @@ "status_label": "OPEN", "status_detail": "This is open, and cannot be resolved with a finite computation.", "prize_amount": "", - "statement": "Let $a_n$ be a sequence of positive integers such that for every sequence of positive integers $b_n$ with $b_n/a_n\\to 1$ the sum\\[\\sum\\frac{1}{b_n}\\]is irrational. Is $a_n=2^{2^n}$ such a sequence? Must such a sequence satisfy $a_n^{1/n}\\to \\infty$?", + "statement": "Let $a_n$ be an increasing sequence of positive integers such that for every sequence of positive integers $b_n$ with $b_n/a_n\\to 1$ the sum\\[\\sum\\frac{1}{b_n}\\]is irrational. Is $a_n=2^{2^n}$ such a sequence? Must such a sequence satisfy $a_n^{1/n}\\to \\infty$?", "tags": [ "irrationality" ], - "last_edited": "20 January 2026", + "last_edited": "02 April 2026", "latex_path": "/latex/263", "formalized": true, "formalized_url": "https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/ErdosProblems/263.lean", "oeis_urls": [], "comments_problem_id": 263, - "comments_count": 3 + "comments_count": 6 }, { "problem_id": 264, @@ -4056,7 +3873,7 @@ "formalized_url": "https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/ErdosProblems/267.lean", "oeis_urls": [], "comments_problem_id": 267, - "comments_count": 2 + "comments_count": 4 }, { "problem_id": 269, @@ -4099,7 +3916,7 @@ "https://oeis.org/A005487" ], "comments_problem_id": 271, - "comments_count": 6 + "comments_count": 7 }, { "problem_id": 272, @@ -4116,8 +3933,8 @@ ], "last_edited": "", "latex_path": "/latex/272", - "formalized": false, - "formalized_url": "", + "formalized": true, + "formalized_url": "https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/ErdosProblems/272.lean", "oeis_urls": [], "comments_problem_id": 272, "comments_count": 7 @@ -4162,7 +3979,7 @@ "formalized_url": "https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/ErdosProblems/274.lean", "oeis_urls": [], "comments_problem_id": 274, - "comments_count": 2 + "comments_count": 3 }, { "problem_id": 276, @@ -4220,13 +4037,13 @@ "covering systems", "primes" ], - "last_edited": "20 January 2026", + "last_edited": "17 April 2026", "latex_path": "/latex/279", - "formalized": false, - "formalized_url": "", + "formalized": true, + "formalized_url": "https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/ErdosProblems/279.lean", "oeis_urls": [], "comments_problem_id": 279, - "comments_count": 0 + "comments_count": 10 }, { "problem_id": 282, @@ -4243,35 +4060,12 @@ ], "last_edited": "", "latex_path": "/latex/282", - "formalized": false, - "formalized_url": "", + "formalized": true, + "formalized_url": "https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/ErdosProblems/282.lean", "oeis_urls": [], "comments_problem_id": 282, "comments_count": 2 }, - { - "problem_id": 283, - "problem_url": "/283", - "status_bucket": "open", - "status_dom_id": "open", - "status_label": "OPEN", - "status_detail": "This is open, and cannot be resolved with a finite computation.", - "prize_amount": "", - "statement": "Let $p:\\mathbb{Z}\\to \\mathbb{Z}$ be a polynomial whose leading coefficient is positive and such that there exists no $d\\geq 2$ with $d\\mid p(n)$ for all $n\\geq 1$. Is it true that, for all sufficiently large $m$, there exist integers $1\\leq n_1<\\cdots 0$ and $k\\geq 2$. Is it true that, for all sufficiently large $n$, there is a sequence of $k$ consecutive integers in $\\{1,\\ldots,n\\}$ all of which are $n^\\epsilon$-smooth?", - "tags": [ - "number theory" - ], - "last_edited": "", - "latex_path": "/latex/369", - "formalized": false, - "formalized_url": "", - "oeis_urls": [], - "comments_problem_id": 369, - "comments_count": 0 - }, { "problem_id": 371, "problem_url": "/371", @@ -5467,7 +5165,7 @@ "https://oeis.org/A389148" ], "comments_problem_id": 374, - "comments_count": 0 + "comments_count": 1 }, { "problem_id": 375, @@ -5535,31 +5233,6 @@ "comments_problem_id": 377, "comments_count": 1 }, - { - "problem_id": 380, - "problem_url": "/380", - "status_bucket": "open", - "status_dom_id": "open", - "status_label": "OPEN", - "status_detail": "This is open, and cannot be resolved with a finite computation.", - "prize_amount": "", - "statement": "We call an interval $[u,v]$ 'bad' if the greatest prime factor of $\\prod_{u\\leq m\\leq v}m$ occurs with an exponent greater than $1$. Let $B(x)$ count the number of $n\\leq x$ which are contained in at least one bad interval. Is it true that\\[B(x)\\sim \\#\\{ n\\leq x: P(n)^2\\mid n\\},\\]where $P(n)$ is the largest prime factor of $n$?", - "tags": [ - "number theory" - ], - "last_edited": "", - "latex_path": "/latex/380", - "formalized": false, - "formalized_url": "", - "oeis_urls": [ - "https://oeis.org/A070003", - "https://oeis.org/A387054", - "https://oeis.org/A388654", - "https://oeis.org/A389100" - ], - "comments_problem_id": 380, - "comments_count": 5 - }, { "problem_id": 382, "problem_url": "/382", @@ -5645,7 +5318,7 @@ "https://oeis.org/A280992" ], "comments_problem_id": 386, - "comments_count": 12 + "comments_count": 13 }, { "problem_id": 387, @@ -5666,7 +5339,7 @@ "formalized_url": "https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/ErdosProblems/387.lean", "oeis_urls": [], "comments_problem_id": 387, - "comments_count": 6 + "comments_count": 8 }, { "problem_id": 388, @@ -5680,13 +5353,13 @@ "tags": [ "number theory" ], - "last_edited": "", + "last_edited": "11 April 2026", "latex_path": "/latex/388", "formalized": false, "formalized_url": "", "oeis_urls": [], "comments_problem_id": 388, - "comments_count": 5 + "comments_count": 10 }, { "problem_id": 389, @@ -5708,7 +5381,7 @@ "https://oeis.org/A375071" ], "comments_problem_id": 389, - "comments_count": 6 + "comments_count": 8 }, { "problem_id": 390, @@ -5731,7 +5404,7 @@ "https://oeis.org/A193429" ], "comments_problem_id": 390, - "comments_count": 1 + "comments_count": 3 }, { "problem_id": 393, @@ -5754,7 +5427,7 @@ "https://oeis.org/A388302" ], "comments_problem_id": 393, - "comments_count": 6 + "comments_count": 8 }, { "problem_id": 394, @@ -5799,7 +5472,7 @@ "https://oeis.org/A375077" ], "comments_problem_id": 396, - "comments_count": 1 + "comments_count": 35 }, { "problem_id": 398, @@ -5814,15 +5487,16 @@ "number theory", "factorials" ], - "last_edited": "", + "last_edited": "01 April 2026", "latex_path": "/latex/398", "formalized": true, "formalized_url": "https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/ErdosProblems/398.lean", "oeis_urls": [ + "https://oeis.org/A141399", "https://oeis.org/A146968" ], "comments_problem_id": 398, - "comments_count": 6 + "comments_count": 10 }, { "problem_id": 400, @@ -5839,11 +5513,11 @@ ], "last_edited": "", "latex_path": "/latex/400", - "formalized": false, - "formalized_url": "", + "formalized": true, + "formalized_url": "https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/ErdosProblems/400.lean", "oeis_urls": [], "comments_problem_id": 400, - "comments_count": 0 + "comments_count": 1 }, { "problem_id": 404, @@ -5879,13 +5553,13 @@ "number theory", "base representations" ], - "last_edited": "30 September 2025", + "last_edited": "13 April 2026", "latex_path": "/latex/406", "formalized": true, "formalized_url": "https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/ErdosProblems/406.lean", "oeis_urls": [], "comments_problem_id": 406, - "comments_count": 4 + "comments_count": 5 }, { "problem_id": 408, @@ -5978,7 +5652,7 @@ "https://oeis.org/A383044" ], "comments_problem_id": 411, - "comments_count": 1 + "comments_count": 4 }, { "problem_id": 412, @@ -6017,7 +5691,7 @@ "number theory", "iterated functions" ], - "last_edited": "28 December 2025", + "last_edited": "17 April 2026", "latex_path": "/latex/413", "formalized": true, "formalized_url": "https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/ErdosProblems/413.lean", @@ -6025,7 +5699,7 @@ "https://oeis.org/A005236" ], "comments_problem_id": 413, - "comments_count": 2 + "comments_count": 3 }, { "problem_id": 414, @@ -6058,17 +5732,17 @@ "status_label": "OPEN", "status_detail": "This is open, and cannot be resolved with a finite computation.", "prize_amount": "", - "statement": "For any $n$ let $F(n)$ be the largest $k$ such that any of the $k!$ possible ordering patterns appears in some sequence of $\\phi(m+1),\\ldots,\\phi(m+k)$ with $m+k\\leq n$. Is it true that\\[F(n)=(c+o(1))\\log\\log\\log n\\]for some constant $c$? Is the first pattern which fails to appear always\\[\\phi(m+1)>\\phi(m+2)>\\cdots \\phi(m+k)?\\]Is it true that 'natural' ordering which mimics what happens to $\\phi(1),\\ldots,\\phi(k)$ is the most likely to appear?", + "statement": "For any $n$ let $F(n)$ be the largest $k$ such that any of the $k!$ possible ordering patterns appears in some sequence of $\\phi(m+1),\\ldots,\\phi(m+k)$ with $m+k\\leq n$. Is it true that\\[F(n)=(c+o(1))\\log\\log\\log n\\]for some constant $c$? Is the first pattern which fails to appear always\\[\\phi(m+1)>\\phi(m+2)>\\cdots >\\phi(m+k)?\\]Is it true that the 'natural' ordering which mimics what happens to $\\phi(1),\\ldots,\\phi(k)$ is the most likely to appear?", "tags": [ "number theory" ], - "last_edited": "", + "last_edited": "19 April 2026", "latex_path": "/latex/415", "formalized": false, "formalized_url": "", "oeis_urls": [], "comments_problem_id": 415, - "comments_count": 0 + "comments_count": 4 }, { "problem_id": 416, @@ -6113,7 +5787,7 @@ "https://oeis.org/A264810" ], "comments_problem_id": 417, - "comments_count": 3 + "comments_count": 5 }, { "problem_id": 420, @@ -6192,7 +5866,7 @@ "tags": [ "number theory" ], - "last_edited": "16 January 2026", + "last_edited": "23 March 2026", "latex_path": "/latex/423", "formalized": false, "formalized_url": "", @@ -6200,7 +5874,7 @@ "https://oeis.org/A005243" ], "comments_problem_id": 423, - "comments_count": 18 + "comments_count": 38 }, { "problem_id": 424, @@ -6214,7 +5888,7 @@ "tags": [ "number theory" ], - "last_edited": "", + "last_edited": "31 March 2026", "latex_path": "/latex/424", "formalized": true, "formalized_url": "https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/ErdosProblems/424.lean", @@ -6237,13 +5911,13 @@ "number theory", "sidon sets" ], - "last_edited": "28 December 2025", + "last_edited": "06 April 2026", "latex_path": "/latex/425", "formalized": false, "formalized_url": "", "oeis_urls": [], "comments_problem_id": 425, - "comments_count": 0 + "comments_count": 5 }, { "problem_id": 428, @@ -6299,7 +5973,7 @@ "number theory", "primes" ], - "last_edited": "", + "last_edited": "08 April 2026", "latex_path": "/latex/431", "formalized": false, "formalized_url": "", @@ -6363,8 +6037,8 @@ ], "last_edited": "27 December 2025", "latex_path": "/latex/445", - "formalized": false, - "formalized_url": "", + "formalized": true, + "formalized_url": "https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/ErdosProblems/445.lean", "oeis_urls": [], "comments_problem_id": 445, "comments_count": 1 @@ -6410,7 +6084,7 @@ "https://oeis.org/A386620" ], "comments_problem_id": 451, - "comments_count": 2 + "comments_count": 4 }, { "problem_id": 452, @@ -6494,29 +6168,7 @@ "formalized_url": "", "oeis_urls": [], "comments_problem_id": 456, - "comments_count": 2 - }, - { - "problem_id": 457, - "problem_url": "/457", - "status_bucket": "open", - "status_dom_id": "open", - "status_label": "OPEN", - "status_detail": "This is open, and cannot be resolved with a finite computation.", - "prize_amount": "", - "statement": "Is there some $\\epsilon>0$ such that there are infinitely many $n$ where all primes $p\\leq (2+\\epsilon)\\log n$ divide\\[\\prod_{1\\leq i\\leq \\log n}(n+i)?\\]", - "tags": [ - "number theory" - ], - "last_edited": "07 October 2025", - "latex_path": "/latex/457", - "formalized": true, - "formalized_url": "https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/ErdosProblems/457.lean", - "oeis_urls": [ - "https://oeis.org/A391668" - ], - "comments_problem_id": 457, - "comments_count": 7 + "comments_count": 3 }, { "problem_id": 458, @@ -6539,7 +6191,7 @@ "https://oeis.org/A056604" ], "comments_problem_id": 458, - "comments_count": 1 + "comments_count": 2 }, { "problem_id": 460, @@ -6580,7 +6232,7 @@ "formalized_url": "", "oeis_urls": [], "comments_problem_id": 461, - "comments_count": 0 + "comments_count": 1 }, { "problem_id": 462, @@ -6669,7 +6321,7 @@ "https://oeis.org/A387503" ], "comments_problem_id": 468, - "comments_count": 7 + "comments_count": 1 }, { "problem_id": 469, @@ -6775,7 +6427,7 @@ "number theory", "additive combinatorics" ], - "last_edited": "", + "last_edited": "05 March 2026", "latex_path": "/latex/475", "formalized": false, "formalized_url": "", @@ -6791,11 +6443,12 @@ "status_label": "OPEN", "status_detail": "This is open, and cannot be resolved with a finite computation.", "prize_amount": "", - "statement": "Is there a polynomial $f:\\mathbb{Z}\\to \\mathbb{Z}$ of degree at least $2$ and a set $A\\subset \\mathbb{Z}$ such that for any $n\\in \\mathbb{Z}$ there is exactly one $a\\in A$ and $b\\in \\{ f(n) : n\\in\\mathbb{Z}\\}$ such that $n=a+b$?", + "statement": "Is there a polynomial $f:\\mathbb{Z}\\to \\mathbb{Z}$ of degree at least $2$ and a set $A\\subset \\mathbb{Z}$ such that for any $n\\in \\mathbb{Z}$ there is exactly one $a\\in A$ and $b\\in \\{ f(k) : k\\in\\mathbb{Z}\\}$ such that $n=a+b$?", "tags": [ - "number theory" + "number theory", + "sidon sets" ], - "last_edited": "29 December 2025", + "last_edited": "11 April 2026", "latex_path": "/latex/477", "formalized": true, "formalized_url": "https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/ErdosProblems/477.lean", @@ -6816,7 +6469,7 @@ "number theory", "factorials" ], - "last_edited": "04 October 2025", + "last_edited": "12 April 2026", "latex_path": "/latex/478", "formalized": false, "formalized_url": "", @@ -6868,7 +6521,7 @@ "additive combinatorics", "ramsey theory" ], - "last_edited": "14 October 2025", + "last_edited": "10 April 2026", "latex_path": "/latex/483", "formalized": false, "formalized_url": "", @@ -6876,7 +6529,7 @@ "https://oeis.org/A030126" ], "comments_problem_id": 483, - "comments_count": 3 + "comments_count": 4 }, { "problem_id": 486, @@ -6891,7 +6544,7 @@ "number theory", "primitive sets" ], - "last_edited": "12 January 2026", + "last_edited": "08 April 2026", "latex_path": "/latex/486", "formalized": true, "formalized_url": "https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/ErdosProblems/486.lean", @@ -6904,20 +6557,20 @@ "problem_url": "/488", "status_bucket": "open", "status_dom_id": "open", - "status_label": "OPEN", - "status_detail": "This is open, and cannot be resolved with a finite computation.", + "status_label": "FALSIFIABLE", + "status_detail": "Open, but could be disproved with a finite counterexample.", "prize_amount": "", "statement": "Let $A$ be a finite set and\\[B=\\{ n \\geq 1 : a\\mid n\\textrm{ for some }a\\in A\\}.\\]Is it true that, for every $m>n\\geq \\max(A)$,\\[\\frac{\\lvert B\\cap [1,m]\\rvert }{m}< 2\\frac{\\lvert B\\cap [1,n]\\rvert}{n}?\\]", "tags": [ "number theory" ], - "last_edited": "31 December 2025", + "last_edited": "08 April 2026", "latex_path": "/latex/488", "formalized": true, "formalized_url": "https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/ErdosProblems/488.lean", "oeis_urls": [], "comments_problem_id": 488, - "comments_count": 14 + "comments_count": 29 }, { "problem_id": 489, @@ -6937,7 +6590,7 @@ "formalized_url": "https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/ErdosProblems/489.lean", "oeis_urls": [], "comments_problem_id": 489, - "comments_count": 0 + "comments_count": 3 }, { "problem_id": 495, @@ -6999,8 +6652,8 @@ ], "last_edited": "25 January 2026", "latex_path": "/latex/501", - "formalized": false, - "formalized_url": "", + "formalized": true, + "formalized_url": "https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/ErdosProblems/501.lean", "oeis_urls": [], "comments_problem_id": 501, "comments_count": 8 @@ -7026,7 +6679,7 @@ "https://oeis.org/A175769" ], "comments_problem_id": 503, - "comments_count": 1 + "comments_count": 6 }, { "problem_id": 507, @@ -7122,13 +6775,13 @@ "tags": [ "analysis" ], - "last_edited": "28 December 2025", + "last_edited": "02 April 2026", "latex_path": "/latex/513", "formalized": true, "formalized_url": "https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/ErdosProblems/513.lean", "oeis_urls": [], "comments_problem_id": 513, - "comments_count": 5 + "comments_count": 6 }, { "problem_id": 514, @@ -7148,7 +6801,7 @@ "formalized_url": "", "oeis_urls": [], "comments_problem_id": 514, - "comments_count": 1 + "comments_count": 12 }, { "problem_id": 517, @@ -7168,7 +6821,7 @@ "formalized_url": "https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/ErdosProblems/517.lean", "oeis_urls": [], "comments_problem_id": 517, - "comments_count": 0 + "comments_count": 1 }, { "problem_id": 520, @@ -7211,7 +6864,7 @@ "formalized_url": "", "oeis_urls": [], "comments_problem_id": 521, - "comments_count": 13 + "comments_count": 20 }, { "problem_id": 522, @@ -7233,7 +6886,7 @@ "formalized_url": "https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/ErdosProblems/522.lean", "oeis_urls": [], "comments_problem_id": 522, - "comments_count": 2 + "comments_count": 24 }, { "problem_id": 524, @@ -7255,7 +6908,7 @@ "formalized_url": "", "oeis_urls": [], "comments_problem_id": 524, - "comments_count": 10 + "comments_count": 11 }, { "problem_id": 528, @@ -7314,7 +6967,7 @@ "number theory", "sidon sets" ], - "last_edited": "16 October 2025", + "last_edited": "08 April 2026", "latex_path": "/latex/530", "formalized": false, "formalized_url": "", @@ -7357,13 +7010,13 @@ "tags": [ "number theory" ], - "last_edited": "", + "last_edited": "29 April 2026", "latex_path": "/latex/535", - "formalized": false, - "formalized_url": "", + "formalized": true, + "formalized_url": "https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/ErdosProblems/535.lean", "oeis_urls": [], "comments_problem_id": 535, - "comments_count": 1 + "comments_count": 9 }, { "problem_id": 536, @@ -7373,17 +7026,17 @@ "status_label": "OPEN", "status_detail": "This is open, and cannot be resolved with a finite computation.", "prize_amount": "", - "statement": "Let $\\epsilon>0$ and $N$ be sufficiently large. Is it true that if $A\\subseteq \\{1,\\ldots,N\\}$ has size at least $\\epsilon N$ then there must be distinct $a,b,c\\in A$ such that\\[[a,b]=[b,c]=[a,c],\\]where $[a,b]$ denotes the least common multiple?", + "statement": "Let $f(N)$ be the largest size of $A\\subseteq \\{1,\\ldots,N\\}$ with the property that there are no distinct $a,b,c\\in A$ such that\\[[a,b]=[b,c]=[a,c],\\]where $[a,b]$ denotes the least common multiple. Estimate $f(N)$ - in particular, is it true that $f(N)=o(N)$?", "tags": [ "number theory" ], - "last_edited": "12 January 2026", + "last_edited": "29 April 2026", "latex_path": "/latex/536", "formalized": true, "formalized_url": "https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/ErdosProblems/536.lean", "oeis_urls": [], "comments_problem_id": 536, - "comments_count": 3 + "comments_count": 6 }, { "problem_id": 538, @@ -7419,11 +7072,11 @@ ], "last_edited": "22 January 2026", "latex_path": "/latex/539", - "formalized": false, - "formalized_url": "", + "formalized": true, + "formalized_url": "https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/ErdosProblems/539.lean", "oeis_urls": [], "comments_problem_id": 539, - "comments_count": 1 + "comments_count": 2 }, { "problem_id": 544, @@ -7438,7 +7091,7 @@ "graph theory", "ramsey theory" ], - "last_edited": "", + "last_edited": "24 April 2026", "latex_path": "/latex/544", "formalized": false, "formalized_url": "", @@ -7446,7 +7099,7 @@ "https://oeis.org/A000791" ], "comments_problem_id": 544, - "comments_count": 0 + "comments_count": 2 }, { "problem_id": 545, @@ -7478,12 +7131,12 @@ "status_dom_id": "open", "status_label": "FALSIFIABLE", "status_detail": "Open, but could be disproved with a finite counterexample.", - "prize_amount": "", + "prize_amount": "$100", "statement": "Let $n\\geq k+1$. Every graph on $n$ vertices with at least $\\frac{k-1}{2}n+1$ edges contains every tree on $k+1$ vertices.", "tags": [ "graph theory" ], - "last_edited": "23 January 2026", + "last_edited": "07 March 2026", "latex_path": "/latex/548", "formalized": false, "formalized_url": "", @@ -7573,9 +7226,11 @@ "latex_path": "/latex/555", "formalized": false, "formalized_url": "", - "oeis_urls": [], + "oeis_urls": [ + "https://oeis.org/A389313" + ], "comments_problem_id": 555, - "comments_count": 1 + "comments_count": 0 }, { "problem_id": 557, @@ -7809,7 +7464,7 @@ "formalized_url": "", "oeis_urls": [], "comments_problem_id": 569, - "comments_count": 0 + "comments_count": 2 }, { "problem_id": 571, @@ -7824,7 +7479,7 @@ "graph theory", "turan number" ], - "last_edited": "18 January 2026", + "last_edited": "07 March 2026", "latex_path": "/latex/571", "formalized": false, "formalized_url": "", @@ -7852,7 +7507,7 @@ "formalized_url": "", "oeis_urls": [], "comments_problem_id": 572, - "comments_count": 1 + "comments_count": 2 }, { "problem_id": 573, @@ -7877,27 +7532,6 @@ "comments_problem_id": 573, "comments_count": 0 }, - { - "problem_id": 574, - "problem_url": "/574", - "status_bucket": "open", - "status_dom_id": "open", - "status_label": "OPEN", - "status_detail": "This is open, and cannot be resolved with a finite computation.", - "prize_amount": "", - "statement": "Is it true that, for $k\\geq 2$,\\[\\mathrm{ex}(n;\\{C_{2k-1},C_{2k}\\})=(1+o(1))(n/2)^{1+\\frac{1}{k}}.\\]", - "tags": [ - "graph theory", - "turan number" - ], - "last_edited": "18 January 2026", - "latex_path": "/latex/574", - "formalized": false, - "formalized_url": "", - "oeis_urls": [], - "comments_problem_id": 574, - "comments_count": 0 - }, { "problem_id": 575, "problem_url": "/575", @@ -7993,13 +7627,13 @@ "tags": [ "graph theory" ], - "last_edited": "05 March 2026", + "last_edited": "01 April 2026", "latex_path": "/latex/583", "formalized": false, "formalized_url": "", "oeis_urls": [], "comments_problem_id": 583, - "comments_count": 1 + "comments_count": 3 }, { "problem_id": 584, @@ -8020,7 +7654,7 @@ "formalized_url": "", "oeis_urls": [], "comments_problem_id": 584, - "comments_count": 1 + "comments_count": 3 }, { "problem_id": 585, @@ -8059,7 +7693,10 @@ "latex_path": "/latex/588", "formalized": false, "formalized_url": "", - "oeis_urls": [], + "oeis_urls": [ + "https://oeis.org/A006065", + "https://oeis.org/A008997" + ], "comments_problem_id": 588, "comments_count": 0 }, @@ -8121,8 +7758,8 @@ ], "last_edited": "", "latex_path": "/latex/593", - "formalized": false, - "formalized_url": "", + "formalized": true, + "formalized_url": "https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/ErdosProblems/593.lean", "oeis_urls": [], "comments_problem_id": 593, "comments_count": 0 @@ -8142,8 +7779,8 @@ ], "last_edited": "", "latex_path": "/latex/595", - "formalized": false, - "formalized_url": "", + "formalized": true, + "formalized_url": "https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/ErdosProblems/595.lean", "oeis_urls": [], "comments_problem_id": 595, "comments_count": 0 @@ -8168,7 +7805,7 @@ "formalized_url": "", "oeis_urls": [], "comments_problem_id": 596, - "comments_count": 0 + "comments_count": 1 }, { "problem_id": 597, @@ -8211,7 +7848,7 @@ "formalized_url": "https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/ErdosProblems/598.lean", "oeis_urls": [], "comments_problem_id": 598, - "comments_count": 0 + "comments_count": 6 }, { "problem_id": 600, @@ -8269,11 +7906,11 @@ ], "last_edited": "", "latex_path": "/latex/602", - "formalized": false, - "formalized_url": "", + "formalized": true, + "formalized_url": "https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/ErdosProblems/602.lean", "oeis_urls": [], "comments_problem_id": 602, - "comments_count": 0 + "comments_count": 1 }, { "problem_id": 603, @@ -8294,7 +7931,7 @@ "formalized_url": "", "oeis_urls": [], "comments_problem_id": 603, - "comments_count": 2 + "comments_count": 12 }, { "problem_id": 604, @@ -8309,7 +7946,7 @@ "geometry", "distances" ], - "last_edited": "15 October 2025", + "last_edited": "23 March 2026", "latex_path": "/latex/604", "formalized": false, "formalized_url": "", @@ -8356,7 +7993,7 @@ "formalized_url": "", "oeis_urls": [], "comments_problem_id": 610, - "comments_count": 0 + "comments_count": 4 }, { "problem_id": 611, @@ -8450,13 +8087,13 @@ "tags": [ "graph theory" ], - "last_edited": "", + "last_edited": "01 April 2026", "latex_path": "/latex/617", "formalized": true, "formalized_url": "https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/ErdosProblems/617.lean", "oeis_urls": [], "comments_problem_id": 617, - "comments_count": 0 + "comments_count": 1 }, { "problem_id": 619, @@ -8476,7 +8113,7 @@ "formalized_url": "", "oeis_urls": [], "comments_problem_id": 619, - "comments_count": 0 + "comments_count": 1 }, { "problem_id": 620, @@ -8516,7 +8153,7 @@ "formalized_url": "https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/ErdosProblems/623.lean", "oeis_urls": [], "comments_problem_id": 623, - "comments_count": 0 + "comments_count": 1 }, { "problem_id": 624, @@ -8642,27 +8279,7 @@ "formalized_url": "", "oeis_urls": [], "comments_problem_id": 629, - "comments_count": 1 - }, - { - "problem_id": 633, - "problem_url": "/633", - "status_bucket": "open", - "status_dom_id": "open", - "status_label": "OPEN", - "status_detail": "This is open, and cannot be resolved with a finite computation.", - "prize_amount": "$25", - "statement": "Classify those triangles which can only be cut into a square number of congruent triangles.", - "tags": [ - "geometry" - ], - "last_edited": "", - "latex_path": "/latex/633", - "formalized": false, - "formalized_url": "", - "oeis_urls": [], - "comments_problem_id": 633, - "comments_count": 2 + "comments_count": 3 }, { "problem_id": 634, @@ -8717,13 +8334,13 @@ "graph theory", "ramsey theory" ], - "last_edited": "", + "last_edited": "10 April 2026", "latex_path": "/latex/638", "formalized": false, "formalized_url": "", "oeis_urls": [], "comments_problem_id": 638, - "comments_count": 1 + "comments_count": 8 }, { "problem_id": 640, @@ -8765,7 +8382,7 @@ "formalized_url": "", "oeis_urls": [], "comments_problem_id": 642, - "comments_count": 2 + "comments_count": 3 }, { "problem_id": 643, @@ -8820,7 +8437,7 @@ "tags": [ "number theory" ], - "last_edited": "05 October 2025", + "last_edited": "07 April 2026", "latex_path": "/latex/647", "formalized": true, "formalized_url": "https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/ErdosProblems/647.lean", @@ -8831,26 +8448,6 @@ "comments_problem_id": 647, "comments_count": 6 }, - { - "problem_id": 650, - "problem_url": "/650", - "status_bucket": "open", - "status_dom_id": "open", - "status_label": "OPEN", - "status_detail": "This is open, and cannot be resolved with a finite computation.", - "prize_amount": "", - "statement": "Let $f(m)$ be such that if $A\\subseteq \\{1,\\ldots,N\\}$ has $\\lvert A\\rvert=m$ then every interval in $[1,\\infty)$ of length $2N$ contains $\\geq f(m)$ many distinct integers $b_1,\\ldots,b_r$ where each $b_i$ is divisible by some $a_i\\in A$, where $a_1,\\ldots,a_r$ are distinct. Estimate $f(m)$. In particular is it true that $f(m)\\ll m^{1/2}$?", - "tags": [ - "number theory" - ], - "last_edited": "", - "latex_path": "/latex/650", - "formalized": false, - "formalized_url": "", - "oeis_urls": [], - "comments_problem_id": 650, - "comments_count": 0 - }, { "problem_id": 653, "problem_url": "/653", @@ -8866,8 +8463,8 @@ ], "last_edited": "", "latex_path": "/latex/653", - "formalized": false, - "formalized_url": "", + "formalized": true, + "formalized_url": "https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/ErdosProblems/653.lean", "oeis_urls": [], "comments_problem_id": 653, "comments_count": 0 @@ -8908,11 +8505,11 @@ ], "last_edited": "", "latex_path": "/latex/655", - "formalized": false, - "formalized_url": "", + "formalized": true, + "formalized_url": "https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/ErdosProblems/655.lean", "oeis_urls": [], "comments_problem_id": 655, - "comments_count": 4 + "comments_count": 5 }, { "problem_id": 657, @@ -8997,7 +8594,7 @@ "formalized_url": "", "oeis_urls": [], "comments_problem_id": 662, - "comments_count": 2 + "comments_count": 4 }, { "problem_id": 663, @@ -9060,7 +8657,7 @@ "formalized_url": "", "oeis_urls": [], "comments_problem_id": 667, - "comments_count": 0 + "comments_count": 1 }, { "problem_id": 668, @@ -9079,7 +8676,9 @@ "latex_path": "/latex/668", "formalized": false, "formalized_url": "", - "oeis_urls": [], + "oeis_urls": [ + "https://oeis.org/A385657" + ], "comments_problem_id": 668, "comments_count": 3 }, @@ -9120,13 +8719,13 @@ "geometry", "distances" ], - "last_edited": "", + "last_edited": "17 April 2026", "latex_path": "/latex/670", "formalized": false, "formalized_url": "", "oeis_urls": [], "comments_problem_id": 670, - "comments_count": 0 + "comments_count": 4 }, { "problem_id": 671, @@ -9186,7 +8785,7 @@ "formalized_url": "", "oeis_urls": [], "comments_problem_id": 675, - "comments_count": 0 + "comments_count": 7 }, { "problem_id": 676, @@ -9200,7 +8799,7 @@ "tags": [ "number theory" ], - "last_edited": "", + "last_edited": "07 April 2026", "latex_path": "/latex/676", "formalized": false, "formalized_url": "", @@ -9208,7 +8807,7 @@ "https://oeis.org/A390181" ], "comments_problem_id": 676, - "comments_count": 10 + "comments_count": 9 }, { "problem_id": 677, @@ -9228,7 +8827,7 @@ "formalized_url": "https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/ErdosProblems/677.lean", "oeis_urls": [], "comments_problem_id": 677, - "comments_count": 0 + "comments_count": 8 }, { "problem_id": 679, @@ -9242,7 +8841,7 @@ "tags": [ "number theory" ], - "last_edited": "12 January 2026", + "last_edited": "17 April 2026", "latex_path": "/latex/679", "formalized": false, "formalized_url": "", @@ -9310,8 +8909,8 @@ ], "last_edited": "31 December 2025", "latex_path": "/latex/683", - "formalized": false, - "formalized_url": "", + "formalized": true, + "formalized_url": "https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/ErdosProblems/683.lean", "oeis_urls": [ "https://oeis.org/A006530", "https://oeis.org/A074399", @@ -9334,7 +8933,7 @@ "primes", "binomial coefficients" ], - "last_edited": "23 January 2026", + "last_edited": "01 April 2026", "latex_path": "/latex/684", "formalized": false, "formalized_url": "", @@ -9342,7 +8941,7 @@ "https://oeis.org/A392019" ], "comments_problem_id": 684, - "comments_count": 23 + "comments_count": 27 }, { "problem_id": 685, @@ -9364,7 +8963,7 @@ "formalized_url": "", "oeis_urls": [], "comments_problem_id": 685, - "comments_count": 1 + "comments_count": 4 }, { "problem_id": 686, @@ -9378,13 +8977,13 @@ "tags": [ "number theory" ], - "last_edited": "", + "last_edited": "11 April 2026", "latex_path": "/latex/686", "formalized": true, "formalized_url": "https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/ErdosProblems/686.lean", "oeis_urls": [], "comments_problem_id": 686, - "comments_count": 16 + "comments_count": 36 }, { "problem_id": 687, @@ -9421,10 +9020,10 @@ "tags": [ "number theory" ], - "last_edited": "", + "last_edited": "07 April 2026", "latex_path": "/latex/688", - "formalized": false, - "formalized_url": "", + "formalized": true, + "formalized_url": "https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/ErdosProblems/688.lean", "oeis_urls": [], "comments_problem_id": 688, "comments_count": 0 @@ -9441,43 +9040,23 @@ "tags": [ "number theory" ], - "last_edited": "06 December 2025", + "last_edited": "08 April 2026", "latex_path": "/latex/689", "formalized": true, "formalized_url": "https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/ErdosProblems/689.lean", "oeis_urls": [], "comments_problem_id": 689, - "comments_count": 17 + "comments_count": 24 }, { - "problem_id": 690, - "problem_url": "/690", + "problem_id": 691, + "problem_url": "/691", "status_bucket": "open", "status_dom_id": "open", "status_label": "OPEN", "status_detail": "This is open, and cannot be resolved with a finite computation.", "prize_amount": "", - "statement": "Let $d_k(p)$ be the density of those integers whose $k$th smallest prime factor is $p$ (i.e. if $p_10$ and let $N$ be large. Let $A\\subseteq \\{2,\\ldots,N\\}$ be such that $(a,b)=1$ for all $a\\neq b\\in A$ and $\\sum_{n\\in A}\\frac{1}{n}\\leq C$. What choice of such an $A$ minimises the number of integers $m\\leq N$ not divisible by any $a\\in A$?", - "tags": [ - "number theory" - ], - "last_edited": "08 February 2026", - "latex_path": "/latex/783", - "formalized": false, - "formalized_url": "", - "oeis_urls": [], - "comments_problem_id": 783, - "comments_count": 28 - }, { "problem_id": 786, "problem_url": "/786", @@ -10480,7 +9980,7 @@ "tags": [ "number theory" ], - "last_edited": "02 February 2026", + "last_edited": "11 April 2026", "latex_path": "/latex/786", "formalized": true, "formalized_url": "https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/ErdosProblems/786.lean", @@ -10544,8 +10044,8 @@ ], "last_edited": "", "latex_path": "/latex/789", - "formalized": false, - "formalized_url": "", + "formalized": true, + "formalized_url": "https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/ErdosProblems/789.lean", "oeis_urls": [], "comments_problem_id": 789, "comments_count": 0 @@ -10700,18 +10200,18 @@ "status_label": "OPEN", "status_detail": "This is open, and cannot be resolved with a finite computation.", "prize_amount": "", - "statement": "Let $k\\geq 3$ and define $F_k(n)$ to be the minimal $r$ such that there is a graph $G$ on $n$ vertices with $\\lfloor n^2/4\\rfloor+1$ many edges such that the edges can be $r$-coloured so that every subgraph isomorphic to $C_{2k+1}$ has no colour repeating on the edges. Is it true that\\[F_k(n)\\sim n^2/8?\\]", + "statement": "Define the anti-Ramsey number $\\chi_S(n,e,G)$ as the smallest $r$ such that there is a graph with $n$ vertices and $e$ edges with an $r$-colouring of its edges in which every copy of $G$ has entirely distinct edge colours. Is it true that, for all $k\\geq 3$,\\[\\chi_S(n, \\lfloor n^2/4\\rfloor+1,C_{2k+1})\\sim n^2/8?\\]", "tags": [ "graph theory", "ramsey theory" ], - "last_edited": "", + "last_edited": "01 April 2026", "latex_path": "/latex/809", "formalized": false, "formalized_url": "", "oeis_urls": [], "comments_problem_id": 809, - "comments_count": 0 + "comments_count": 1 }, { "problem_id": 810, @@ -10726,13 +10226,13 @@ "graph theory", "ramsey theory" ], - "last_edited": "", + "last_edited": "01 April 2026", "latex_path": "/latex/810", "formalized": false, "formalized_url": "", "oeis_urls": [], "comments_problem_id": 810, - "comments_count": 7 + "comments_count": 9 }, { "problem_id": 811, @@ -10770,9 +10270,11 @@ ], "last_edited": "", "latex_path": "/latex/812", - "formalized": false, - "formalized_url": "", - "oeis_urls": [], + "formalized": true, + "formalized_url": "https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/ErdosProblems/812.lean", + "oeis_urls": [ + "https://oeis.org/A059442" + ], "comments_problem_id": 812, "comments_count": 0 }, @@ -10834,7 +10336,7 @@ "formalized_url": "", "oeis_urls": [], "comments_problem_id": 819, - "comments_count": 0 + "comments_count": 2 }, { "problem_id": 820, @@ -10912,13 +10414,13 @@ "tags": [ "number theory" ], - "last_edited": "", + "last_edited": "17 April 2026", "latex_path": "/latex/826", "formalized": true, "formalized_url": "https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/ErdosProblems/826.lean", "oeis_urls": [], "comments_problem_id": 826, - "comments_count": 1 + "comments_count": 2 }, { "problem_id": 827, @@ -10932,13 +10434,13 @@ "tags": [ "geometry" ], - "last_edited": "24 October 2025", + "last_edited": "11 May 2026", "latex_path": "/latex/827", "formalized": false, "formalized_url": "", "oeis_urls": [], "comments_problem_id": 827, - "comments_count": 1 + "comments_count": 3 }, { "problem_id": 828, @@ -11041,7 +10543,7 @@ "formalized_url": "https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/ErdosProblems/835.lean", "oeis_urls": [], "comments_problem_id": 835, - "comments_count": 4 + "comments_count": 6 }, { "problem_id": 836, @@ -11063,7 +10565,7 @@ "formalized_url": "", "oeis_urls": [], "comments_problem_id": 836, - "comments_count": 0 + "comments_count": 3 }, { "problem_id": 837, @@ -11105,7 +10607,7 @@ "formalized_url": "", "oeis_urls": [], "comments_problem_id": 838, - "comments_count": 0 + "comments_count": 3 }, { "problem_id": 839, @@ -11125,7 +10627,7 @@ "formalized_url": "", "oeis_urls": [], "comments_problem_id": 839, - "comments_count": 1 + "comments_count": 2 }, { "problem_id": 840, @@ -11148,26 +10650,6 @@ "comments_problem_id": 840, "comments_count": 0 }, - { - "problem_id": 846, - "problem_url": "/846", - "status_bucket": "open", - "status_dom_id": "open", - "status_label": "DISPROVED (LEAN)", - "status_detail": "This has been solved in the negative and the proof verified in Lean.", - "prize_amount": "", - "statement": "Let $A\\subset \\mathbb{R}^2$ be an infinite set for which there exists some $\\epsilon>0$ such that in any subset of $A$ of size $n$ there are always at least $\\epsilon n$ with no three on a line. Is it true that $A$ is the union of a finite number of sets where no three are on a line?", - "tags": [ - "geometry" - ], - "last_edited": "", - "latex_path": "/latex/846", - "formalized": true, - "formalized_url": "https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/ErdosProblems/846.lean", - "oeis_urls": [], - "comments_problem_id": 846, - "comments_count": 8 - }, { "problem_id": 849, "problem_url": "/849", @@ -11220,26 +10702,6 @@ "comments_problem_id": 850, "comments_count": 4 }, - { - "problem_id": 851, - "problem_url": "/851", - "status_bucket": "open", - "status_dom_id": "open", - "status_label": "PROVED", - "status_detail": "This has been solved in the affirmative.", - "prize_amount": "", - "statement": "Let $\\epsilon>0$. Is there some $r\\ll_\\epsilon 1$ such that the density of integers of the form $2^k+n$, where $k\\geq 0$ and $n$ has at most $r$ prime divisors, is at least $1-\\epsilon$?", - "tags": [ - "number theory" - ], - "last_edited": "", - "latex_path": "/latex/851", - "formalized": true, - "formalized_url": "https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/ErdosProblems/851.lean", - "oeis_urls": [], - "comments_problem_id": 851, - "comments_count": 5 - }, { "problem_id": 852, "problem_url": "/852", @@ -11263,7 +10725,7 @@ "https://oeis.org/A078515" ], "comments_problem_id": 852, - "comments_count": 1 + "comments_count": 6 }, { "problem_id": 853, @@ -11287,7 +10749,7 @@ "https://oeis.org/A390769" ], "comments_problem_id": 853, - "comments_count": 2 + "comments_count": 3 }, { "problem_id": 854, @@ -11317,15 +10779,15 @@ "problem_url": "/855", "status_bucket": "open", "status_dom_id": "open", - "status_label": "FALSIFIABLE", - "status_detail": "Open, but could be disproved with a finite counterexample.", + "status_label": "OPEN", + "status_detail": "This is open, and cannot be resolved with a finite computation.", "prize_amount": "", "statement": "If $\\pi(x)$ counts the number of primes in $[1,x]$ then is it true that (for large $x$ and $y$)\\[\\pi(x+y) \\leq \\pi(x)+\\pi(y)?\\]", "tags": [ "number theory", "primes" ], - "last_edited": "12 January 2026", + "last_edited": "08 April 2026", "latex_path": "/latex/855", "formalized": true, "formalized_url": "https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/ErdosProblems/855.lean", @@ -11353,7 +10815,7 @@ "formalized_url": "", "oeis_urls": [], "comments_problem_id": 856, - "comments_count": 12 + "comments_count": 17 }, { "problem_id": 857, @@ -11369,33 +10831,12 @@ ], "last_edited": "", "latex_path": "/latex/857", - "formalized": false, - "formalized_url": "", + "formalized": true, + "formalized_url": "https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/ErdosProblems/857.lean", "oeis_urls": [], "comments_problem_id": 857, "comments_count": 1 }, - { - "problem_id": 858, - "problem_url": "/858", - "status_bucket": "open", - "status_dom_id": "open", - "status_label": "OPEN", - "status_detail": "This is open, and cannot be resolved with a finite computation.", - "prize_amount": "", - "statement": "Let $A\\subseteq \\{1,\\ldots,N\\}$ be such that there is no solution to $at=b$ with $a,b\\in A$ and the smallest prime factor of $t$ is $>a$. Estimate the maximum of\\[\\frac{1}{\\log N}\\sum_{n\\in A}\\frac{1}{n}.\\]", - "tags": [ - "number theory", - "primitive sets" - ], - "last_edited": "", - "latex_path": "/latex/858", - "formalized": false, - "formalized_url": "", - "oeis_urls": [], - "comments_problem_id": 858, - "comments_count": 0 - }, { "problem_id": 859, "problem_url": "/859", @@ -11441,28 +10882,6 @@ "comments_problem_id": 860, "comments_count": 3 }, - { - "problem_id": 863, - "problem_url": "/863", - "status_bucket": "open", - "status_dom_id": "open", - "status_label": "OPEN", - "status_detail": "This is open, and cannot be resolved with a finite computation.", - "prize_amount": "", - "statement": "Let $r\\geq 2$ and let $A\\subseteq \\{1,\\ldots,N\\}$ be a set of maximal size such that there are at most $r$ solutions to $n=a+b$ with $a\\leq b$ for any $n$. (That is, $A$ is a $B_2[r]$ set.) Similarly, let $B\\subseteq \\{1,\\ldots,N\\}$ be a set of maximal size such that there are at most $r$ solutions to $n=a-b$ for any $n$. If $\\lvert A\\rvert\\sim c_rN^{1/2}$ as $N\\to \\infty$ and $\\lvert B\\rvert \\sim c_r'N^{1/2}$ as $N\\to \\infty$ then is it true that $c_r\\neq c_r'$ for $r\\geq 2$? Is it true that $c_r'k$, then is it true that, for every $k\\geq 1$,\\[\\liminf_{n\\to \\infty}\\sum_{0\\leq ik$. Estimate $k(n)$.", + "statement": "Let $k(n)$ be the maximal $k$ such that there exists $m\\leq n$ such that each of the integers\\[m+1,\\ldots,m+k\\]are divisible by at least one prime $>k$. Estimate $k(n)$ - in particular, is it true that\\[\\log k(n) \\leq (\\log n)^{1/2+o(1)}?\\]", "tags": [ "number theory" ], - "last_edited": "30 December 2025", + "last_edited": "03 April 2026", "latex_path": "/latex/962", - "formalized": false, - "formalized_url": "", + "formalized": true, + "formalized_url": "https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/ErdosProblems/962.lean", "oeis_urls": [ "https://oeis.org/A327909" ], "comments_problem_id": 962, - "comments_count": 3 + "comments_count": 5 }, { "problem_id": 963, @@ -12830,7 +12167,7 @@ "tags": [ "number theory" ], - "last_edited": "", + "last_edited": "31 March 2026", "latex_path": "/latex/968", "formalized": true, "formalized_url": "https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/ErdosProblems/968.lean", @@ -12944,7 +12281,7 @@ "formalized_url": "", "oeis_urls": [], "comments_problem_id": 973, - "comments_count": 7 + "comments_count": 6 }, { "problem_id": 975, @@ -12988,7 +12325,7 @@ "formalized_url": "", "oeis_urls": [], "comments_problem_id": 976, - "comments_count": 1 + "comments_count": 3 }, { "problem_id": 978, @@ -12998,17 +12335,17 @@ "status_label": "OPEN", "status_detail": "This is open, and cannot be resolved with a finite computation.", "prize_amount": "", - "statement": "Let $f\\in \\mathbb{Z}[x]$ be an irreducible polynomial of degree $k>2$ (and suppose that $k\\neq 2^l$ for any $l\\geq 1$) such that the leading coefficient of $f$ is positive. Does the set of integers $n\\geq 1$ for which $f(n)$ is $(k-1)$-power-free have positive density? If $k>3$ then are there infinitely many $n$ for which $f(n)$ is $(k-2)$-power-free? In particular, does\\[n^4+2\\]represent infinitely many squarefree numbers?", + "statement": "Let $f\\in \\mathbb{Z}[x]$ be an irreducible polynomial of degree $k>2$ (and suppose that $k\\neq 2^l$ for any $l\\geq 1$) such that the leading coefficient of $f$ is positive. Does the set of integers $n\\geq 1$ for which $f(n)$ is $(k-1)$-power-free have positive density? If $k>3$, and for all primes $p$ there exists $n$ such that $p^{k-2}\\nmid f(n)$, then are there infinitely many $n$ for which $f(n)$ is $(k-2)$-power-free? In particular, does\\[n^4+2\\]represent infinitely many squarefree numbers?", "tags": [ "number theory" ], - "last_edited": "05 March 2026", + "last_edited": "31 March 2026", "latex_path": "/latex/978", "formalized": true, "formalized_url": "https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/ErdosProblems/978.lean", "oeis_urls": [], "comments_problem_id": 978, - "comments_count": 8 + "comments_count": 16 }, { "problem_id": 979, @@ -13030,7 +12367,7 @@ "https://oeis.org/A385316" ], "comments_problem_id": 979, - "comments_count": 12 + "comments_count": 11 }, { "problem_id": 982, @@ -13074,7 +12411,7 @@ "formalized_url": "", "oeis_urls": [], "comments_problem_id": 983, - "comments_count": 4 + "comments_count": 7 }, { "problem_id": 985, @@ -13098,7 +12435,7 @@ "https://oeis.org/A219429" ], "comments_problem_id": 985, - "comments_count": 1 + "comments_count": 2 }, { "problem_id": 986, @@ -13124,47 +12461,6 @@ "comments_problem_id": 986, "comments_count": 0 }, - { - "problem_id": 987, - "problem_url": "/987", - "status_bucket": "open", - "status_dom_id": "open", - "status_label": "OPEN", - "status_detail": "This is open, and cannot be resolved with a finite computation.", - "prize_amount": "", - "statement": "Let $x_1,x_2,\\ldots \\in (0,1)$ be an infinite sequence and let\\[A_k=\\limsup_{n\\to \\infty}\\left\\lvert \\sum_{j\\leq n} e(kx_j)\\right\\rvert,\\]where $e(x)=e^{2\\pi ix}$. Is it true that\\[\\limsup_{k\\to \\infty} A_k=\\infty?\\]Is it possible for $A_k=o(k)$?", - "tags": [ - "analysis", - "discrepancy" - ], - "last_edited": "29 December 2025", - "latex_path": "/latex/987", - "formalized": false, - "formalized_url": "", - "oeis_urls": [], - "comments_problem_id": 987, - "comments_count": 7 - }, - { - "problem_id": 990, - "problem_url": "/990", - "status_bucket": "open", - "status_dom_id": "open", - "status_label": "OPEN", - "status_detail": "This is open, and cannot be resolved with a finite computation.", - "prize_amount": "", - "statement": "Let $f=a_0+\\cdots+a_dx^d\\in \\mathbb{C}[x]$ be a polynomial. Is it true that, if $f$ has roots $z_1,\\ldots,z_d$ with corresponding arguments $\\theta_1,\\ldots,\\theta_d\\in [0,2\\pi]$, then for all intervals $I\\subseteq [0,2\\pi]$\\[\\left\\lvert (\\# \\theta_i \\in I) - \\frac{\\lvert I\\rvert}{2\\pi}d\\right\\rvert \\ll \\left(n\\log M\\right)^{1/2},\\]where $n$ is the number of non-zero coefficients of $f$ and\\[M=\\frac{\\lvert a_0\\rvert+\\cdots +\\lvert a_d\\rvert}{(\\lvert a_0\\rvert\\lvert a_d\\rvert)^{1/2}}.\\]", - "tags": [ - "analysis" - ], - "last_edited": "", - "latex_path": "/latex/990", - "formalized": false, - "formalized_url": "", - "oeis_urls": [], - "comments_problem_id": 990, - "comments_count": 1 - }, { "problem_id": 993, "problem_url": "/993", @@ -13183,7 +12479,7 @@ "formalized_url": "", "oeis_urls": [], "comments_problem_id": 993, - "comments_count": 1 + "comments_count": 3 }, { "problem_id": 995, @@ -13204,7 +12500,7 @@ "formalized_url": "", "oeis_urls": [], "comments_problem_id": 995, - "comments_count": 0 + "comments_count": 7 }, { "problem_id": 996, @@ -13224,29 +12520,7 @@ "formalized_url": "https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/ErdosProblems/996.lean", "oeis_urls": [], "comments_problem_id": 996, - "comments_count": 0 - }, - { - "problem_id": 997, - "problem_url": "/997", - "status_bucket": "open", - "status_dom_id": "open", - "status_label": "OPEN", - "status_detail": "This is open, and cannot be resolved with a finite computation.", - "prize_amount": "", - "statement": "Call $x_1,x_2,\\ldots \\in (0,1)$ well-distributed if, for every $\\epsilon>0$, if $k$ is sufficiently large then, for all $n>0$ and intervals $I\\subseteq [0,1]$,\\[\\lvert \\# \\{ n0$ such that\\[\\lim_k \\frac{R(k+1,k)}{R(k,k)}> 1+c.\\]", + "statement": "Let $R(k,l)$ be the usual Ramsey number: the smallest $n$ such that if the edges of $K_n$ are coloured red and blue then there exists either a red $K_k$ or a blue $K_l$. Prove the existence of some $c>0$ such that\\[\\lim_{k\\to \\infty}\\frac{R(k+1,k)}{R(k,k)}> 1+c.\\]", "tags": [ "graph theory", "ramsey theory" ], - "last_edited": "03 December 2025", + "last_edited": "23 March 2026", "latex_path": "/latex/1030", "formalized": false, "formalized_url": "", @@ -13504,7 +12759,7 @@ "https://oeis.org/A059442" ], "comments_problem_id": 1030, - "comments_count": 1 + "comments_count": 2 }, { "problem_id": 1032, @@ -13525,7 +12780,7 @@ "formalized_url": "", "oeis_urls": [], "comments_problem_id": 1032, - "comments_count": 1 + "comments_count": 3 }, { "problem_id": 1033, @@ -13539,7 +12794,7 @@ "tags": [ "graph theory" ], - "last_edited": "", + "last_edited": "03 April 2026", "latex_path": "/latex/1033", "formalized": false, "formalized_url": "", @@ -13585,7 +12840,7 @@ "formalized_url": "https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/ErdosProblems/1038.lean", "oeis_urls": [], "comments_problem_id": 1038, - "comments_count": 127 + "comments_count": 137 }, { "problem_id": 1039, @@ -13606,7 +12861,7 @@ "formalized_url": "", "oeis_urls": [], "comments_problem_id": 1039, - "comments_count": 1 + "comments_count": 15 }, { "problem_id": 1040, @@ -13626,7 +12881,7 @@ "formalized_url": "", "oeis_urls": [], "comments_problem_id": 1040, - "comments_count": 0 + "comments_count": 1 }, { "problem_id": 1041, @@ -13647,27 +12902,27 @@ "formalized_url": "https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/ErdosProblems/1041.lean", "oeis_urls": [], "comments_problem_id": 1041, - "comments_count": 2 + "comments_count": 46 }, { "problem_id": 1045, "problem_url": "/1045", "status_bucket": "open", "status_dom_id": "open", - "status_label": "FALSIFIABLE", - "status_detail": "Open, but could be disproved with a finite counterexample.", + "status_label": "OPEN", + "status_detail": "This is open, and cannot be resolved with a finite computation.", "prize_amount": "", "statement": "Let $z_1,\\ldots,z_n\\in \\mathbb{C}$ with $\\lvert z_i-z_j\\rvert\\leq 2$ for all $i,j$, and\\[\\Delta(z_1,\\ldots,z_n)=\\prod_{i\\neq j}\\lvert z_i-z_j\\rvert.\\]What is the maximum possible value of $\\Delta$? Is it maximised by taking the $z_i$ to be the vertices of a regular polygon?", "tags": [ "analysis" ], - "last_edited": "30 December 2025", + "last_edited": "02 April 2026", "latex_path": "/latex/1045", "formalized": false, "formalized_url": "", "oeis_urls": [], "comments_problem_id": 1045, - "comments_count": 43 + "comments_count": 47 }, { "problem_id": 1049, @@ -13687,7 +12942,7 @@ "formalized_url": "https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/ErdosProblems/1049.lean", "oeis_urls": [], "comments_problem_id": 1049, - "comments_count": 1 + "comments_count": 3 }, { "problem_id": 1052, @@ -13754,7 +13009,7 @@ "https://oeis.org/A167485" ], "comments_problem_id": 1054, - "comments_count": 5 + "comments_count": 22 }, { "problem_id": 1055, @@ -13910,7 +13165,7 @@ "https://oeis.org/A038372" ], "comments_problem_id": 1062, - "comments_count": 4 + "comments_count": 13 }, { "problem_id": 1063, @@ -13976,7 +13231,7 @@ "formalized_url": "", "oeis_urls": [], "comments_problem_id": 1066, - "comments_count": 1 + "comments_count": 2 }, { "problem_id": 1068, @@ -14087,7 +13342,7 @@ "https://oeis.org/A064164" ], "comments_problem_id": 1074, - "comments_count": 3 + "comments_count": 5 }, { "problem_id": 1075, @@ -14122,7 +13377,7 @@ "geometry", "distances" ], - "last_edited": "20 December 2025", + "last_edited": "11 April 2026", "latex_path": "/latex/1082", "formalized": true, "formalized_url": "https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/ErdosProblems/1082.lean", @@ -14147,7 +13402,9 @@ "latex_path": "/latex/1083", "formalized": false, "formalized_url": "", - "oeis_urls": [], + "oeis_urls": [ + "https://oeis.org/A186704" + ], "comments_problem_id": 1083, "comments_count": 0 }, @@ -14187,13 +13444,15 @@ "geometry", "distances" ], - "last_edited": "17 October 2025", + "last_edited": "23 May 2026", "latex_path": "/latex/1085", "formalized": true, "formalized_url": "https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/ErdosProblems/1085.lean", - "oeis_urls": [], + "oeis_urls": [ + "https://oeis.org/A186705" + ], "comments_problem_id": 1085, - "comments_count": 0 + "comments_count": 1 }, { "problem_id": 1086, @@ -14249,7 +13508,7 @@ "tags": [ "geometry" ], - "last_edited": "16 October 2025", + "last_edited": "08 April 2026", "latex_path": "/latex/1088", "formalized": false, "formalized_url": "", @@ -14258,88 +13517,46 @@ "comments_count": 0 }, { - "problem_id": 1091, - "problem_url": "/1091", + "problem_id": 1093, + "problem_url": "/1093", "status_bucket": "open", "status_dom_id": "open", "status_label": "OPEN", "status_detail": "This is open, and cannot be resolved with a finite computation.", "prize_amount": "", - "statement": "Let $G$ be a $K_4$-free graph with chromatic number $4$. Must $G$ contain an odd cycle with at least two diagonals? More generally, is there some $f(r)\\to \\infty$ such that every graph with chromatic number $4$, in which every subgraph on $\\leq r$ vertices has chromatic number $\\leq 3$, contains an odd cycle with at least $f(r)$ diagonals?", + "statement": "For $n\\geq 2k$ we define the deficiency of $\\binom{n}{k}$ as follows. If $\\binom{n}{k}$ is divisible by a prime $p\\leq k$ then the deficiency is undefined. Otherwise, the deficiency is the number of $0\\leq i1$?", "tags": [ - "graph theory", - "chromatic number" + "number theory", + "binomial coefficients" ], - "last_edited": "06 December 2025", - "latex_path": "/latex/1091", - "formalized": false, - "formalized_url": "", + "last_edited": "27 December 2025", + "latex_path": "/latex/1093", + "formalized": true, + "formalized_url": "https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/ErdosProblems/1093.lean", "oeis_urls": [], - "comments_problem_id": 1091, + "comments_problem_id": 1093, "comments_count": 2 }, { - "problem_id": 1092, - "problem_url": "/1092", + "problem_id": 1094, + "problem_url": "/1094", "status_bucket": "open", "status_dom_id": "open", "status_label": "OPEN", "status_detail": "This is open, and cannot be resolved with a finite computation.", "prize_amount": "", - "statement": "Let $f_r(n)$ be maximal such that, if a graph $G$ has the property that every subgraph $H$ on $m$ vertices is the union of a graph with chromatic number $r$ and a graph with $\\leq f_r(m)$ edges, then $G$ has chromatic number $\\leq r+1$. Is it true that $f_2(n) \\gg n$? More generally, is $f_r(n)\\gg_r n$?", + "statement": "For all $n\\geq 2k$ the least prime factor of $\\binom{n}{k}$ is $\\leq \\max(n/k,k)$, with only finitely many exceptions.", "tags": [ - "graph theory", - "chromatic number" + "number theory", + "binomial coefficients" ], - "last_edited": "06 December 2025", - "latex_path": "/latex/1092", + "last_edited": "24 October 2025", + "latex_path": "/latex/1094", "formalized": true, - "formalized_url": "https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/ErdosProblems/1092.lean", + "formalized_url": "https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/ErdosProblems/1094.lean", "oeis_urls": [], - "comments_problem_id": 1092, - "comments_count": 2 - }, - { - "problem_id": 1093, - "problem_url": "/1093", - "status_bucket": "open", - "status_dom_id": "open", - "status_label": "OPEN", - "status_detail": "This is open, and cannot be resolved with a finite computation.", - "prize_amount": "", - "statement": "For $n\\geq 2k$ we define the deficiency of $\\binom{n}{k}$ as follows. If $\\binom{n}{k}$ is divisible by a prime $p\\leq k$ then the deficiency is undefined. Otherwise, the deficiency is the number of $0\\leq i1$?", - "tags": [ - "number theory", - "binomial coefficients" - ], - "last_edited": "27 December 2025", - "latex_path": "/latex/1093", - "formalized": true, - "formalized_url": "https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/ErdosProblems/1093.lean", - "oeis_urls": [], - "comments_problem_id": 1093, - "comments_count": 2 - }, - { - "problem_id": 1094, - "problem_url": "/1094", - "status_bucket": "open", - "status_dom_id": "open", - "status_label": "OPEN", - "status_detail": "This is open, and cannot be resolved with a finite computation.", - "prize_amount": "", - "statement": "For all $n\\geq 2k$ the least prime factor of $\\binom{n}{k}$ is $\\leq \\max(n/k,k)$, with only finitely many exceptions.", - "tags": [ - "number theory", - "binomial coefficients" - ], - "last_edited": "24 October 2025", - "latex_path": "/latex/1094", - "formalized": true, - "formalized_url": "https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/ErdosProblems/1094.lean", - "oeis_urls": [], - "comments_problem_id": 1094, - "comments_count": 1 + "comments_problem_id": 1094, + "comments_count": 1 }, { "problem_id": 1095, @@ -14362,27 +13579,7 @@ "https://oeis.org/A003458" ], "comments_problem_id": 1095, - "comments_count": 6 - }, - { - "problem_id": 1096, - "problem_url": "/1096", - "status_bucket": "open", - "status_dom_id": "open", - "status_label": "OPEN", - "status_detail": "This is open, and cannot be resolved with a finite computation.", - "prize_amount": "", - "statement": "Let $10$ is sufficiently small, $x_{k+1}-x_k \\to 0$?", - "tags": [ - "number theory" - ], - "last_edited": "19 October 2025", - "latex_path": "/latex/1096", - "formalized": false, - "formalized_url": "", - "oeis_urls": [], - "comments_problem_id": 1096, - "comments_count": 0 + "comments_count": 8 }, { "problem_id": 1097, @@ -14392,18 +13589,18 @@ "status_label": "OPEN", "status_detail": "This is open, and cannot be resolved with a finite computation.", "prize_amount": "", - "statement": "Let $A$ be a set of $n$ integers. How many distinct $d$ can occur as the common difference of a three-term arithmetic progression in $A$? Are there always $O(n^{3/2})$ many such $d$?", + "statement": "Let $A$ be a set of $n$ integers. How many distinct $d$ can occur as the common difference of a three-term arithmetic progression in $A$? In particular, are there always $O(n^{3/2})$ many such $d$?", "tags": [ "number theory", "additive combinatorics" ], - "last_edited": "03 December 2025", + "last_edited": "01 April 2026", "latex_path": "/latex/1097", "formalized": true, "formalized_url": "https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/ErdosProblems/1097.lean", "oeis_urls": [], "comments_problem_id": 1097, - "comments_count": 11 + "comments_count": 17 }, { "problem_id": 1100, @@ -14426,7 +13623,7 @@ "https://oeis.org/A325864" ], "comments_problem_id": 1100, - "comments_count": 2 + "comments_count": 5 }, { "problem_id": 1101, @@ -14446,7 +13643,7 @@ "formalized_url": "https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/ErdosProblems/1101.lean", "oeis_urls": [], "comments_problem_id": 1101, - "comments_count": 1 + "comments_count": 3 }, { "problem_id": 1103, @@ -14468,7 +13665,7 @@ "https://oeis.org/A392164" ], "comments_problem_id": 1103, - "comments_count": 4 + "comments_count": 5 }, { "problem_id": 1104, @@ -14487,7 +13684,9 @@ "latex_path": "/latex/1104", "formalized": true, "formalized_url": "https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/ErdosProblems/1104.lean", - "oeis_urls": [], + "oeis_urls": [ + "https://oeis.org/A292528" + ], "comments_problem_id": 1104, "comments_count": 2 }, @@ -14599,13 +13798,13 @@ "tags": [ "number theory" ], - "last_edited": "22 January 2026", + "last_edited": "01 April 2026", "latex_path": "/latex/1110", "formalized": false, "formalized_url": "", "oeis_urls": [], "comments_problem_id": 1110, - "comments_count": 3 + "comments_count": 4 }, { "problem_id": 1111, @@ -14662,8 +13861,8 @@ ], "last_edited": "29 December 2025", "latex_path": "/latex/1113", - "formalized": false, - "formalized_url": "", + "formalized": true, + "formalized_url": "https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/ErdosProblems/1113.lean", "oeis_urls": [ "https://oeis.org/A076336" ], @@ -14722,13 +13921,13 @@ "tags": [ "number theory" ], - "last_edited": "30 December 2025", + "last_edited": "01 April 2026", "latex_path": "/latex/1122", "formalized": false, "formalized_url": "", "oeis_urls": [], "comments_problem_id": 1122, - "comments_count": 0 + "comments_count": 1 }, { "problem_id": 1131, @@ -14749,7 +13948,7 @@ "formalized_url": "", "oeis_urls": [], "comments_problem_id": 1131, - "comments_count": 2 + "comments_count": 5 }, { "problem_id": 1132, @@ -14764,13 +13963,13 @@ "analysis", "polynomials" ], - "last_edited": "23 January 2026", + "last_edited": "01 April 2026", "latex_path": "/latex/1132", "formalized": false, "formalized_url": "", "oeis_urls": [], "comments_problem_id": 1132, - "comments_count": 1 + "comments_count": 2 }, { "problem_id": 1133, @@ -14791,7 +13990,7 @@ "formalized_url": "", "oeis_urls": [], "comments_problem_id": 1133, - "comments_count": 0 + "comments_count": 2 }, { "problem_id": 1135, @@ -14846,8 +14045,8 @@ "problem_url": "/1138", "status_bucket": "open", "status_dom_id": "open", - "status_label": "OPEN", - "status_detail": "This is open, and cannot be resolved with a finite computation.", + "status_label": "DISPROVED (LEAN)", + "status_detail": "This has been solved in the negative and the proof verified in Lean.", "prize_amount": "", "statement": "Let $x/21$. If $d=\\max_{p_n \\left(\\frac{2}{\\pi}-o(1)\\right)\\log n?\\]", - "tags": [ - "analysis", - "polynomials" - ], - "last_edited": "01 February 2026", - "latex_path": "/latex/1153", - "formalized": false, - "formalized_url": "", - "oeis_urls": [], - "comments_problem_id": 1153, - "comments_count": 7 - }, { "problem_id": 1154, "problem_url": "/1154", @@ -15234,7 +14366,7 @@ "tags": [ "combinatorics" ], - "last_edited": "23 January 2026", + "last_edited": "10 April 2026", "latex_path": "/latex/1159", "formalized": false, "formalized_url": "", @@ -15302,7 +14434,7 @@ "formalized_url": "", "oeis_urls": [], "comments_problem_id": 1163, - "comments_count": 1 + "comments_count": 2 }, { "problem_id": 1167, @@ -15319,8 +14451,8 @@ ], "last_edited": "23 January 2026", "latex_path": "/latex/1167", - "formalized": false, - "formalized_url": "", + "formalized": true, + "formalized_url": "https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/ErdosProblems/1167.lean", "oeis_urls": [], "comments_problem_id": 1167, "comments_count": 3 @@ -15344,7 +14476,7 @@ "formalized_url": "", "oeis_urls": [], "comments_problem_id": 1168, - "comments_count": 0 + "comments_count": 1 }, { "problem_id": 1169, @@ -15417,18 +14549,18 @@ "status_label": "OPEN", "status_detail": "This is open, and cannot be resolved with a finite computation.", "prize_amount": "", - "statement": "Establish whether the following are true assuming the generalised continuum hypothesis:\\[\\omega_3 \\to (\\omega_2,\\omega_1+2)^2,\\]\\[\\omega_3\\to (\\omega_2+\\omega_1,\\omega_2+\\omega)^2,\\]\\[\\omega_2\\to (\\omega_1^{\\omega+2}+2, \\omega_1+2)^2.\\]Establish whether the following is true assuming the continuum hypothesis:\\[\\omega_2\\to (\\omega_1+\\omega)_2^2.\\]", + "statement": "Establish whether the following are true assuming the generalised continuum hypothesis:\\[\\omega_3 \\to (\\omega_2,\\omega_1+2)^2,\\]\\[\\omega_3\\to (\\omega_2+\\omega_1,\\omega_2+\\omega)^2,\\]\\[\\omega_2\\to (\\omega_1^{\\omega+2}+2, \\omega_1+2)^2.\\]Establish whether the following is consistent with the generalised continuum hypothesis:\\[\\omega_2\\to (\\omega_1+\\omega)_2^2,\\]or even $\\omega_2 \\to (\\xi)_2^2$ for all $\\xi<\\omega_2$.", "tags": [ "set theory", "ramsey theory" ], - "last_edited": "23 January 2026", + "last_edited": "11 April 2026", "latex_path": "/latex/1172", "formalized": false, "formalized_url": "", "oeis_urls": [], "comments_problem_id": 1172, - "comments_count": 1 + "comments_count": 2 }, { "problem_id": 1173, @@ -15456,8 +14588,8 @@ "problem_url": "/1174", "status_bucket": "open", "status_dom_id": "open", - "status_label": "OPEN", - "status_detail": "This is open, and cannot be resolved with a finite computation.", + "status_label": "NOT DISPROVABLE", + "status_detail": "Open in general, but there exist models of set theory where the result is true.", "prize_amount": "", "statement": "Does there exist a graph $G$ with no $K_4$ such that every edge colouring of $G$ with countably many colours contains a monochromatic $K_3$? Does there exist a graph $G$ with no $K_{\\aleph_1}$ such that every edge colouring of $G$ with countably many colours contains a monochromatic $K_{\\aleph_0}$?", "tags": [ @@ -15470,7 +14602,7 @@ "formalized_url": "", "oeis_urls": [], "comments_problem_id": 1174, - "comments_count": 1 + "comments_count": 2 }, { "problem_id": 1175, @@ -15487,8 +14619,8 @@ ], "last_edited": "", "latex_path": "/latex/1175", - "formalized": false, - "formalized_url": "", + "formalized": true, + "formalized_url": "https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/ErdosProblems/1175.lean", "oeis_urls": [], "comments_problem_id": 1175, "comments_count": 1 @@ -15498,8 +14630,8 @@ "problem_url": "/1176", "status_bucket": "open", "status_dom_id": "open", - "status_label": "OPEN", - "status_detail": "This is open, and cannot be resolved with a finite computation.", + "status_label": "NOT DISPROVABLE", + "status_detail": "Open in general, but there exist models of set theory where the result is true.", "prize_amount": "", "statement": "Let $G$ be a graph with chromatic number $\\aleph_1$. Is it true that there is a colouring of the edges with $\\aleph_1$ many colours such that, in any countable colouring of the vertices, there exists a vertex colour containing all edge colours?", "tags": [ @@ -15512,7 +14644,7 @@ "formalized_url": "https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/ErdosProblems/1176.lean", "oeis_urls": [], "comments_problem_id": 1176, - "comments_count": 0 + "comments_count": 1 }, { "problem_id": 1177, @@ -15556,6 +14688,468 @@ "oeis_urls": [], "comments_problem_id": 1178, "comments_count": 1 + }, + { + "problem_id": 1181, + "problem_url": "/1181", + "status_bucket": "open", + "status_dom_id": "open", + "status_label": "OPEN", + "status_detail": "This is open, and cannot be resolved with a finite computation.", + "prize_amount": "", + "statement": "Let $q(n,k)$ denote the least prime which does not divide $\\prod_{1\\leq i\\leq k}(n+i)$. Is it true that there exists some $c>0$ such that, for all large $n$,\\[q(n,\\log n)<(1-c)(\\log n)^2?\\]", + "tags": [ + "number theory" + ], + "last_edited": "07 March 2026", + "latex_path": "/latex/1181", + "formalized": false, + "formalized_url": "", + "oeis_urls": [], + "comments_problem_id": 1181, + "comments_count": 0 + }, + { + "problem_id": 1182, + "problem_url": "/1182", + "status_bucket": "open", + "status_dom_id": "open", + "status_label": "OPEN", + "status_detail": "This is open, and cannot be resolved with a finite computation.", + "prize_amount": "", + "statement": "Let $f(n)$ be maximal such that there is a connected graph $G$ with $n$ vertices and $f(n)$ edges such that\\[R(K_3,G)= 2n-1.\\]Let $F(n)$ be maximal such that every connected graph $G$ with $n$ vertices and $\\leq F(n)$ edges has\\[R(K_3,G)= 2n-1.\\]Estimate $f(n)$ and $F(n)$. In particular, is it true that $F(n)/n\\to \\infty$?", + "tags": [ + "graph theory", + "ramsey theory" + ], + "last_edited": "11 April 2026", + "latex_path": "/latex/1182", + "formalized": false, + "formalized_url": "", + "oeis_urls": [], + "comments_problem_id": 1182, + "comments_count": 3 + }, + { + "problem_id": 1183, + "problem_url": "/1183", + "status_bucket": "open", + "status_dom_id": "open", + "status_label": "OPEN", + "status_detail": "This is open, and cannot be resolved with a finite computation.", + "prize_amount": "", + "statement": "Let $f(n)$ be maximal such that in any $2$-colouring of the subsets of $\\{1,\\ldots,n\\}$ there is always a monochromatic family of at least $f(n)$ sets which is closed under taking unions and intersections. Estimate $f(n)$. Let $F(n)$ be defined similarly, except that we only require the family be closed under taking unions. Estimate $F(n)$. In particular, is it true that $F(n)\\geq n^{\\omega(n)}$ for some $\\omega(n)\\to \\infty$ as $n\\to \\infty$, and $F(n)<(1+o(1))^n$?", + "tags": [ + "combinatorics", + "ramsey theory" + ], + "last_edited": "", + "latex_path": "/latex/1183", + "formalized": false, + "formalized_url": "", + "oeis_urls": [], + "comments_problem_id": 1183, + "comments_count": 10 + }, + { + "problem_id": 1184, + "problem_url": "/1184", + "status_bucket": "open", + "status_dom_id": "open", + "status_label": "OPEN", + "status_detail": "This is open, and cannot be resolved with a finite computation.", + "prize_amount": "", + "statement": "Let $f(n,k)$ count the number of $1\\leq i\\leq k$ such that $P(n+i)>k$ (where $P(m)$ is the largest prime divisor of $m$). Is it true that, if $\\alpha>1$ is such that $n=k^{\\alpha+o(1)}$, then\\[f(n,k)=(1-\\rho(\\alpha)+o(1))k,\\]where $\\rho$ is the Dickman function ?", + "tags": [ + "number theory", + "primes" + ], + "last_edited": "06 April 2026", + "latex_path": "/latex/1184", + "formalized": false, + "formalized_url": "", + "oeis_urls": [], + "comments_problem_id": 1184, + "comments_count": 1 + }, + { + "problem_id": 1186, + "problem_url": "/1186", + "status_bucket": "open", + "status_dom_id": "open", + "status_label": "OPEN", + "status_detail": "This is open, and cannot be resolved with a finite computation.", + "prize_amount": "", + "statement": "Let $\\delta_k$ be such that in any $2$-colouring of $\\{1,\\ldots,n\\}$ there exist at least $(\\delta_k+o(1))n^2$ many monochromatic $k$-term arithmetic progressions. Give reasonable bounds (or even an asymptotic formula) for $\\delta_k$.", + "tags": [ + "additive combinatorics", + "arithmetic progressions" + ], + "last_edited": "08 April 2026", + "latex_path": "/latex/1186", + "formalized": false, + "formalized_url": "", + "oeis_urls": [], + "comments_problem_id": 1186, + "comments_count": 0 + }, + { + "problem_id": 1188, + "problem_url": "/1188", + "status_bucket": "open", + "status_dom_id": "open", + "status_label": "OPEN", + "status_detail": "This is open, and cannot be resolved with a finite computation.", + "prize_amount": "", + "statement": "Call a set of distinct integers $11$) form an irreducible covering set?", + "tags": [ + "number theory", + "covering systems" + ], + "last_edited": "08 April 2026", + "latex_path": "/latex/1189", + "formalized": false, + "formalized_url": "", + "oeis_urls": [], + "comments_problem_id": 1189, + "comments_count": 6 + }, + { + "problem_id": 1190, + "problem_url": "/1190", + "status_bucket": "open", + "status_dom_id": "open", + "status_label": "OPEN", + "status_detail": "This is open, and cannot be resolved with a finite computation.", + "prize_amount": "", + "statement": "Let\\[\\epsilon_m=\\max \\sum \\frac{1}{n_i}\\]where the maximum is taken over all finite sequences $m0\\]for some $c>0$?", + "tags": [ + "additive combinatorics", + "sidon sets" + ], + "last_edited": "06 April 2026", + "latex_path": "/latex/1191", + "formalized": false, + "formalized_url": "", + "oeis_urls": [], + "comments_problem_id": 1191, + "comments_count": 0 + }, + { + "problem_id": 1192, + "problem_url": "/1192", + "status_bucket": "open", + "status_dom_id": "open", + "status_label": "OPEN", + "status_detail": "This is open, and cannot be resolved with a finite computation.", + "prize_amount": "", + "statement": "For $A\\subset \\mathbb{N}$ let $f_r(n)$ count the number of solutions to $n=a_1+\\cdots+a_r$ with $a_i\\in A$. Does there exist, for all $r\\geq 2$, a basis $A$ of order $r$ (so that $f_r(n)>0$ for all large $n$) such that\\[\\sum_{n\\leq x}f_r(n)^2 \\ll x\\]for all $x$?", + "tags": [ + "additive combinatorics", + "additive basis" + ], + "last_edited": "06 April 2026", + "latex_path": "/latex/1192", + "formalized": false, + "formalized_url": "", + "oeis_urls": [], + "comments_problem_id": 1192, + "comments_count": 0 + }, + { + "problem_id": 1194, + "problem_url": "/1194", + "status_bucket": "open", + "status_dom_id": "open", + "status_label": "OPEN", + "status_detail": "This is open, and cannot be resolved with a finite computation.", + "prize_amount": "", + "statement": "Let $A\\subset\\mathbb{N}$ be such that every integer $n\\geq 1$ can be written uniquely as $a_n-b_n$ for some $a_n,b_n\\in A$. How fast must $a_n/n$ increase?", + "tags": [ + "additive combinatorics", + "additive basis", + "sidon sets" + ], + "last_edited": "24 April 2026", + "latex_path": "/latex/1194", + "formalized": false, + "formalized_url": "", + "oeis_urls": [], + "comments_problem_id": 1194, + "comments_count": 8 + }, + { + "problem_id": 1199, + "problem_url": "/1199", + "status_bucket": "open", + "status_dom_id": "open", + "status_label": "OPEN", + "status_detail": "This is open, and cannot be resolved with a finite computation.", + "prize_amount": "", + "statement": "Is it true that in any $2$-colouring of $\\mathbb{N}$ there exists an infinite set $A$ such that all elements of $A+A$ are the same colour?", + "tags": [ + "additive combinatorics", + "ramsey theory" + ], + "last_edited": "", + "latex_path": "/latex/1199", + "formalized": true, + "formalized_url": "https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/ErdosProblems/1199.lean", + "oeis_urls": [], + "comments_problem_id": 1199, + "comments_count": 3 + }, + { + "problem_id": 1200, + "problem_url": "/1200", + "status_bucket": "open", + "status_dom_id": "open", + "status_label": "OPEN", + "status_detail": "This is open, and cannot be resolved with a finite computation.", + "prize_amount": "", + "statement": "There exists a constant $C$ such that for all large $x$ there is a collection of primes $p_1<\\ldots0$ there exists a $k$ such that the density of $n$ for which\\[P(n(n+1)\\cdots(n+k))>n^{1-\\epsilon}\\]is at least $1-\\eta$ (where $P(m)$ is the greatest prime divisor of $m$)?", + "tags": [ + "number theory", + "primes" + ], + "last_edited": "", + "latex_path": "/latex/1201", + "formalized": false, + "formalized_url": "", + "oeis_urls": [], + "comments_problem_id": 1201, + "comments_count": 10 + }, + { + "problem_id": 1203, + "problem_url": "/1203", + "status_bucket": "open", + "status_dom_id": "open", + "status_label": "OPEN", + "status_detail": "This is open, and cannot be resolved with a finite computation.", + "prize_amount": "", + "statement": "If $\\omega(n)$ counts the number of distinct prime divisors of $n$ then let\\[F(n)=\\max_k \\omega(n+k)\\frac{\\log\\log k}{\\log k}.\\]Prove that $F(n)\\to \\infty$ as $n\\to \\infty$.", + "tags": [ + "number theory" + ], + "last_edited": "07 April 2026", + "latex_path": "/latex/1203", + "formalized": true, + "formalized_url": "https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/ErdosProblems/1203.lean", + "oeis_urls": [], + "comments_problem_id": 1203, + "comments_count": 0 + }, + { + "problem_id": 1204, + "problem_url": "/1204", + "status_bucket": "open", + "status_dom_id": "open", + "status_label": "OPEN", + "status_detail": "This is open, and cannot be resolved with a finite computation.", + "prize_amount": "", + "statement": "We call a sequence of integers $0\\leq a_1<\\cdots 0$?", + "tags": [ + "geometry", + "distances" + ], + "last_edited": "08 April 2026", + "latex_path": "/latex/1207", + "formalized": false, + "formalized_url": "", + "oeis_urls": [], + "comments_problem_id": 1207, + "comments_count": 0 + }, + { + "problem_id": 1208, + "problem_url": "/1208", + "status_bucket": "open", + "status_dom_id": "open", + "status_label": "OPEN", + "status_detail": "This is open, and cannot be resolved with a finite computation.", + "prize_amount": "", + "statement": "For $d\\geq 2$ let $F_d(n)$ be minimal such that every set of $n$ points in $\\mathbb{R}^d$ contains a set of $F_d(n)$ points with distinct distances. Estimate $F_d(n)$ for fixed $d$ as $n\\to \\infty$.", + "tags": [ + "geometry", + "distances" + ], + "last_edited": "08 April 2026", + "latex_path": "/latex/1208", + "formalized": false, + "formalized_url": "", + "oeis_urls": [], + "comments_problem_id": 1208, + "comments_count": 0 + }, + { + "problem_id": 1209, + "problem_url": "/1209", + "status_bucket": "open", + "status_dom_id": "open", + "status_label": "OPEN", + "status_detail": "This is open, and cannot be resolved with a finite computation.", + "prize_amount": "", + "statement": "Let $A=\\{a_11$ and at least one of $x$ or $y$ is composite?", + "tags": [ + "number theory", + "primes" + ], + "last_edited": "08 April 2026", + "latex_path": "/latex/1212", + "formalized": false, + "formalized_url": "", + "oeis_urls": [], + "comments_problem_id": 1212, + "comments_count": 0 } ] }