conjectures.cpp

#include <iostream>
using namespace std;
#include <ctime>
#include <cmath>

//QUESTION NO 1
//PART a)
//  .
bool IsPrime(long long Number)
{

  if (Number == 2)
  {
    return true;
  }
  else if (Number < 2)
  {
    return false;
  }

  long long Limit = Number / 2;

  for (int counter = 2; counter <= Limit; counter++)
  {
    if (Number % counter == 0)
    {
      return false;
    }
  }

  return true;
}
//  .

bool IsPrimeFaster(long long Number)
{
  if (Number == 2)
  {
    return true;
  }
  else if (Number < 2)
  {
    return false;
  }

  long long Limit = sqrt(Number);

  for (int counter = 2; counter <= Limit; counter++)
  {
    if (Number % counter == 0)
    {
      return false;
    }
  }

  return true;
}
//PART b)

bool P(long long n)
{
  long long N = (n * n) + n + 41;

  if (IsPrimeFaster(N) == true)
  {
    return true;
  }

  else if (IsPrimeFaster(N) == false)
  {
    return false;
  }
}
//PART c)
void twinprime(long long range)
{
  for (long long start = range; start > 0; start--)
  {
    if (IsPrimeFaster(start) == true && IsPrimeFaster(start - 2) == true)
    {
      cout << "Twin Primes: " << start << "," << start - 2 << endl;
      break;
    }
  }
}
//Part c

//Problem 2

bool EularConjucture(long long a, long long b, long long c, long long d)
{

  if (pow(a, 4) + pow(b, 4) + pow(c, 4) == pow(d, 4))
  {
    return true;
  }
  else if (pow(a, 4) + pow(b, 4) + pow(c, 4) != pow(d, 4))
  {
    return false;
  }
}

//problem 3

//part a
bool ellipticcurve(long long a, long long b, long long c)
{
  if (313 * (pow(a, 3) + pow(b, 3)) == pow(c, 3))
  {
    return true;
  }
  else if (313 * (pow(a, 3) + pow(b, 3)) != pow(c, 3))
  {
    return false;
  }
}
//part a

//part b

void goldbach(long long even)
{
  long long half = even / 2;

  if (IsPrimeFaster(half) == true)
  {
    cout << "The even number with the sum of two prime numbers is: " << half << "+" << half << endl;
  }

  else if (IsPrimeFaster(half) == false)
  {
    long long p1, p2 = 0;
    long long countx = 0, county = 0;
    for (long long x = half, y = half; x > 0, y < INT_MAX; x--, y++, countx++, county++)
    {
      if (IsPrimeFaster(x) == true && IsPrimeFaster(y) == true && countx == county)
      {
        p1 = x;
        p2 = y;
        break;
      }
    }

    cout << "The even number with the sum of two prime numbers: " << p1 << "+" << p2 << endl;
  }
}

//part b

//part c

bool IsEven(long long Number)
{
  if (Number % 2 == 0)
  {
    return true;
  }

  return false;
}

int Collatz(long long Number)
{
  if (Number / 2 == 1)
  {
    cout << "The number that is shrunked by dividing it by 2 is " << Number << endl;
    cout << "It shrunk to 1" << endl;
    return 0;
  }

  else if (IsEven(Number) == true)
  {
    cout << "The number that is shrunked by dividing it by 2 is " << Number << endl;
    long long Temp_for_Even = Number / 2;
    cout << "The number shrunked is " << Temp_for_Even << endl;
    return (Collatz(Temp_for_Even));
  }

  else if (IsEven(Number) == false)
  {
    cout << "The number that is tripled by multiplying it by 3 and adding 1 is " << Number << endl;
    long long Temp_for_ODD = (3 * Number) + 1;
    cout << "The number increased is" << Temp_for_ODD;
    return (Collatz(Temp_for_ODD));
  }
}

//part c

//problem 3

int main()
{

  cout << "Enter the NUMER of the problem to check that problem (1 to 3)................." << endl;
  int problem;
  cin >> problem;

  system("cls");

  //problem 1
  if (problem == 1)
  {
    cout << "Enter the part of the question you want to check. Enter a,b or c to check." << endl;
    char part;
    cin >> part;

    system("cls");

    //part a
    if (part == 'a')
    {
      cout << "Enter n if you want to check by n/2 method otherwise enter s if you want to check by square root method: " << endl;

      char check;
      cin >> check;

      if (check == 'n')
      {
        system("cls");

        cout << "Enter the number you want to check" << endl;

        long long number = 0;
        cin >> number;

        time_t begin, end;
        time(&begin);

        system("cls");

        if (IsPrime(number) == true)
        {
          cout << "its a prime number" << endl;
        }
        else if (IsPrime(number) == false)
        {
          cout << "its not a prime number" << endl;
        }

        time(&end);

        double difference = difftime(end, begin);

        printf("time taken by function to get executed function() %.2lf seconds.\n", difference);
      }
      else if (check == 's')
      {

        system("cls");

        cout << "Enter the number you want to check" << endl;

        long long number = 0;
        cin >> number;

        time_t begin, end;
        time(&begin);

        system("cls");

        if (IsPrimeFaster(number) == true)
        {
          cout << "its a prime number" << endl;
        }
        else if (IsPrimeFaster(number) == false)
        {
          cout << "its not a prime number" << endl;
        }

        time(&end);

        double difference = difftime(end, begin);

        printf("time taken by function to get executed function() %.2lf seconds.\n", difference);
      }

      //part a
    }
    //part b
    else if (part == 'b')
    {
      cout << "Press u to enter user defined values to check if the proposition p(n)=n^2+n+41 is prime or not else press c to check at which number the proposition fails:" << endl;

      char choice;
      cin >> choice;

      system("cls");
      if (choice == 'u')
      {

        cout << "Enter the number n to check  p(n)=n^2+n+41 is prime or not" << endl;
        long long n;
        cin >> n;

        if (P(n) == true)
        {
          cout << "its a prime number" << endl;
        }
        else if (P(n) == false)
        {
          cout << "its not a prime number" << endl;
        }
      }
      else if (choice == 'c')
      {
        for (int n = 0; n < INT_MAX; n++)
        {
          if (P(n) == false)
          {
            cout << "Claim failed at the number:" << n << endl;
            break;
          }
        }
      }
    }
    //part b

    //part c
    else if (part == 'c')
    {
      cout << "Enter the maximum range under which you want to get twin primes " << endl;
      long long range;
      cin >> range;
      twinprime(range);
    }
    //partc
  }
  //problem 1

  //Problem 2
  else if (problem == 2)
  {
    cout << "Enter a to check first part of the question otherwise enter b to check second part" << endl;
    char choice;
    cin >> choice;
    //part a
    system("cls");
    if (choice == 'a')
    {
      system("cls");
      cout << "Enter a b c and d" << endl;
      long long a, b, c, d;
      cout << "Enter a:";
      cin >> a;
      cout << "Enter b:";
      cin >> b;
      cout << "Enter c:";
      cin >> c;
      cout << "Enter d:";
      cin >> d;
      system("cls");

      if (EularConjucture(a, b, c, d) == true)
      {
        cout << "Conjucture satisfied" << endl;
        cout << "a=" << a << endl;
        cout << "b=" << b << endl;
        cout << "c=" << c << endl;
        cout << "d=" << d << endl;
      }
      else if (EularConjucture(a, b, c, d) == false)
      {
        cout << "Conjucture Failed" << endl;
        cout << "a=" << a << endl;
        cout << "b=" << b << endl;
        cout << "c=" << c << endl;
        cout << "d=" << d << endl;
      }
    }
    //part a

    //part b
    else if (choice == 'b')
    {
      long long range = 2147383647;

      for (int a = range; a > 0; a--)
      {
        for (int b = range; b > 0; b--)
        {
          for (int c = range; c > 0; c--)
          {
            for (int d = range; d > 0; d--)
            {
              if (EularConjucture(a, b, c, d) == true)
              {
                cout << "Finally" << endl;
                cout << "Conjucture satisfied" << endl;
                cout << "a=" << a << endl;
                cout << "b=" << b << endl;
                cout << "c=" << c << endl;
                cout << "d=" << d << endl;
              }
            }
          }
        }
      }

      // 2682440, 15365639, 18796760, 20615673. Solution after long term testing.
    }
    //part b
  }
  //problem2

  //problem 3
  else if (problem == 3)
  {
    cout << "Enter The part of the question you want to check (a,b,c)" << endl;
    char part;
    cin >> part;

    system("cls");

    //part a
    if (part == 'a')
    {
      cout << "Press e to enter triptlet to check if it satisfies 313(a^3+b^3)=c^3.\n\nElse press t to find 3 tuple which satisfies given proposition:" << endl;

      char option;
      cin >> option;

      system("cls");

      if (option == 'e')
      {
        cout << "Enter a,b and c" << endl;
        long long a, b, c;
        cout << "Enter a:";
        cin >> a;
        cout << "Enter b:";
        cin >> b;
        cout << "Enter c:";
        cin >> c;
        system("cls");
        if (ellipticcurve(a, b, c) == true)
        {
          cout << "Conjucture satisfied" << endl;
          cout << "a=" << a << endl;
          cout << "b=" << b << endl;
          cout << "c=" << c << endl;
        }
        else if (ellipticcurve(a, b, c) == false)
        {
          cout << "Conjucture Failed" << endl;
          cout << "a=" << a << endl;
          cout << "b=" << b << endl;
          cout << "c=" << c << endl;
        }
      }
      else if (option == 't')
      {
        long long range = 2147383647;

        for (int a = range; a > 0; a--)
        {
          for (int b = range; b > 0; b--)
          {
            for (int c = range; c > 0; c--)
            {
              if (ellipticcurve(a, b, c) == true)
              {
                cout << "Finally!!" << endl;
                cout << "Conjucture satisfied" << endl;
                cout << "a=" << a << endl;
                cout << "b=" << b << endl;
                cout << "c=" << c << endl;
              }
            }
          }
        }
      }
    }
    //part b
    else if (part == 'b')
    {
      cout << "How many numbers you want to test :" << endl;
      int limit;
      cin >> limit;
      system("cls");

      for (int start = 0; start < limit; start++)
      {
        cout << "Enter only the even number you want to test: " << endl;
        long long number;
        cin >> number;

        goldbach(number);
        cout << endl;
      }
    }
    //part b

    else if (part == 'c')
    {
      cout << "How many numbers you want to test :" << endl;
      int limit;
      cin >> limit;
      system("cls");

      for (int start = 0; start < limit; start++)
      {
        cout << "Enter number you want to test: " << endl;
        long long number;
        cin >> number;

        Collatz(number);
        cout << endl;
      }
    }
  }
  //problem 3

  system("pause");
}