# Find the prime P using given four integers

Given four integers X, Y, X2%P, Y2%P, where P is a prime number. The task is to find prime P.

Examples:

Input : X = 3, XsqmodP = 0, Y = 5, YsqmodP = 1

Output : 3
When x = 3, x2 = 9, and 9 modulo P is 0. So possible value of p is 3
When x = 5, x2 = 25, and 25 modulo P is 1. So possible value of p is 3

Input : X = 4, XsqmodP = 1, Y = 5, YsqmodP = 0
Output : 5

## Recommended: Please try your approach on {IDE} first, before moving on to the solution.

Approach :
From above numbers we get,

X2 – XsqmodP = 0 mod P
Y2 – YsqmodP = 0 mod P

Now find all the common prime factors from both the equation, and check if it satisfies the original equation, If it does (one of them will since answer always exists) then that’s the answer.

Below is the implementation of the above approach :

## C++

 `// CPP program to possible prime number ` `#include ` `using` `namespace` `std; ` ` `  `// Function to check if a  ` `// number is prime or not ` `bool` `Prime(``int` `n) ` `{ ` `    ``for` `(``int` `j = 2; j <= ``sqrt``(n); j++) ` `        ``if` `(n % j == 0) ` `            ``return` `false``; ` ` `  `    ``return` `true``;  ` `} ` ` `  ` `  `// Function to find possible prime number ` `int` `find_prime(``int` `x, ``int` `xsqmodp , ``int` `y, ``int` `ysqmodp) ` `{ ` `     `  `    ``int` `n = x*x - xsqmodp; ` `    ``int` `n1 = y*y - ysqmodp; ` `     `  `    ``// Find a possible prime number ` `    ``for` `(``int` `j = 2; j <= max(``sqrt``(n), ``sqrt``(n1)); j++)  ` `    ``{ ` `        ``if` `(n % j == 0 && (x * x) % j == xsqmodp &&  ` `                  ``n1 % j == 0 && (y * y) % j == ysqmodp) ` `            ``if` `(Prime(j)) ` `                ``return` `j; ` `                 `  `        ``int` `j1 = n / j; ` `        ``if` `(n % j1 == 0 && (x * x) % j1 == xsqmodp  ` `             ``&& n1 % j1 == 0 && (y * y) % j1 == ysqmodp) ` `            ``if` `(Prime(j1)) ` `                ``return` `j1; ` `         `  `        ``j1 = n1 / j; ` `        ``if` `(n % j1 == 0 && (x * x) % j1 == xsqmodp &&  ` `                ``n1 % j1 == 0 && (y * y) % j1 == ysqmodp) ` `            ``if` `(Prime(j1)) ` `                ``return` `j1; ` `    ``} ` `     `  `    ``// Last condition ` `    ``if` `(n == n1) ` `        ``return` `n; ` `} ` ` `  `// Driver code ` `int` `main() ` `{ ` `    ``int` `x = 3, xsqmodp = 0, y = 5, ysqmodp = 1; ` `     `  `    ``// Function call ` `    ``cout << find_prime(x, xsqmodp, y, ysqmodp); ` ` `  `    ``return` `0; ` `} `

## Java

 `// Java program to possible prime number ` `import` `java.util.*; ` `class` `GFG ` `{ ` ` `  `// Function to check if a  ` `// number is prime or not ` `static` `boolean` `Prime(``int` `n) ` `{ ` `    ``for` `(``int` `j = ``2``;  ` `             ``j <= Math.sqrt(n); j++) ` `        ``if` `(n % j == ``0``) ` `            ``return` `false``; ` ` `  `    ``return` `true``;  ` `} ` ` `  `// Function to find possible prime number ` `static` `int` `find_prime(``int` `x, ``int` `xsqmodp ,  ` `                      ``int` `y, ``int` `ysqmodp) ` `{ ` `    ``int` `n = x * x - xsqmodp; ` `    ``int` `n1 = y * y - ysqmodp; ` `     `  `    ``// Find a possible prime number ` `    ``for` `(``int` `j = ``2``;  ` `             ``j <= Math.max(Math.sqrt(n),  ` `                           ``Math.sqrt(n1)); j++)  ` `    ``{ ` `        ``if` `(n % j == ``0` `&& (x * x) % j == xsqmodp &&  ` `            ``n1 % j == ``0` `&& (y * y) % j == ysqmodp) ` `            ``if` `(Prime(j)) ` `                ``return` `j; ` `                 `  `        ``int` `j1 = n / j; ` `        ``if` `(n % j1 == ``0` `&& (x * x) % j1 == xsqmodp &&  ` `            ``n1 % j1 == ``0` `&& (y * y) % j1 == ysqmodp) ` `            ``if` `(Prime(j1)) ` `                ``return` `j1; ` `         `  `        ``j1 = n1 / j; ` `        ``if` `(n % j1 == ``0` `&& (x * x) % j1 == xsqmodp &&  ` `            ``n1 % j1 == ``0` `&& (y * y) % j1 == ysqmodp) ` `            ``if` `(Prime(j1)) ` `                ``return` `j1; ` `    ``} ` `     `  `    ``// Last condition ` `    ``if` `(n == n1) ` `        ``return` `n; ` `    ``return` `Integer.MIN_VALUE; ` `} ` ` `  `// Driver code ` `public` `static` `void` `main(String[] args)  ` `{ ` `    ``int` `x = ``3``, xsqmodp = ``0``,  ` `        ``y = ``5``, ysqmodp = ``1``; ` `     `  `    ``// Function call ` `    ``System.out.println(find_prime(x, xsqmodp,  ` `                                  ``y, ysqmodp)); ` `} ` `} ` ` `  `// This code is contributed by PrinciRaj1992  `

## Python3

 `# Pyhton 3 program to possible prime number ` `from` `math ``import` `sqrt ` ` `  `# Function to check if a  ` `# number is prime or not ` `def` `Prime(n): ` `    ``for` `j ``in` `range``(``2``, ``int``(sqrt(n)) ``+` `1``, ``1``): ` `        ``if` `(n ``%` `j ``=``=` `0``): ` `            ``return` `False` ` `  `    ``return` `True` ` `  `# Function to find possible prime number ` `def` `find_prime(x, xsqmodp, y, ysqmodp): ` `    ``n ``=` `x ``*` `x ``-` `xsqmodp ` `    ``n1 ``=` `y ``*` `y ``-` `ysqmodp ` `     `  `    ``# Find a possible prime number ` `    ``for` `j ``in` `range``(``2``, ``max``(``int``(sqrt(n)),  ` `                          ``int``(sqrt(n1))), ``1``): ` `        ``if` `(n ``%` `j ``=``=` `0` `and` `(x ``*` `x) ``%` `j ``=``=` `xsqmodp ``and`  `            ``n1 ``%` `j ``=``=` `0` `and` `(y ``*` `y) ``%` `j ``=``=` `ysqmodp): ` `            ``if` `(Prime(j)): ` `                ``return` `j ` `                 `  `        ``j1 ``=` `n ``/``/` `j ` `        ``if` `(n ``%` `j1 ``=``=` `0` `and` `(x ``*` `x) ``%` `j1 ``=``=` `xsqmodp ``and` `            ``n1 ``%` `j1 ``=``=` `0` `and` `(y ``*` `y) ``%` `j1 ``=``=` `ysqmodp): ` `            ``if` `(Prime(j1)): ` `                ``return` `j1 ` `         `  `        ``j1 ``=` `n1 ``/``/` `j ` `        ``if` `(n ``%` `j1 ``=``=` `0` `and` `(x ``*` `x) ``%` `j1 ``=``=` `xsqmodp ``and`  `            ``n1 ``%` `j1 ``=``=` `0` `and` `(y ``*` `y) ``%` `j1 ``=``=` `ysqmodp): ` `            ``if` `(Prime(j1)): ` `                ``return` `j1 ` `     `  `    ``# Last condition ` `    ``if` `(n ``=``=` `n1): ` `        ``return` `n ` ` `  `# Driver code ` `if` `__name__ ``=``=` `'__main__'``: ` `    ``x ``=` `3` `    ``xsqmodp ``=` `0` `    ``y ``=` `5` `    ``ysqmodp ``=` `1` `     `  `    ``# Function call ` `    ``print``(find_prime(x, xsqmodp, y, ysqmodp)) ` ` `  `# This code is contributed by ` `# Surendra_Gangwar `

## C#

 `// C# program to possible prime number ` `using` `System; ` ` `  `class` `GFG ` `{ ` ` `  `// Function to check if a  ` `// number is prime or not ` `static` `bool` `Prime(``int` `n) ` `{ ` `    ``for` `(``int` `j = 2;  ` `             ``j <= Math.Sqrt(n); j++) ` `        ``if` `(n % j == 0) ` `            ``return` `false``; ` ` `  `    ``return` `true``;  ` `} ` ` `  `// Function to find possible prime number ` `static` `int` `find_prime(``int` `x, ``int` `xsqmodp ,  ` `                      ``int` `y, ``int` `ysqmodp) ` `{ ` `    ``int` `n = x * x - xsqmodp; ` `    ``int` `n1 = y * y - ysqmodp; ` `     `  `    ``// Find a possible prime number ` `    ``for` `(``int` `j = 2;  ` `            ``j <= Math.Max(Math.Sqrt(n),  ` `                          ``Math.Sqrt(n1)); j++)  ` `    ``{ ` `        ``if` `(n % j == 0 && (x * x) % j == xsqmodp &&  ` `            ``n1 % j == 0 && (y * y) % j == ysqmodp) ` `            ``if` `(Prime(j)) ` `                ``return` `j; ` `                 `  `        ``int` `j1 = n / j; ` `        ``if` `(n % j1 == 0 && (x * x) % j1 == xsqmodp &&  ` `            ``n1 % j1 == 0 && (y * y) % j1 == ysqmodp) ` `            ``if` `(Prime(j1)) ` `                ``return` `j1; ` `         `  `        ``j1 = n1 / j; ` `        ``if` `(n % j1 == 0 && (x * x) % j1 == xsqmodp &&  ` `            ``n1 % j1 == 0 && (y * y) % j1 == ysqmodp) ` `            ``if` `(Prime(j1)) ` `                ``return` `j1; ` `    ``} ` `     `  `    ``// Last condition ` `    ``if` `(n == n1) ` `        ``return` `n; ` `    ``return` `int``.MinValue; ` `} ` ` `  `// Driver code ` `public` `static` `void` `Main()  ` `{ ` `    ``int` `x = 3, xsqmodp = 0,  ` `        ``y = 5, ysqmodp = 1; ` `     `  `    ``// Function call ` `    ``Console.WriteLine(find_prime(x, xsqmodp,  ` `                                 ``y, ysqmodp)); ` `} ` `} ` ` `  `// This code is contributed by anuj_67.. `

Output:

```3
```

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