Check n^2 – m^2 is prime or not

Given two integers n and m. Check n^2 – m^2 is prime or not. n and m can be very large.


Examples:

Input : n = 6, m = 5
Output : YES

Input : n = 16, m = 13
Output : NO

A simple solution is to fist compute n^2 – m^2, then check if it is prime or not. n^2 – m^2 might be very large – it might not even fit into 64-bit integer. Checking primality for it certainly cannot be performed naively.



A better solution is to express n^2 – m^2 as (n-m)(n+m). This is prime if and only if n-m = 1 and n+m is a prime.

C++

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// CPP program to find n^2 - m^2
// is prime or not.
#include <bits/stdc++.h>
using namespace std;
  
// Check a number is prime or not
bool isprime(int x)
{
    // run a loop upto square of given number
    for (int i = 2; i * i <= x; i++)
        if (x % i == 0)
            return false;
    return true;
}
  
// Check if n^2 - m^2 is prime
bool isNSqMinusnMSqPrime(int m, int n)
{
    if (n - m == 1 and isprime(m + n))
        return true;
    else
        return false;
}
  
// Driver code
int main()
{
    int m = 13, n = 16;
    if (isNSqMinusnMSqPrime(m, n))
        cout << "YES";
    else
        cout << "NO";
  
    return 0;
}

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Java














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// Java program to find n^2 - m^2 


// is prime or not. 


  


class GFG 


{ 


        // Check if a number is prime or not 


        static boolean isprime(int x) 


        { 


            // run a loop upto square of given number 


            for (int i = 2; i * i <= x; i++) 


                if (x % i == 0) 


                    return false; 


            return true; 


        } 


          


        // Check if n^2 - m^2 is prime 


        static boolean isNSqMinusnMSqPrime(int m, int n) 


        { 


            if (n - m == 1 && isprime(m + n)) 


                return true; 


            else


                return false; 


        } 


          


        // Driver code 


        public static void  main(String [] args) 


        { 


            int m = 13, n = 16; 


            if (isNSqMinusnMSqPrime(m, n)) 


                System.out.println("YES"); 


            else


                System.out.println("NO"); 


          


        } 


} 


  


// This code is contributed 


// by ihritik 



















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Python3














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# Python program to find n^2 - m^2  


# is prime or not.  


  


# Check a number is prime or not  


def isprime(x):  


  


    # run a loop upto square  


    # of given number  


    for i in range(2, math.sqrt(x)):  


        if (x % i == 0) : 


            return False;  


    return True;  


  


# Check if n^2 - m^2 is prime  


def isNSqMinusnMSqPrime( m, n):  


  


    if (n - m == 1 and isprime(m + n)):  


        return True;  


    else: 


        return False;  


  


# Driver code  


m = 13; 


n = 16;  


if (isNSqMinusnMSqPrime(m, n)) : 


    print ( "YES");  


else: 


    print ("NO");  


  


# This code is contributed  


# by Shivi_Aggarwal 



















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C#














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// C# program to find n^2 - m^2  


// is prime or not.  


using System; 


  


class GFG  


{  


// Check if a number is prime or not  


static bool isprime(int x)  


{  


    // run a loop upto square 


    // of given number  


    for (int i = 2; i * i <= x; i++)  


        if (x % i == 0)  


            return false;  


    return true;  


}  


  


// Check if n^2 - m^2 is prime  


static bool isNSqMinusnMSqPrime(int m,  


                                int n)  


{  


    if (n - m == 1 && isprime(m + n))  


        return true;  


    else


        return false;  


}  


  


// Driver code  


public static void Main()  


{  


    int m = 13, n = 16;  


    if (isNSqMinusnMSqPrime(m, n))  


        Console.Write("YES");  


    else


        Console.Write("NO");  


}  


}  


  


// This code is contributed  


// by Smitha 



















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PHP














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<?php 


//PHP program to find n^2 - m^2 


// is prime or not. 


  


// Check a number is prime or not 


function isprime($x) 


{ 


    // run a loop upto square  


    // of given number 


    for ( $i = 2; $i * $i <= $x; $i++) 


        if ($x % i == 0) 


            return false; 


    return true; 


} 


  


// Check if n^2 - m^2 is prime 


function isNSqMinusnMSqPrime($m, $n) 


{ 


    if ($n - $m == 1 and isprime($m + $n)) 


        return true; 


    else


        return false; 


} 


  


// Driver code 


$m = 13; $n = 16; 


if (isNSqMinusnMSqPrime($m, $n)) 


    echo "YES"; 


else


    echo "NO"; 


      


// This code is contributed  


// by inder_verma 


?> 



















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Output:


NO



Time Complexity: O(sqrt(n+m))



          
          
          
          

            

    

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