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 first 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++

// 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;
}

C

// C program to find n^2 - m^2
// is prime or not.
#include <stdio.h>

// Check a number is prime or not
int isprime(int x)
{

    // run a loop upto square of given number
    for (int i = 2; i * i <= x; i++)
        if (x % i == 0)
            return 0;
    return 1;
}

// Check if n^2 - m^2 is prime
int isNSqMinusnMSqPrime(int m, int n)
{
    if (n - m == 1 && isprime(m + n))
        return 1;
    else
        return 0;
}

// Driver code
int main()
{
    int m = 13, n = 16;
    if (isNSqMinusnMSqPrime(m, n))
        printf("YES");
    else
        printf("NO");
    return 0;
}

Java

// 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

Python3

# 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

C#

// 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

Javascript

<script>

// JavaScript program to find n^2 - m^2
// is prime or not.

// Check if a number is prime or not
function isprime(x)
{

    // Run a loop upto square of given number
    for(var 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 && isprime(m + n))
        return true;
    else
        return false;
}

// Driver Code
var m = 13, n = 16;

if (isNSqMinusnMSqPrime(m, n))
    document.write("YES");
else
    document.write("NO");

// This code is contributed by Khushboogoyal499

</script>

PHP

<?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
?>

Output

NO

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

Auxiliary Space: O(1) 

Approach:

The approach is to use a property of prime numbers which states that if a number p is prime, then p^2 – 1 is divisible by 24. So, if n^2 – m^2 is prime, then it should satisfy this property as well.

We can check if n^2 – m^2 is divisible by 24 or not. If it is, then it can be a prime number, otherwise, it is definitely not a prime number.

So, in the code, we first calculate n^2 – m^2 and then check if it is less than or equal to 1. If it is, then it is not a prime number. Otherwise, we check if it is divisible by 24 by checking if (n^2 – m^2 + 1) is divisible by 24 or not. If it is, then we return true, else we return false.

Below is the implementation of the above approach:

C++

#include <bits/stdc++.h>
using namespace std;

// Check if n^2 - m^2 is prime
bool isNSqMinusnMSqPrime(int m, int n)
{
    int num = n*n - m*m;

    if (num <= 1)
        return false;

    // Check if p^2 - 1 is divisible by 24
    if ((num+1)%24 == 0)
        return true;

    return false;
}

// Driver code
int main()
{
    int m = 13, n = 16;

    if (isNSqMinusnMSqPrime(m, n))
        cout << "YES";
    else
        cout << "NO";

    return 0;
}

Java

import java.util.*;

public
class Main {

    // Check if n^2 - m^2 is prime
    static boolean isNSqMinusnMSqPrime(int m, int n)
    {
        int num = n * n - m * m;

        if (num <= 1)
            return false;

        // Check if p^2 - 1 is divisible by 24
        if ((num + 1) % 24 == 0)
            return true;

        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 Prajwal Kandekar

Python3

# Check if n^2 - m^2 is prime
def isNSqMinusnMSqPrime(m, n):
    num = n*n - m*m
    if num <= 1:
        return False

    # Check if p^2 - 1 is divisible by 24
    if (num+1) % 24 == 0:
        return True
    return False

# Driver code
if __name__ == '__main__':

    m, n = 13, 16
    if isNSqMinusnMSqPrime(m, n):
        print("YES")
    else:
        print("NO")

C#

using System;

class Program
{
    // Check if n^2 - m^2 is prime
    static bool IsNSqMinusnMSqPrime(int m, int n)
    {
        int num = n * n - m * m;

        if (num <= 1)
            return false;

        // Check if p^2 - 1 is divisible by 24
        if ((num + 1) % 24 == 0)
            return true;

        return false;
    }

    // Driver code
    static void Main()
    {
        int m = 13, n = 16;

        if (IsNSqMinusnMSqPrime(m, n))
            Console.WriteLine("YES");
        else
            Console.WriteLine("NO");
    }
}

Javascript

function isNSqMinusnMSqPrime(m, n) {
var num = n * n - m * m;
if (num <= 1) {
return false;
}

// Check if p^2 - 1 is divisible by 24
if ((num + 1) % 24 == 0) {
return true;
}
return false;
}

// Driver code
var m = 13,
n = 16;
if (isNSqMinusnMSqPrime(m, n)) {
console.log("YES");
} else {
console.log("NO");
}

Output

NO

Time Complexity: O(1)

Auxiliary Space: O(1)