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

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

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++

// 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; 
} 

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 

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))

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My Personal Notes

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

C++

Java

Python3

C#

PHP

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