Check if a prime number can be expressed as sum of two Prime Numbers

Given a prime number $N$ . The task is to check if it is possible to express $N$ as sum of two separate prime numbers.

Note: The range of N is less than 10⁸.

Examples:

Input : N = 13
Output : Yes
Explanation : The number 13 can be written as 11 + 2, 
here 11 and 2 are both prime.

Input : N = 11
Output : No

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

Simple Solution: A simple solution is to create a sieve to store all the prime numbers less than the number N. Then run a loop from 1 to N and check whether $i$ and $n-i$ are both prime or not. If yes then print Yes, else No.

Efficient solution: Apart from 2, all of the prime numbers are odd. So it is not possible to represent a prime number (which is odd) to be written as a sum of two odd prime numbers, so we are sure that one of the two prime number should be 2. So we have to check whether n-2 is prime or not. If it holds we print Yes else No.

For example, if the number is 19 then we have to check whether 19-2 = 17 is a prime number or not. If 17 is a prime number then print yes otherwise print no.

Below is the implementation of the above approach:

C++

// C++ program to check if a prime number 
// can be expressed as sum of 
// two Prime Numbers 
#include <bits/stdc++.h> 
using namespace std; 
  
// Function to check whether a number 
// is prime or not 
bool isPrime(int n) 
{ 
    if (n <= 1) 
        return false; 
  
    for (int i = 2; i <= sqrt(n); i++) { 
        if (n % i == 0) 
            return false; 
    } 
  
    return true; 
} 
  
// Function to check if a prime number 
// can be expressed as sum of 
// two Prime Numbers 
bool isPossible(int N) 
{ 
    // if the number is prime, 
    // and number-2 is also prime 
    if (isPrime(N) && isPrime(N - 2)) 
        return true; 
    else
        return false; 
} 
  
// Driver code 
int main() 
{ 
    int n = 13; 
  
    if (isPossible(n)) 
        cout << "Yes"; 
    else
        cout << "No"; 
  
    return 0; 
} 

Java

// Java program to check if a prime number 
// can be expressed as sum of 
// two Prime Numbers 
  
public class GFG{ 
      
    // Function to check whether a number 
    // is prime or not 
    static boolean isPrime(int n) 
    { 
        if (n <= 1) 
            return false; 
      
        for (int i = 2; i <= Math.sqrt(n); i++) { 
            if (n % i == 0) 
                return false; 
        } 
      
        return true; 
    } 
      
    // Function to check if a prime number 
    // can be expressed as sum of 
    // two Prime Numbers 
    static boolean isPossible(int N) 
    { 
        // if the number is prime, 
        // and number-2 is also prime 
        if (isPrime(N) && isPrime(N - 2)) 
            return true; 
        else
            return false; 
    } 
      
     // Driver code 
     public static void main(String []args){ 
           
        int n = 13; 
      
        if (isPossible(n) == true) 
            System.out.println("Yes"); 
        else
            System.out.println("No"); 
     } 
     // This code is contributed by ANKITRAI1 
} 

Python3

# Python 3 program to check if a prime  
# number can be expressed as sum of  
# two Prime Numbers  
import math 
  
# Function to check whether a number  
# is prime or not  
def isPrime(n): 
    if n <= 1: 
        return False
    for i in range(2, (int)(math.sqrt(n))): 
        if n % i == 0: 
            return False
    return True
  
# Function to check if a prime number  
# can be expressed as sum of  
# two Prime Numbers  
def isPossible(n): 
  
    # if the number is prime,  
    # and number-2 is also prime  
    if isPrime(n) and isPrime(n - 2): 
        return True
    else: 
        return False
  
# Driver code 
n = 13
if isPossible(n) == True: 
    print("Yes") 
else: 
    print("No") 
      
# This code is contributed by Shrikant13 

C#

// C# program to check if a prime  
// number can be expressed as sum  
// of two Prime Numbers 
using System; 
  
class GFG 
{ 
  
// Function to check whether a  
// number is prime or not 
static bool isPrime(int n) 
{ 
    if (n <= 1) 
        return false; 
  
    for (int i = 2;  
             i <= Math.Sqrt(n); i++)  
    { 
        if (n % i == 0) 
            return false; 
    } 
  
    return true; 
} 
  
// Function to check if a prime  
// number can be expressed as sum  
// of two Prime Numbers 
static bool isPossible(int N) 
{ 
    // if the number is prime, 
    // and number-2 is also prime 
    if (isPrime(N) && isPrime(N - 2)) 
        return true; 
    else
        return false; 
} 
  
// Driver code 
public static void Main() 
{ 
    int n = 13; 
  
    if (isPossible(n) == true) 
        Console.Write("Yes"); 
    else
        Console.Write("No"); 
} 
} 
  
// This code is contributed  
// by ChitraNayal 

PHP

<?php 
// PHP program to check if a prime  
// number can be expressed as sum  
// of two Prime Numbers 
  
// Function to check whether a  
// number is prime or not 
function isPrime($n) 
{ 
    if ($n <= 1) 
        return false; 
  
    for ($i = 2; $i <= sqrt($n); $i++)  
    { 
        if ($n % $i == 0) 
            return false; 
    } 
  
    return true; 
} 
  
// Function to check if a prime  
// number can be expressed as sum  
// of two Prime Numbers 
function isPossible($N) 
{ 
    // if the number is prime, 
    // and number-2 is also prime 
    if (isPrime($N) && isPrime($N - 2)) 
        return true; 
    else
        return false; 
} 
  
// Driver code 
$n = 13; 
  
if (isPossible($n)) 
    echo ("Yes"); 
else
    echo("No"); 
  
// This code is contributed  
// by Shivi_Aggarwal  
?> 

Output:

Yes

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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