C++ Program To Check If a Prime Number Can Be Expressed as Sum of Two Prime Numbers

A prime number greater than 2 is odd, while 2 is the only even prime number. This property can be used to efficiently determine whether a given prime number can be represented as the sum of two prime numbers.

• For an odd prime N, one of the two prime numbers must be 2.
• Therefore, we only need to check whether N - 2 is also prime.

Illustration

Input: N = 13
Output: Yes

Explanation: 13 = 11 + 2, and both 11 and 2 are prime.

Input: N = 11
Output: No

Explanation: 11 - 2 = 9, which is not prime, so 11 cannot be expressed as the sum of two prime numbers.

Approach

The efficient approach uses the fact that every prime number except 2 is odd.

• Check whether N is prime.
• Compute N - 2.
• Check whether N - 2 is prime.
• If both are prime, N can be expressed as 2 + (N - 2).
• Otherwise, no such representation exists.

For example, for N = 19, we check 19 - 2 = 17. Since both 19 and 17 are prime, the answer is Yes because 19 = 2 + 17.

#include <iostream>
#include <cmath>
using namespace std;

// Checks whether a number is prime
bool isPrime(int n)
{
    if (n <= 1)
        return false;

    for (int i = 2; i * i <= n; i++) {
        if (n % i == 0)
            return false;
    }

    return true;
}

// Checks whether N can be expressed as
// the sum of two prime numbers
bool isPossible(int N)
{
    return isPrime(N) && isPrime(N - 2);
}

int main()
{
    int N = 13;

    if (isPossible(N))
        cout << "Yes";
    else
        cout << "No";

    return 0;
}

Yes

Explanation: For N = 13:

• 13 is prime.
• 13 - 2 = 11.
• 11 is also prime.
• Therefore, 13 = 2 + 11, so the answer is Yes.