C program for a number to be expressed as a sum of two prime numbers.

In C, we can check if a given positive integer can be expressed as the sum of two prime numbers. This concept is related to Goldbach's conjecture, which states that every even integer greater than 2 can be expressed as the sum of two primes.

Syntax

int isPrime(int n);
// Returns 1 if n is prime, 0 otherwise

Algorithm

The algorithm to check if a number can be expressed as sum of two primes is −

• Step 1: Input the number to be checked
• Step 2: Iterate from i = 2 to (num/2)
• Step 3: Check if i is a prime number
• Step 4: If i is prime, check if (num - i) is also prime
• Step 5: If both i and (num - i) are primes, the number can be expressed as their sum

Example

The following C program checks if a given number can be expressed as the sum of two prime numbers −

#include <stdio.h>

int isPrime(int n) {
    if (n <= 1) return 0;
    if (n == 2) return 1;
    if (n % 2 == 0) return 0;
    
    for (int i = 3; i * i <= n; i += 2) {
        if (n % i == 0) {
            return 0;
        }
    }
    return 1;
}

int main() {
    int num = 34;
    int flag = 0;
    
    printf("Number: %d<br>", num);
    printf("Prime pairs that sum to %d:<br>", num);
    
    for (int i = 2; i <= num/2; i++) {
        if (isPrime(i) && isPrime(num - i)) {
            printf("%d + %d = %d<br>", i, num - i, num);
            flag = 1;
        }
    }
    
    if (flag == 0) {
        printf("The number %d cannot be expressed as sum of two primes<br>", num);
    }
    
    return 0;
}

Number: 34
Prime pairs that sum to 34:
3 + 31 = 34
5 + 29 = 34
11 + 23 = 34
17 + 17 = 34

Key Points

• The isPrime() function efficiently checks primality by testing divisibility up to ?n
• We only check up to num/2 to avoid duplicate pairs (e.g., 3+31 and 31+3)
• Even numbers greater than 2 usually have multiple prime pair representations
• Odd numbers can only be expressed as sum of two primes if one of them is 2

Conclusion

This program demonstrates how to find all possible ways to express a number as the sum of two prime numbers. The algorithm uses an efficient prime checking function and iterates through potential pairs systematically.