In the field of programming, exploring solutions to mathematical problems is both an interesting and challenging task. Today, we will implement a specific mathematical problem using C language: determining whether a given positive integer can be expressed as the sum of two prime numbers. This problem is known in mathematics as “Goldbach’s conjecture”. Although this conjecture has not been completely proven, it is believed to be true for all known even numbers greater than 2. However, to simplify the problem, we will not limit ourselves to even numbers but will attempt to verify all positive integers.

1. Problem Background and Understanding

Goldbach’s conjecture is an unsolved problem in number theory that asserts every even number greater than 2 can be expressed as the sum of two prime numbers. Although this conjecture has not been proven, extensive verification has shown that it holds for all known even numbers greater than 2. However, our task is not limited to even numbers; we will examine all positive integers to see if they can be expressed as the sum of two prime numbers.

1.1 Definition of Prime Numbers

A prime number is a natural number greater than 1 that has no divisors other than 1 and itself. For example, 2, 3, 5, 7, 11, etc., are all prime numbers.

1.2 Problem Analysis

To determine if a number N can be expressed as the sum of two prime numbers, we need to follow these steps:

1. 1. Generate a list of prime numbers: First, we need a method to generate or identify all prime numbers within a certain range.

2. 2. Iterate through pairs of prime numbers: For the given number N, we will iterate through all prime numbers less than or equal to N/2 (since the sum of two prime numbers cannot exceed N, and the larger prime must be less than or equal to N/2), checking if there exists another prime number such that their sum equals N.

3. 3. Output the result: If such a pair of prime numbers is found, output these two prime numbers and their sum, indicating that N can be expressed as the sum of these two primes; if not, output the corresponding information.

2. C Language Implementation

Next, we will implement the above algorithm using C language.

2.1 Generating a List of Prime Numbers

There are many methods to generate a list of prime numbers; here we will use a simple yet efficient method: the Sieve of Eratosthenes. However, considering that we only need to determine if a number can be expressed as the sum of two primes, we can adopt a more space-efficient method by generating primes on demand rather than generating a complete list of primes at once.

2.2 Iterating and Checking Pairs of Prime Numbers

For the given number N, we start from the smallest prime number 2 and incrementally check up to N/2 (or a slightly smaller prime, depending on the implementation), checking if each prime can be added to another prime to equal N.

2.3 Complete Code

Below is the complete implementation code in C language:

#include <stdio.h>
#include <stdbool.h>

// Function prototype declarations
bool isPrime(int num);
void findPrimePair(int N);

int main() {
    int N;
    printf("Please enter a positive integer N: ");
    scanf("%d", &N);

    if (N <= 2) {
        printf("%d cannot be expressed as the sum of two primes (due to the definition of primes and input range limitations).\n", N);
        return 0;
    }

    findPrimePair(N);

    return 0;
}

// Function to check if a number is prime
bool isPrime(int num) {
    if (num <= 1) return false;
    for (int i = 2; i * i <= num; i++) {
        if (num % i == 0) return false;
    }
    return true;
}

// Function to find two primes whose sum equals N
void findPrimePair(int N) {
    for (int i = 2; i <= N / 2; i++) {
        if (isPrime(i) && isPrime(N - i)) {
            printf("%d can be expressed as the sum of two primes %d and %d.\n", N, i, N - i);
            return; // Return after finding a solution
        }
    }
    printf("%d cannot be expressed as the sum of two primes.\n", N);
}

2.4 Code Explanation

• • The isPrime function checks if a number is prime by iterating through all integers from 2 to the square root of that number, checking for divisibility.

• • The findPrimePair function finds two primes whose sum equals the given number N. It uses a loop starting from 2 and incrementing up to N/2 (inclusive), checking if each number is prime and if N minus that number is also prime. If such a pair is found, it outputs the pair and their sum, then exits the function. If the loop ends without finding a pair, it outputs that N cannot be expressed as the sum of two primes.

• • The main function is the entry point of the program. It first prompts the user to input a positive integer N, then calls the findPrimePair function to check if two primes can sum to N and outputs the result.

3. Performance Considerations and Improvements

Although the above implementation is simple and straightforward, it may encounter performance issues when dealing with very large numbers. This is because for each number, we may need to call the isPrime function multiple times (within the findPrimePair function), and the isPrime function itself has a certain computational overhead.

3.1 Improvement Methods

1. 1. Optimize Prime Checking: Although the isPrime function is already quite efficient (using square root optimization), in some cases, we can further reduce unnecessary checks by maintaining a list of known primes to avoid redundant checks.

2. 2. Memoization: For numbers that have already been checked for primality, we can store their results for quick lookup later, avoiding recalculation.

3. 3. Parallelization: For multi-core processors, we can consider parallelizing the loop in the findPrimePair function to speed up the search.

However, it is important to note that for most practical applications, the above implementation is already efficient enough and easy to understand and implement.

Through the implementation in C language, we have successfully solved the problem of determining whether a number can be expressed as the sum of two prime numbers. This not only deepens our understanding of C programming skills but also enhances our knowledge of prime numbers, algorithm optimization, and other mathematical concepts. We hope this article can assist you, whether as a reference for learning C language or as inspiration for solving mathematical problems.