Step by Step solution about How to resolve the algorithm Goldbach's comet step by step in the C++ programming language

The given code is a C++ program that calculates the Goldbach function for even numbers up to a specified limit. The Goldbach function, G(n), represents the number of ways to express the even number n as the sum of two prime numbers.

The program begins by initializing a vector primes with a size equal to the specified limit and initializing all its elements to True. This vector will be used to determine if a number is prime or not.

Next, the program iterates through all numbers from 2 to the square root of the limit and marks all multiples of these numbers as non-prime in the primes vector. This is done to efficiently identify prime numbers using the Sieve of Eratosthenes algorithm.

The goldbach_function function is defined to calculate the Goldbach function for a given even number. It first checks if the input number is valid (even and greater than 2) and throws an exception if not.

Within the goldbach_function function, it iterates from 1 to half of the input number and checks if both the current number and the difference between the input number and the current number are prime. If both are prime, it increments the result count.

In the main function, the program initializes the primes vector up to a limit of 2,000,000 using the initialise_primes function.

It then prints the first 100 Goldbach numbers, with each number on a new line if it's a multiple of 10.

Finally, the program prints the 1,000,000th Goldbach number.

#include <cmath>
#include <cstdint>
#include <iomanip>
#include <iostream>
#include <stdexcept>
#include <vector>

std::vector<bool> primes;

void initialise_primes(const int32_t& limit) {
	primes.resize(limit);
	for ( int32_t i = 2; i < limit; ++i ) {
		primes[i] = true;
	}

	for ( int32_t n = 2; n < sqrt(limit); ++n ) {
		for ( int32_t k = n * n; k < limit; k += n ) {
			primes[k] = false;
		}
	}
}

int32_t goldbach_function(const int32_t& number) {
	if (  number <= 2 || number % 2 == 1 ) {
		throw std::invalid_argument("Argument must be even and greater than 2: " + std::to_string(number));
	}

	int32_t result = 0;
	for ( int32_t i = 1; i <= number / 2; ++i ) {
		if ( primes[i] && primes[number - i] ) {
			result++;
		}
	}
	return result;
}

int main() {
	initialise_primes(2'000'000);

	std::cout << "The first 100 Goldbach numbers:" << std::endl;
	for ( int32_t n = 2; n < 102; ++n ) {
		std::cout << std::setw(3) << goldbach_function(2 * n) << ( n % 10 == 1 ? "\n" : "" );
	}

	std::cout << "\n" << "The 1,000,000th Goldbach number = " << goldbach_function(1'000'000) << std::endl;
}