The twin prime constant is defined as the limit of the sum of the reciprocals of twin primes, given by the formula: C2=p≥3∏(1−(p−1)21), where p ranges over all prime numbers such that p and p+2 are both prime.
It's value is C2≈0.660161815846869573927812110014 and the code below quickly converges to that number even for a low number of primes.
The fact that this number exists and is stable means there has to be a pattern in primes.
#include <iostream>
#include <vector>
#include <cmath>
// Function to check if a number is prime
bool is_prime(int n) {
if (n < 2) return false;
if (n == 2 || n == 3) return true;
if (n % 2 == 0 || n % 3 == 0) return false;
for (int i = 5; i * i <= n; i += 6) {
if (n % i == 0 || n % (i + 2) == 0) return false;
}
return true;
}
// Function to compute the twin prime constant
double compute_twin_prime_constant(int limit) {
double C2 = 1.0; // Start with multiplicative identity
for (int p = 3; p <= limit; p += 2) {
if (is_prime(p)) {
double factor = 1.0 - 1.0 / ((p - 1) * (p - 1));
C2 *= factor;
}
}
return C2;
}
int main() {
int limit;
std::cout << "Enter the upper limit for prime numbers: ";
std::cin >> limit;
if (limit < 3) {
std::cout << "Please enter a limit greater than or equal to 3." << std::endl;
return 1;
}
double result = compute_twin_prime_constant(limit);
std::cout.precision(15);
std::cout << "Approximated Twin Prime Constant (C2) with limit " << limit << ": "
<< result << std::endl;
std::cout << "Known value (approx): 0.6601618158468" << std::endl;
return 0;
}