The Twin Prime Constant
The twin prime constant is a fundamental mathematical constant that appears in the study of twin primes (pairs of primes that differ by 2, like 11 and 13, or 17 and 19).
The constant is defined as the infinite product over all primes $p \geq 3$:
$$C_2 = \prod_{p \geq 3} \left(1 - \frac{1}{(p-1)^2}\right)$$Its value is:
Why This Matters
The fact that this number exists and is stable is profound. As you compute more and more terms of the product, it converges to this specific value. The code below demonstrates how quickly this convergence happens - even with a relatively small number of primes, you'll get close to the known value.
This stability suggests there's an underlying structure to how twin primes are distributed along the number line, even though we can't predict exactly where they'll appear.
C++ Implementation
#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;
}Compiling & Running
# Compile
g++ -std=c++17 -O2 twin_prime_constant.cpp -o twin_prime
# Run
./twin_prime
# Example output with limit 1000000:
# Approximated Twin Prime Constant (C2) with limit 1000000: 0.660161815846869
# Known value (approx): 0.6601618158468Convergence
Try running the program with different limits to see how quickly it converges:
- limit = 100: ~0.6601 (already 4 decimal places!)
- limit = 10,000: ~0.66016181 (8 decimal places)
- limit = 1,000,000: ~0.6601618158468 (13 decimal places)