Strike a tuning fork, and it rings at a single frequency. Prime numbers have their own ‘tone’ — the number λ∞, approximately 0,674036. Although primes are scattered seemingly at random, their behavior follows strict laws. If you take each prime p, form the fraction 1/(λ∞·p² − p + 1), and add up all such fractions, the sum is exactly 1. This constant is the principal number of a vast network of connections among primes.

To find λ∞, mathematician Alessandro Munari built such a network from the first 10 million primes. Beyond that, the contribution becomes tiny, but it must still be accounted for. For that, they used a an ingenious approximate counting method and a a powerful computer method — Newton’s algorithm. 11 steps, 336 seconds — done: 500 exact decimal places. The result was double-checked with statistical tests and pattern search algorithms, but no simple closed-form expression for λ∞ was found.

This constant is to primes what π²/6 is to all integers. Interestingly, the prime 2 contributes the most: without it, the equation wouldn’t balance at all. It’s not known whether λ∞ can be expressed as a simple fraction, but one thing is clear: the mysterious tuning fork of primes has sounded, and its melody remains to be deciphered.