The Prime Number Theorem

The primes look utterly random — no formula, no pattern in their last digits, gaps that come and go. But step back and a law emerges: around a large number x, primes thin out like 1/ln(x), and the total count of primes below x, written π(x), is almost exactly x/ln(x). Gauss guessed it as a teenager in 1792; it took a century to prove. The Prime Number Theorem is the discovery that chaos, at scale, has a shape.