The primes are infinite, but they grow scarcer as you climb. Among the first ten numbers, four are prime; among ten numbers near a million, usually only one is. Is there any order to this thinning, or are the primes scattered at random? Remarkably, there is a precise law — one of the most beautiful in mathematics.

Counting the primes: \pi(x)

Write \pi(x) for the number of primes up to x. So \pi(10) = 4 (namely 2, 3, 5, 7) and \pi(100) = 25. It is a staircase that jumps by one at every prime and is flat in between. Watch how its rising shape is shadowed almost exactly by a smooth curve:

The smooth curve: x / \ln x

Gauss, as a teenager, noticed that \pi(x) is tracked closely by x / \ln x. Equivalently, the density of primes near a large number x is about 1/\ln x — so a number near x has roughly a 1 / \ln x chance of being prime. This is the celebrated Prime Number Theorem:

Proving it took a century after Gauss's guess and required the deep machinery of complex analysis — we will return to it in the analytic stage.

Gaps, twins, and stubborn mysteries

The average behaviour is law-like, but the local behaviour is wild. There are arbitrarily long runs with no primes at all (the numbers n! + 2, n! + 3, \dots, n! + n are all composite). Yet primes also keep clustering as close as possible — twin primes like (11, 13) and (101, 103) differing by 2. Whether there are infinitely many twin primes is still unknown — a simple-sounding question that has resisted every assault for centuries.