Numberland

Chapter VI · the atoms of arithmetic

Primes

Some numbers refuse to be divided. 2, 3, 5, 7, 11, 13… — the primes, the indivisible atoms from which every other number is built.

A prime has exactly two divisors: itself and 1. Everything else is composite — a product of smaller things — and every composite number breaks apart into primes in exactly one way: 60 is 2 × 2 × 3 × 5, and no other combination of primes will ever build it. That is the fundamental theorem of arithmetic, and it is why primes are called the atoms of the number system.

How many primes are there? Euclid proved the answer is: infinitely many. Suppose the list were finite; multiply every prime on it together and add 1 — the result is divisible by none of them, so it must carry a brand-new prime factor. And yet the primes thin out as they go: 25 below 100, 168 below 1,000 — and two thousand years of asking later, the law of their spacing still hides the deepest unsolved mystery in mathematics, the Riemann hypothesis.

Eratosthenes' sieve (c. 240 BCE): cross out every multiple of 2, then 3, then 5, then 7 — whatever survives is prime. Watch the strikes fall, and the 25 survivors light up.
The fundamental theorem in action: 60, 84, 90 and 72 split apart, level by level, until only primes remain — and the primes at the leaves rebuild the number as a product.