Prime numbers

What makes a number prime

A prime number is a whole number greater than 1 that has exactly two divisors: 1 and itself. Nothing else divides into it evenly.

Take 7. You can't split it into equal whole groups except as one group of 7 or seven groups of 1. That stubbornness is what makes it prime. Compare 12, which falls apart into 2 times 6, or 3 times 4 — so 12 is not prime. We call numbers like 12 composite, because they're composed from smaller pieces.

The primes start 2, 3, 5, 7, 11, 13, 17, 19, 23, and keep going forever. They never settle into a neat pattern, which is a big part of why they're so fascinating.

Why 1 doesn't count

It's tempting to call 1 prime — it only divides by 1 and itself, after all. But 1 and itself are the same thing, so it has just one divisor, not two. That single difference matters.

Mathematicians give 1 a special name: a unit. Keeping it out of the club isn't fussiness. If 1 were prime, every number could be written as a product of primes in endless ways (12 = 2 × 2 × 3 = 1 × 2 × 2 × 3 = 1 × 1 × 2 × 2 × 3…). Leaving 1 out keeps each number's prime recipe unique.

The building blocks of every number

Primes are the atoms of arithmetic. Every whole number bigger than 1 is either a prime itself or can be built by multiplying primes together — and in exactly one way (apart from the order you write them in).

So 60 is 2 × 2 × 3 × 5, and there's no other set of primes that does it. This is called a number's prime factorisation, and it's so dependable it has a grand name: the Fundamental Theorem of Arithmetic. Knowing the primes inside a number tells you almost everything about how it behaves.

Infinitely many, but they thin out

The primes never run out. Around 300 BC the Greek mathematician Euclid proved there are infinitely many, with an argument so clean it still feels like magic: if you had a complete list of all primes, you could multiply them all together and add 1 — and that new number is divisible by none of them, so a prime is always missing. The list can never be finished.

And yet primes grow rarer the higher you climb. There are plenty below 100, fewer between 1,000 and 1,100, and out among enormous numbers you can walk a long way without meeting one. They thin out, but they never stop appearing. The largest primes anyone has found are Mersenne primes (numbers one less than a power of two), discovered by the worldwide volunteer project GIMPS.

One last quirk: 2 is the only even prime. Every other even number is divisible by 2, so it can't be prime. That makes 2 the loneliest, oddest prime of all — the only one that isn't odd.