Concept

Euclid lemma

The statement that a prime dividing a product must divide one of the factors. It is the step that proves factorisation into primes unique, and it fails exactly where an irreducible number is not prime, as in the numbers of the form 4k + 1.

Named by 2 essays across one field — each of them below, with the objects they name alongside it.

Two factor trees of 360. The same number split two different ways, both ending in the same primes.

One way to factor, and no other

Every number breaks into primes in exactly one way. That is so familiar it is hard to see as a claim at all — until it is put beside an arithmetic where it is false, and where six has two different factorisations that cannot be reconciled.

number · Unique factorisation

441 factored two ways among the numbers of the form 4k + 1. Two factor trees for 441 among the Hilbert numbers, those one more than a multiple of four: one splits it as 9 times 49, the other as 21 times 21, and every factor is irreducible there.

A factorisation that hides its primes

Keep only the whole numbers one more than a multiple of four. They multiply among themselves and nothing is lost — yet 441 is 9 × 49 and also 21 × 21, and every one of those factors is unbreakable there. Unique factorisation turns out not to be a fact about multiplication at all.

number · Unique factorisation

Named alongside it

The objects these essays reach for when they reach for this one.

Greatest common divisor Irreducible Primes Unique factorisation Congruence Counting two ways Dirichlet theorem Divisibility Modular arithmetic Norm Proof by contradiction

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