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Unique factorisation — the series

4 essays on one idea, from the one that introduces it to the one that assumes the rest.
  1. 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.

    part 1 · number
  2. The divisors of 60. Every divisor as a lattice point, one axis per prime, joined when one divides the other by a single prime.

    The shape of a number's divisors

    Lay a number's divisors out as a lattice with one axis per prime, and two of the most useful facts in arithmetic stop being formulas and become the width and the corner of a rectangle.

    part 2 · number
  3. 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.

    part 3 · number
  4. Where discs of radius one cover the lattice, and where they leave holes. Six lattices of algebraic integers drawn as points in the plane with a unit disc around each; for five of them the discs cover the whole plane and for the sixth, the integers of the field of the square root of minus nineteen, uncovered holes remain.

    Factoring uniquely with no way to divide

    Unique factorisation is proved by dividing with a small remainder, and in the Gaussian integers that works because discs of radius one cover the plane. In the integers of ℚ(√−19) the discs leave holes, no division algorithm of any kind can be made to work — and factorisation is unique anyway. The same field is why n² + n + 41 is prime forty times running.

    part 4 · number

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