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Concept

Unique factorisation — where it appears

3 essays name this object, across one field. What follows is each of them, and the objects they name alongside it.
first split 2 × 180first split 18 × 203602180229022245222315222335360182029210233225both end in 2 × 2 × 2 × 3 × 3 × 5the same primes, the same number of times, in a different order — and that is the theorem

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 124361251020153060× 2 →× 3 ↑2^2 × 3 × 5 — 3 × 2 × 2 = 12 divisors

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.

number · unique factorisation 12 lattice points sit on the circle — and 4 × (3 − 0) = 12divisors of 25: 1, 5, 25 are 1 mod 4, none are 3 mod 4

Two squares, and a lattice

Whether a prime is the sum of two squares is decided entirely by its remainder on division by four. A fact about circles is settled by a fact about remainders, and neither statement contains any hint of the other.

number · sums of two squares

Named alongside it

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

Counting two waysLatticeNormPrimesDivisibilityDivisor functionDivisor sumEuclid lemmaGaussian integersGreatest common divisorGeometric seriesHasse diagram

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