Lattice — where it appears
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.
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.
Two dials at once
Watch one number on two clocks with different faces. If the faces share no factor, every pair of readings occurs exactly once — so two remainders name a number, and a hard calculation can be split into two easy ones.
Counting one rectangle, twice
Whether seven is a square modulo eleven, and whether eleven is a square modulo seven, are two unrelated-looking questions. Their answers are linked, and the link is a rectangle of dots counted along its rows and then along its columns.
Named alongside it
The objects these essays reach for when they reach for this one.
Counting two waysModular arithmeticPrimesUnique factorisationBijectionChinese remainder theoremCyclic groupDivisor functionDivisor sumGauss lemmaGaussian integersGreatest common divisor