Concept

Gauss lemma

The rule that the Legendre symbol of a is minus one to the number of the multiples a, 2a, … that fold down from the top half of the residues. It is the count that reaches the two supplements, which the lattice-point proof of reciprocity cannot.

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

Counting a 5 by 3 rectangle two ways. Lattice points in a rectangle cut by a diagonal of slope q over p, coloured by which side they fall.

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.

number · Quadratic reciprocity

The two supplements, and the residue classes that decide them. A table of odd primes with the Legendre symbols of minus one and two beside the residue of p modulo four and modulo eight.

The two supplements, and where the eight comes from

The main law relates two odd primes to each other and says nothing about −1 or about 2. Those two are settled separately, by their own counts, and the answers arrive modulo four and modulo eight — which is a clue about where the whole subject is really taking place.

number · Quadratic reciprocity

Named alongside it

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

Counting-two ways Legendre symbol Modular arithmetic Parity Primes Quadratic reciprocity Quadratic residue Gaussian integers Lattice Sums of two squares

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