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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