Concept

Legendre symbol

A quantity that is plus one when a is a non-zero square modulo the prime p, minus one when it is not, and nought when p divides a. It multiplies over its top argument, which reduces any such question to the primes and the two supplements.

Named by 5 essays across one field — each of them below, with the objects they name alongside it.

Also named here as quadratic reciprocity — the same set of essays touches all of them, so they are one junction rather than several.

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

Multiplication by 3 modulo 11, and the sign of the shuffle. Residues in two rows joined by strings showing where multiplication sends each one, with a strip beneath comparing the sign of the shuffle to the Legendre symbol for every multiplier.

The symbol is the sign of a shuffle

Multiplying every residue modulo p by a fixed number rearranges them. That rearrangement is a permutation, permutations have a sign, and the sign is exactly the Legendre symbol — so a question about squares becomes a question about crossings.

number · Quadratic reciprocity

The Gauss sum for 13, added one root at a time. Partial sums of the p-th roots of unity signed by the Legendre symbol, drawn as a walk closing on a point at distance root p from the origin.

One sum, squared two ways

Add the p-th roots of unity, each taken with a plus or a minus according to whether its index is a square. The walk that results closes on a point at distance √p from the origin — and squaring that one number, evaluated two different ways, is the reciprocity law.

number · Quadratic reciprocity

Where a congruence decides which primes a form represents, and where it does not. Rows of primes marked by whether each is represented by x squared plus n y squared, with the residue classes that decide it where such classes exist.

Which primes a form takes

A prime is the sum of two squares exactly when it is 1 modulo 4. Change the form slightly, to x² + 27y², and no congruence on p decides it at all — which is where the elementary subject ends and its successor begins.

number · Quadratic reciprocity

Named alongside it

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

Quadratic reciprocity Quadratic residue Modular arithmetic Counting-two ways Primes Gauss lemma Gaussian integers Parity Primitive element Sums of two squares Complex numbers Cyclic group

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