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Concept

Gaussian integers

The complex numbers with whole real and imaginary parts, forming a ring with unique factorisation and four units. A prime of the ordinary integers stops being prime there exactly when it is a sum of two squares.

Named by 3 essays across one field — each of them below, with the objects they name alongside it.

The circle of radius √25 on the integer lattice. A circle drawn on the whole-number grid, with the lattice points it passes through marked.

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 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 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.

Modular arithmeticPrimesSums of two squaresCounting-two waysLegendre symbolQuadratic reciprocityQuadratic residueUnique factorisationDescentGauss lemmaLatticeNorm

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