Concept

Gauss sum

A sum of roots of unity weighted by a character, such as the quadratic symbol, over all remainders modulo a prime. Its size is exactly the square root of the prime, which is what bounds character sums and proves reciprocity.

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

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

Forty-two squares in a row. p = 366791: partial sums of (n/p) for n ≤ 200: 0, 10, 20, 30, 40, 46, 54, 60, 64, 72, 78, 78, 88, 92, 98, 102, 108, 116, 122, 126, 132; least non-residue 43.

The first number that is not a square

Modulo a prime, half the numbers are perfect squares. Count up from 1 and the first non-square is usually 2 or 3 — but modulo 366,791 the numbers 1 to 42 are all squares. How far the run of squares can last is one of the oldest open questions about primes: the truth is tiny, the proofs are huge, and the Riemann hypothesis would close most of the gap.

number · Quadratic reciprocity

Named alongside it

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

Legendre symbol Quadratic reciprocity Quadratic residue Complex numbers Conjecture Cyclotomic polynomial Primes Primitive element Primitive root Random walk Riemann hypothesis Roots of unity

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