Legendre symbol
Named by 8 essays across 2 fields — each of them below, with the objects they name alongside it.
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.
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.
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.
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.
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.
Give or take twice the square root
A cubic curve over the integers mod 43 should have about 44 points — one for each value of x, on average, and one at infinity. No curve misses by more than 13, the largest whole number below 2√43, and every count from 31 to 57 belongs to some curve. The first fact is Hasse's theorem, the second Deuring's, and the way the counts spread between the limits is a semicircle.
When minus one can be reached
x² − Dy² = 1 always has a solution. Put −1 on the right and it sometimes does and sometimes does not. A remainder rules out most D at a glance, every prime of the form 4k + 1 is guaranteed one, and between the two lies a set of D decided by arithmetic several layers deep — of which, a million numbers in, 78% are solvable, on the way to a limit of 58% that was proved only in 2022.
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.
Named alongside it
The objects these essays reach for when they reach for this one.
Quadratic residueQuadratic reciprocityModular arithmeticPrimesCounting two waysGauss lemmaGauss sumGaussian integersParityPrimitive elementRandom walkSums of two squares