Quadratic residue
Named by 18 essays across 5 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.
Eighteen people, and the seventeen that escape
Among any eighteen people, four are mutual acquaintances or four are mutual strangers. Seventeen can be arranged so that neither happens, and the arrangement is not a lucky find — it is a rule about squares.
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.
Randomness that has to be earned
A generator that resists prediction cannot be built out of a rule anybody can fit. It has to be built out of a computation believed hard to undo, and the belief is the load-bearing part — which makes cryptographic randomness a conditional statement rather than a construction.
Infinitely many of one kind
Euclid's argument produces a prime nobody had listed, and says nothing about what it looks like. Ask for infinitely many primes ending in 3, or leaving a remainder of 1 on division by 4, and the same construction has to be aimed — and for most targets nobody knows how to aim it.
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.
No set with a line in every direction is small
In the plane over the integers modulo 7 there are 49 points and lines in 8 directions. A set holding a whole line in every direction needs 31 of the points — more than half — and in any dimension such a set fills a fixed share of the space. In the real plane the same sets can have area zero. Over a finite field one polynomial of low degree shows they cannot be small.
A root lifted one digit at a time
On a dial of seven, 3 × 3 is 2. On a dial of forty-nine the square root of 2 must reduce to 3, so there are only seven candidates, and exactly one of them works: 10. On a dial of 343 exactly one lift of 10 works: 108. Each step adds one digit on the left, found by solving a linear equation, and the digits go on for ever — a number …21216213 whose square is 2, in a world where closeness means divisibility by seven.
The biggest box built from signs
Fill a square table with plus and minus ones and ask how large its determinant can be. The columns all have the same length, so the answer is a box with fixed edges — largest when every corner is square, which is possible only when the size is a multiple of four.
How often two generates every remainder
For some primes p, the powers of 2 run through every non-zero remainder before repeating; for others they get stuck in a smaller cycle. Among the primes up to two million the first kind make up 37.43% — and Artin predicted 37.40% in 1927 by treating each way of getting stuck as an independent accident. For the base 5 the same prediction is wrong, because two of the accidents are secretly the same one, and nobody can yet prove that 2 works for infinitely many primes at all.
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 squares that answer every request
Rado's graph has, for any finite sets U and V, a vertex joined to all of U and none of V. A finite graph can only answer the small requests, and the Paley graphs — residues modulo a prime, joined when their difference is a square — are the classic way to build one. Checking every request exactly finds the thresholds: 13 residues answer every request of two, 29 every request of three, 89 every request of four. The theorem that guarantees it asks for 64, 576 and 4,096. Coin-toss graphs of the same sizes almost never manage it.
Is the partition count even half the time?
The number of partitions of n is even for 50.0% of the n up to half a million, its runs of one parity are as long as a coin's, and nothing proves that the share is a half — the best theorems only show there are at least about √n of each. Modulo 5 and 7 the zeros carry Ramanujan's congruences and something more: an excess that follows whether 1 − 24n is a square.
The same bell on thinner sets
A typical whole number has about log log N prime factors, spread in a bell curve. That was a theorem about all numbers, built on the picture of each prime dividing independently. The numbers one less than a prime, and the numbers n² + 1, are far too thin for that picture to have any obvious claim on them, and on both the same bell appears — with means shifted by constants that come straight from how often each small prime divides them. The squares plus one that are prime, the case of a single factor, number 102,205 up to n = two million against 102,302 predicted; whether there are infinitely many is open.
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.
Modular arithmeticLegendre symbolQuadratic reciprocityPrimesCounting argumentCounting two waysParityCongruenceConjectureDensityFinite fieldGauss lemma