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Concept

Arithmetic progression

A sequence of numbers each a fixed step above the last. Which of them contain infinitely many primes is decided entirely by whether the step and the starting value share a factor, and the answer is that every progression not obviously ruled out does.

Named by 4 essays across 3 fields — each of them below, with the objects they name alongside it.

Two colours avoid a progression up to 8, and no further. The numbers 1 to 8 in the two colours that avoid three equally spaced numbers in one colour, with the number 9 beside them in both colours and the pattern each choice forces.

Three in a row on the number line

Colour the numbers one to eight in two colours and it can be arranged that no three equally spaced numbers agree. Add the ninth and it cannot. The structure being forced is arithmetic rather than graphical, and the proof is a different proof.

discrete · Ramsey theory The primes 3 mod 4 against the primes 1 mod 4, out to 30,000. A plot of the difference between the counts of primes in two residue classes, against the bound, showing a persistent lead for one class and where it is lost.

Every class, and in equal shares

Euclid's argument aimed at a residue class reaches some classes and stalls at others. The theorem covering all of them is Dirichlet's, its proof abandons arithmetic entirely for analysis, and what it proves is stronger than infinitude — the classes are equal, though not at any point anybody has counted.

number · Infinitude of primes A + B modulo 13: 4 and 3 residues make 9. A clock face of residues with two sets marked on an inner ring and their sumset marked on an outer ring.

A sum of two sets modulo a prime cannot be small

Add every element of one set of residues to every element of another. Over the whole numbers the sums always number at least |A| + |B| − 1. Modulo a prime the sums can wrap round and collide, and still they never number fewer — the theorem Cauchy proved in 1813 and Davenport again in 1935. Modulo 12 they can. A polynomial of low degree explains the difference in a paragraph.

computation · Finite fields Eighty-one cards and twenty with no SET among them. A three-by-three arrangement of three-by-three grids covering the eighty-one points of four-dimensional space modulo three, with twenty cells marked that contain no three on a line.

Twenty cards with no set among them

The card game SET is a four-dimensional space over the integers mod 3, and a set is a line in it. Twenty cards can avoid every line and twenty-one cannot — a fact that took a proof in 1970 — while laying cards down at random and stopping when nothing more fits reaches twenty about once in two thousand tries.

computation · Finite fields

Named alongside it

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

Modular arithmeticCounting argumentExhaustive searchFinite fieldAffine planeBiasBinomial coefficientCap setCompactnessDensityDirichlet theoremExistence proof

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