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 2 essays across 2 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

Named alongside it

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

Modular arithmetic Bias Compactness Counting argument Density Dirichlet theorem Exhaustive search Existence proof Fermats little theorem Prime Ramsey number Schur theorem

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