Concept

Dirichlet theorem

The statement that every arithmetic progression whose step and start share no factor contains infinitely many primes. Its proof abandons arithmetic for analysis, and what it establishes is stronger than infinitude: the admissible classes hold equal shares.

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

The primes below 100,000, by remainder mod 4. A bar for each remainder on division by 4, showing how many primes below 100000 leave it. The 2 classes sharing no factor with 4 hold near-equal counts; the rest are empty or hold one prime.

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.

number · Infinitude of primes

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 Prime Arithmetic progression Bias Congruence Density Proof by contradiction Quadratic residue Residue class Unique factorisation

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