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 4 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

441 factored two ways among the numbers of the form 4k + 1. Two factor trees for 441 among the Hilbert numbers, those one more than a multiple of four: one splits it as 9 times 49, the other as 21 times 21, and every factor is irreducible there.

A factorisation that hides its primes

Keep only the whole numbers one more than a multiple of four. They multiply among themselves and nothing is lost — yet 441 is 9 × 49 and also 21 × 21, and every one of those factors is unbreakable there. Unique factorisation turns out not to be a fact about multiplication at all.

number · Unique factorisation

Each prime of the form 4k + 1 as a point on its own circle. Points (a, b) with a > b > 0 and a² + b² prime, for the 1125 such primes below 20000, filling the sector between 0° and 45°.

Which way a prime's two squares point

A prime one more than a multiple of four is a sum of two squares in exactly one way, and the two squares make a point on a circle. Draw that point for every such prime and ask which way it faces. It faces every way equally — the angles spread evenly over the sector, which Hecke proved by giving each angle a remainder and copying Dirichlet. Measured over seventy-four thousand primes, they are even more evenly spread than random angles would be.

number · Sums of two squares

Named alongside it

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

Modular arithmetic Congruence Prime Primes Unique factorisation Arithmetic progression Bias Density Discrepancy Equidistribution Euclid lemma Gaussian integers

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