Concept

Residue class

All the whole numbers that leave the same remainder when divided by a fixed number, such as the numbers ending in 7 in base ten. Questions about primes are often asked class by class, since a class can hold infinitely many primes only if its remainder shares no factor with the divisor.

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

Prime pairs a fixed distance apart, counted, against Hardy and Littlewood's prediction. Bars for the number of prime pairs (p, p + d) up to 2000000 for even d from 2 to 60, jagged with peaks at multiples of 6 and 30, each with the Hardy–Littlewood prediction marked.

The patterns primes are allowed to make

Primes two apart are twin primes; primes six apart are about twice as common; three primes two apart happen exactly once, at 3, 5, 7. Which patterns of primes can recur for ever is decided by remainders alone — a pattern must leave some remainder free modulo every prime — and how often each one recurs is predicted, to within a few per cent, by a constant computed from nothing else.

number · Infinitude of primes

Named alongside it

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

Bounded gaps Congruence Dirichlet theorem Hardy littlewood conjecture Modular arithmetic Prime Prime constellation Primes Proof by contradiction Quadratic residue Sieve Twin primes

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