Concept

Prime — where it appears

A whole number above one whose only divisors are one and itself. Every whole number above one is a product of primes in exactly one way, which is why they are the objects arithmetic is built from rather than a curiosity about divisibility.

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

Five families of primes, counted below 100,000. Five counting curves on logarithmic axes: every prime, the primes one more than a multiple of four, the twin pairs, the primes one more than a square, and the Mersenne primes. Two of the five families are known to be infinite and three are open questions.

Which infinitudes are proved

The primes never stop, and neither — apparently — do the twin pairs, the primes one more than a square, or the Mersenne primes. Three of those four statements are theorems and one is not, and counting the members of each family tells nobody which.

number · Infinitude of primes

Named alongside it

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

Asymptotic Congruence Conjecture Counting argument Dirichlet theorem Divergence Heuristic Modular arithmetic Open problem Proof by contradiction Quadratic residue Residue class

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