Theme

Decided by exhaustion — page 11

Questions with finitely many cases, settled by going through all of them — and what changes when a claim about every argument becomes a count.

Twelve's divisors make every total up to twenty-eight. Stacked bars for each total from 1 to 28, each built from distinct divisors of 12.

Number

Divisors that make every amount

Twelve's divisors — 1, 2, 3, 4, 6, 12 — can be chosen to add to every whole number from one to twenty-eight. Numbers like that are called practical, a test on their primes decides them, and they turn out to be counted like the primes: about 1.336 x / log x of them below x. For practical numbers Goldbach's conjecture and the twin conjecture are theorems.

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