Concept

Subset sum

The problem of deciding whether some selection from a list of numbers adds up exactly to a given target. It is hard in general, and the best known methods split the list and look for a collision between the halves' sums.

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

Seventy, and the totals its divisors can make. A strip of the totals 0 to 74 marking those reachable as sums of distinct proper divisors of 70; 70 itself and the excess 4 are not.

Abundant, and still not a sum of its parts

Seventy's proper divisors add to seventy-four, more than seventy itself — and yet no selection of them adds to exactly seventy. The reason is that the excess, four, cannot be made from the small divisors one, two and five. Numbers like seventy are called weird; there are thirty-six below twenty thousand, every one of them even, and nobody knows whether an odd one exists.

number · Perfect numbers

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.

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.

number · Perfect numbers

Named alongside it

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

Density Divisor sum Exhaustive search Perfect number Abundance Asymptotics Counterexample Divisibility Egyptian fraction Goldbach Open problem Parity

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