Concept

Mersenne prime

A prime of the form 2ⁿ − 1, such as 3, 7, 31 and 127; n must itself be prime for it to have a chance. Each one gives an even perfect number, 2ⁿ⁻¹(2ⁿ − 1), and every even perfect number arises this way — so the two lists are the same length, and nobody knows whether either is infinite.

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

28 is perfect, because its divisors form this rectangle. Two rows of divisors: the powers of two, and the same powers multiplied by the Mersenne prime.

Numbers that are their own parts

Six is one plus two plus three. Twenty-eight is one plus two plus four plus seven plus fourteen. Euclid explained where such numbers come from; Euler proved there are no others of that kind; and whether an odd one exists has been open for two thousand years.

number · Perfect numbers

Numbers whose divisors add to two, three, four, five and six times themselves. A table of multiperfect numbers with the multiple their divisor sum makes of them, the number itself and its factorisation into prime powers.

Divisors that add to three times the number

The divisors of 6 add up to 12, twice 6: a perfect number. The divisors of 120 add up to 360, three times 120, and those of 30,240 to four times it. Numbers like these were a sport for Fermat and Descartes, and they are held together by one fact — the ratio σ(n)/n is a product over the primes, and each prime can add only a little.

number · Perfect numbers

Named alongside it

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

Abundance Divisor function Divisor sum Multiplicative function Perfect number Primes Counting two ways Exhaustive search Existence proof

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