Concept

Divisor-sum

The total of all a number's divisors, the number itself included. Comparing it with twice the number itself is what sorts numbers into deficient, perfect and abundant, and Euclid's rule builds a perfect one from every Mersenne prime.

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

Also named here as perfect number — the same set of essays touches all of them, so they are one junction rather than several.

The divisors of 60. Every divisor as a lattice point, one axis per prime, joined when one divides the other by a single prime.

The shape of a number's divisors

Lay a number's divisors out as a lattice with one axis per prime, and two of the most useful facts in arithmetic stop being formulas and become the width and the corner of a rectangle.

number · Unique factorisation

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

Named alongside it

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

Counting two ways Divisor function Multiplicative function Perfect number Abundance Existence proof Geometric series Hasse diagram Lattice Mersenne prime Order lattice Primes

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