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Concept

Divisor function

The count of a number's divisors, which comes out as the product of one more than each exponent in its factorisation. Its multiplicativity is what makes it computable from the factorisation rather than by trial, and it is odd exactly for square numbers.

Named by 5 essays across 2 fields — each of them below, with the objects they name alongside it.

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 The lattice-point count inside a circle, less its area, out to radius 160. A plot of the difference between the number of lattice points in a disc and the disc's area, against radius, with envelopes proportional to the square root and the two-thirds power drawn.

The dots a circle catches

Pick's theorem gives a lattice polygon's area exactly, with no error term anywhere. Ask a circle the same question and the exactness is gone: the count is the area plus something, the something has been measured for two centuries, and nobody knows how big it is.

discrete · Pick theorem 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 Numbers up to 600 that share their abundancy. A scatter of abundancy against n with horizontal segments joining numbers that have exactly the same abundancy.

A ratio nobody else has

Divide the sum of a number's divisors by the number and you get its abundancy: 2 for every perfect number, 12/5 for both 30 and 140. Numbers that share an abundancy are called friends. Some numbers provably have no friend at all, most have friends only far away — and for 10, whose abundancy is 9/5, nobody knows whether a friend exists.

number · Perfect numbers

Named alongside it

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

Divisor sumMultiplicative functionPerfect numberAbundanceExhaustive searchPrimesCounting two waysLatticeMersenne primeAsymptoticsError termExistence proof

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