Generator

Two factor trees of 360

A generator in the number library, called 21 times across 7 essays. Below: what it draws at its defaults and at each mode an essay asks for, what it checks while drawing, and everywhere it is used.

factor is one function. Everything below came out of it during this build, at parameters taken from the essays rather than invented for this page — so a figure here is the same figure a reader meets in an essay, and if the generator changes, this page changes with it.

At its defaults

Two factor trees of 360. The same number split two different ways, both ending in the same primes.

`show: "lattice"`

The divisors of 30. Every divisor as a lattice point, one axis per prime, joined when one divides the other by a single prime.

`show: "mersenne"`

496 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.

`show: "aliquot"`

Proper divisors, added up. For each number, its proper divisors laid end to end against the number itself.

`show: "grid"`

The divisors of 72 as a rectangle. Powers of one prime across, powers of the other down; every divisor appears once.

`show: "failure"`

Six, factored two ways. The lattice of numbers a + b√−5, each point labelled with its norm a² + 5b².

What it checks while it draws

Collected by running the family and recording what it asserted, not written here. The count is how many separate times the claim was put to the test while these drawings were made.

6 is called what it is ×7
the bar for 6 is its divisors laid end to end ×7
the lattice draws every divisor of 30 exactly once ×5
496 is perfect ×2
the number is a whole number between 4 and 100000 ×2
and so do the norms of two and three ×1
between one and eight whole numbers above one ×1
both first splits divide the number ×1
both trees end in the same multiset of primes ×1
every divisor including the number itself adds to twice the number ×1
no divisor is drawn twice ×1
no element of this ring has norm two or three, so neither factor can split further ×1
the cells add up to the divisor sum ×1
the divisor count is the product of one more than each exponent ×1
the divisor sum factorises as the two row totals multiplied ×1
the exponent is a whole number between 2 and 7 ×1
the first tree's leaves multiply back to the number ×1
the lattice half-width is a whole number between 2 and 5 ×1
the Mersenne number for this exponent is prime ×1
the number has a factor tree to draw ×1
the number has at least three prime factors, so its trees can differ ×1
the number has at most three distinct primes ×1
the number is a product of exactly two prime powers ×1
the powers of two below 2^k add up to the Mersenne number ×1
the rectangle holds every divisor ×1
the second tree's leaves multiply back to the number ×1
the two conjugate factors have norms multiplying to 36 ×1
the two trees start differently ×1

Where it is called

Changing this generator changes every figure on this list. That is what makes the list worth publishing rather than keeping in a check script.

Number

Always one before the double

A density says what happens on average and permits long empty stretches. This says something a density cannot — that the stretch from any number to twice it contains a prime, at every scale, without exception.

Number

The primes are what is left over

Eratosthenes' sieve does not find the primes. It removes everything else, one prime at a time, and whatever survives is prime by default — which is a strange way to reach the most studied objects in arithmetic.

Number

There is no last prime

Euclid's argument is often described as producing a new prime from any finite list. It does not, and the number it builds is frequently composite — which makes the proof more interesting rather than less.

Number

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

One way to factor, and no other

Every number breaks into primes in exactly one way. That is so familiar it is hard to see as a claim at all — until it is put beside an arithmetic where it is false, and where six has two different factorisations that cannot be reconciled.

Algebra

The identity that multiplies sums of squares

A sum of two squares times a sum of two squares is a sum of two squares, and the same is true of four and of eight. It is true of three of nothing: 3 and 5 are each a sum of three squares, 15 is not, and one small pair of numbers rules the case out forever.

Number

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.

The whole library · What the figures prove