Generator

Two factor trees of 360

A generator in the number library, called 56 times across 13 essays. Below: what it draws with nothing chosen 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.

With nothing chosen

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

441 factored two ways among the numbers of the form 4k + 1

Two factorisations of 441 are one factorisation grouped two ways

The Hilbert numbers up to 441, and which of them are irreducible

Irreducible Hilbert numbers that divide a product and neither factor

The first failure of unique factorisation in four multiplicative sets

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.

every factorisation of 5 has the same number of factors ×4999
the divisor sum of 12 from its factorisation matches the list ×1635
n² + n + 2 is prime for every n up to 0 exactly when 7 is on the list ×78
1 has no friend below the bound ×51
n² + n + 41 is prime at n = 0 ×40
9 is irreducible because it is two primes of the form 4k + 3 ×32
5 fails Euclid's lemma exactly when it is not an ordinary prime ×28
6 has abundancy 2/1 ×23
6 is called what it is ×10
the bar for 6 is its divisors laid end to end ×10
the discs cover the plane exactly for the five Euclidean fields (1) ×9
the lattice for −1 leaves points at most 0.707 from it ×9
every known 2-perfect number has enough primes ×5
the count for 2 is found ×5
the lattice draws every divisor of 30 exactly once ×5
σ(n) is exactly 2 times n ×5
the range is a whole number between 100 and 2000 ×4
9 cannot be split into two numbers of the form 4k + 1 ×3
9 is a Hilbert number ×3
the factors multiply to 3 ×3
3 leaves remainder 3 on division by 4 ×2
496 is perfect ×2
the bound is a whole number between 100 and 3000 ×2
the number is a whole number between 4 and 100000 ×2
the σ of the pieces multiply to 3 times n ×2
10 is the first number of unknown status ×1
441 has exactly two factorisations into Hilbert irreducibles ×1
441 is 3 × 3 × 7 × 7 ×1
5/4 is ruled out: its denominator forces n even ×1
6 and 220 are classified as what they are ×1
6 and 28 sit on the line ×1
a held term leads to a term at least as large that is again held by the perfect number ×1
a proper multiple of a perfect or abundant number is abundant ×1
a term held by the perfect driver never shrinks ×1
a visible share of the range sits just above one ×1
an odd perfect number needs at least three distinct primes ×1
and at n = 40 it is 41² ×1
and so do the norms of two and three ×1
and the next term is again the perfect number times an odd number ×1
and they are the same numbers ×1
below 1000, the sequences that pass 10^22 are the twelve undecided ones and 840 ×1
below a million only 10 itself has abundancy 9/5 ×1
between one and eight whole numbers above one ×1
between one and five starting values from 2 to ten million ×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
every one of these monoids has an element with two factorisations ×1
every sequence is decided or passes the bound within the step budget ×1
for 4k + 1 it is 441 ×1
in fact at most half the divisor's norm, because the square lattice is covered by discs of radius √2/2 ×1
no divisor is drawn twice ×1
no element of the ring has norm 2 or 3 ×1
no element of this ring has norm two or three, so neither factor can split further ×1
rounding the exact quotient leaves a remainder of smaller norm ×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 driver actually holds somewhere on the drawn sequence ×1
the driver is one of the first four perfect numbers ×1
the exponent is a whole number between 2 and 7 ×1
the first odd abundant number is 945 ×1
the first step shown is a whole number between 0 and 200 ×1
the first tree's leaves multiply back to the number ×1
the largest denominator is a whole number between 4 and 16 ×1
the largest number is a whole number between 101 and 600 ×1
the last step shown is a whole number between 10 and 37 ×1
the lattice half-width is a whole number between 2 and 5 ×1
the Mersenne number for this exponent is prime ×1
the nearest are at distance √(5/4), so every remainder has norm 4 × 5/4 = 5, more than N(2) = 4 ×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 number is inside the range the primality test decides ×1
the perfect numbers below a million form one club of four ×1
the powers of two below 2^k add up to the Mersenne number ×1
the rectangle holds every divisor ×1
the search radius is a whole number between 3 and 8 ×1
the second tree's leaves multiply back to the number ×1
the sequence is long enough for the rows asked for ×1
the share at the end of the range is close to a quarter ×1
the share falls as the threshold rises and stays positive through four ×1
the sieve finds exactly the known multiperfect numbers below the bound ×1
the smallest Hilbert number with two factorisations is 441 ×1
the smallest norms are 1, 4, 5 ×1
the step budget is a whole number between 10 and 800 ×1
the stretch shows the driver holding ×1
the two conjugate factors have norms multiplying to 36 ×1
the two trees start differently ×1
the view is one of lattice, grid, aliquot, mersenne, failure, trajectory, driver, fates, abundancy, primitive, abundantdensity, abundancydist, primitivesum, hilbert, hilbertgroup, hilbertatoms, hilbertmulti, monoids, covering, divfail, euclidgi, motzkin, hilbertlemma, euler41, rabinowitsch, multitable, abundancybars, primesneeded, multiscan, sigmachain, solitary, friends, clubs, ten, outlaws ×1
there are primitive abundant numbers in the range ×1
σ(m) is odd exactly for squares and twice squares, up to 2,000 ×1
ω/2 is more than 1 from every integer of the field ×1

Where it is called

Every figure on this list is drawn by the same rule, so a change to the rule changes all of them at once. That is why the list is published.

Number

A factorisation that hides its primes

Keep only the whole numbers one more than a multiple of four. They multiply among themselves and nothing is lost — yet 441 is 9 × 49 and also 21 × 21, and every one of those factors is unbreakable there. Unique factorisation turns out not to be a fact about multiplication at all.

Number

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

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

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

Factoring uniquely with no way to divide

Unique factorisation is proved by dividing with a small remainder, and in the Gaussian integers that works because discs of radius one cover the plane. In the integers of ℚ(√−19) the discs leave holes, no division algorithm of any kind can be made to work — and factorisation is unique anyway. The same field is why n² + n + 41 is prime forty times running.

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.

Number

The sum of the parts, taken again

Replace a number by the sum of its proper divisors and do it again. Most numbers fall to 1, a few land on a perfect number or a cycle, and some climb for hundreds of steps. Below a thousand there are twelve whose fate nobody knows — and the thing that keeps them climbing is a perfect number hiding in their factorisation.

Number

Why a quarter of numbers overshoot

About one number in four has proper divisors adding to more than itself. That a proportion exists at all is not automatic — there are sets defined just as simply that have no proportion — and the reason this one does is that abundance is inherited by multiples, and the numbers it is first inherited from are sparse enough to add up.

The whole library · What the figures prove