Generator

Ulam's spiral to 900

A generator in the discrete library, called 8 times across 2 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.

number-spiral 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

Ulam's spiral to 900. The integers up to 900 laid out in a square spiral, with the primes marked; they crowd onto diagonal lines.

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.

a marked square holds a prime, and an unmarked one does not ×1
and 41² is where it fails ×1
consecutive integers are neighbours on the spiral ×1
Euler's polynomial is prime for forty values in a row and then is not ×1
no two integers land on the same square ×1
the first polynomial really is the prime-richer one over the range drawn ×1
the sieve starts where the primes do ×1
there are primes to compare against ×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

Counting what has no formula

There is no expression that gives the nth prime, and yet the number of primes below a bound is predictable to within a fraction of a per cent — by a function that is not a formula for the primes but an integral of the wrong-looking quantity.

Discrete

The primes on a spiral, and a pattern nobody ordered

Wind the whole numbers outward in a square spiral, mark the primes, and they line up on diagonals. The observation is a hundred years old and there is still no proof it means anything.

The whole library · What the figures prove