The Ulam Spiral: Primes Hiding in Diagonals

A doodle that became a discovery

In 1963, mathematician Stanislaw Ulam was reportedly sitting through a long, uninteresting presentation and began doodling: he wrote the integers 1, 2, 3… in a square spiral starting at the centre, and circled every prime as he went. What should have been a scatter of dots turned out to fall along diagonal lines far more often than chance would suggest. He and colleagues Myron Stein and Mark Wells plotted a much larger version by computer soon after, and the pattern held: even out to hundreds of thousands of integers, primes visibly cluster along certain diagonals and avoid others.

Building the spiral

The construction is entirely mechanical: place 1 at the centre of a grid, then walk outward in a square spiral — right, up, left, left, down, down, right, right, right… — placing each successive integer on the next cell. Every straight segment of the spiral of a fixed direction corresponds to the values of a specific quadratic polynomial in the step count, because the spiral's arm lengths grow by exactly one cell every half-turn. That is the algebraic seed of the whole pattern: the numbers running along any diagonal of the spiral are literally the outputs of some polynomial 4n² + bn + c for fixed integers b and c, evaluated at consecutive n.

Why some diagonals are so much richer in primes

Not all quadratics produce primes equally often. The classic example, discovered by Euler in 1772, is:

f(n) = n² + n + 41

f(0)=41  f(1)=43  f(2)=47  f(3)=53 … all prime for every
integer n from 0 to 39 — 40 consecutive prime values.

It eventually fails — f(40) = 41², obviously composite — but the run of 40 primes in a row is far longer than a "random" quadratic would give. The reason is arithmetic, not magic: for a given small prime p, count how many residues n mod p make f(n) divisible by p. A quadratic like n² + n + 41 has an unusually low count of such "bad" residues across many small primes p, meaning fewer of its outputs get automatically ruled out as composite by small-prime divisibility, so a larger fraction of its values survive as primes over any given stretch.

Hardy-Littlewood conjecture F

This intuition was made precise by G. H. Hardy and John Littlewood's 1923 conjecture F, which predicts the asymptotic density of primes among the values of a quadratic an² + bn + c in terms of a product over primes p measuring exactly this residue-avoidance effect. Diagonals of the Ulam spiral corresponding to polynomials with a high Hardy-Littlewood constant are predicted — and observed — to be visibly denser in primes than diagonals with a low constant, which is precisely the striped pattern the spiral makes visible to the eye. The conjecture remains formally unproven, but it matches empirical prime counts extremely closely.

The Sacks spiral: a smoother cousin

Robert Sacks's 1994 variant replaces the square spiral's fixed-length steps with a true Archimedean spiral, placing integer n at angle 2π√n and radius proportional to √n, so perfect squares always land on the same ray. This trades the Ulam spiral's straight diagonal lines for curved arcs, but the same underlying mechanism — quadratics tracing curves, some curves prime-rich, others prime-poor — produces an equally striking, if more organic-looking, pattern.

Frequently asked questions

Try it live

Everything above runs in your browser — open Ulam Spiral and change the parameters while it is running. Nothing is installed, nothing is uploaded, the whole model lives in one tab.