where the primes fall

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three notes

1 — A doodle, 1963 Stanisław Ulam, sitting through a long and very boring paper at a scientific meeting, began writing the integers in a square spiral on a sheet of paper and circling the primes. They did not scatter the way he expected. They fell into diagonal lines, some of them crowded, some of them bare.

2 — Diagonals are quadratics Walk outward along any diagonal of the spiral and the numbers you pass are the values of a polynomial 4t² + bt + c. Touch a point to see the one it belongs to. Euler’s k² + k + 41 is prime for every k from 0 to 39; switch it on and it runs through the spiral as two parallel threads, and through the Sacks spiral as a single curve.

3 — Nobody knows why Hardy and Littlewood conjectured in 1923 how rich in primes each quadratic ought to be, and their prediction agrees with every count ever made. It remains a conjecture. Nobody has proved that even one quadratic polynomial takes infinitely many prime values, so every line you see here is, strictly, unexplained.

every dot is one integer