The Ulam Spiral
Primes lining up on mysterious diagonals
Stanisław Ulam, bored at a 1963 lecture, doodled the integers in a square spiral and circled the primes. The primes did not scatter; they crowded along visible diagonals. Many of those diagonals correspond to prime-rich quadratics like Euler's n² − n + 41, which is prime for every n from 0 to 40. Why primes prefer certain quadratic forms over others is part of the deepest unsolved area of number theory — Ulam saw it on a napkin.
Highlight a quadratic
Pink = the diagonal the quadratic carves out.
