Prime Sieve & Ulam Spiral

The precise distribution of primes is controlled by the zeros of the Riemann zeta function ζ(s). All 10¹³ known non-trivial zeros lie on the critical line Re(s) = 1/2. Proving this for all zeros would give the sharpest possible prime-counting bounds — and win a $1 million Millennium Prize. As of 2025, it is unproved.

The Sieve of Eratosthenes finds primes by elimination rather than by testing each number. Start at 2, cross out every multiple, jump to the next surviving number, and repeat; whatever is never crossed out is prime. The clever shortcut: to sieve all numbers up to n, you only need to remove multiples of primes up to √n. So sieving everything below 100 requires crossing out multiples of just 2, 3, 5 and 7.

In 1963 the mathematician Stanisław Ulam, bored during a talk, scribbled the integers in a square spiral and shaded the primes — and startlingly clear diagonal streaks appeared. Those diagonals trace prime-rich quadratics like Euler's n² + n + 41, which yields a prime for every n from 0 to 39. Why certain diagonals stay so dense is still not fully understood, which is why the pattern keeps fascinating number theorists.