In 1963, Stanislaw Ulam was doodling during a boring lecture. He wrote integers in a square spiral and circled the primes. Unexpected diagonal lines appeared. Each line traces a quadratic polynomial whose values happen to be prime more often than average.
k² ± 0..(2k) — they're quadratic polynomials in k. Many such polynomials happen to evaluate to primes for many small inputs.
n² + n + 41, which is prime for n = 0 to 39 — an unreasonable run, related to the imaginary quadratic field ℚ(√−163). Toggle it on and you'll see one of the boldest diagonals in the whole picture. Other polynomials trace fainter diagonals depending on their density of prime values.