Wilson's theorem says that for every prime \(p\), the quantity \((p-1)! + 1\) is divisible by \(p\). A Wilson prime is rarer: one for which \((p-1)! + 1\) is divisible by \(p^2\). Astonishingly, only three are known — 5, 13, and 563 — and there are no others below \(2 \times 10^{13}\).
It is conjectured there are infinitely many Wilson primes, but they should be extraordinarily sparse (about \(\log \log x\) of them below \(x\)). The hunt for a fourth has consumed serious computing effort with nothing to show — one of the great "is that really all?" puzzles of number theory.