Pillai prime

In number theory, a Pillai prime is a prime number p for which there is an integer n > 0 such that the factorial of n is one less than a multiple of the prime, but the prime is not one more than a multiple of n. To put it algebraically, $$n! \equiv -1 \mod p$$ but $$p \not\equiv 1 \mod n$$. The first few Pillai primes are


 * 23, 29, 59, 61, 67, 71, 79, 83, 109, 137, 139, 149, 193, ...

Pillai primes are named after the mathematician Subbayya Sivasankaranarayana Pillai, who studied these numbers. Their infinitude has been proven several times, by Subbarao, Erdős, and Hardy & Subbarao.