Pillai prime

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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, ... (sequence A063980 in OEIS)

Sarinya Intaraprasert proved that there are infinitely many Pillai primes. Pillai primes are named after the mathematician Subbayya Sivasankaranarayana Pillai.