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, $n!\equiv -1modp$ but $p\equiv ̸1modn$. The first few Pillai primes are

23, 29, 59, 61, 67, 71, 79, 83, 109, 137, 139, 149, 193, ... Template:OEIS

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.

References

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• https://planetmath.org/pillaiprime, PlanetMath