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, formula_1 but formula_2. The first few Pillai primes are Pillai primes are named after the mathematician Subbayya Sivasankaranarayana Pillai, who studied these numbers. Their infinitude has been proved several times, by Subbarao, Erdős, and Hardy & Subbarao.