Pillai prime

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 $p\; otequiv\; 1\; mod\; n$. The first few Pillai primes are

:23, 29, 59, 61, 67, 71, 79, 83, 109, 137, 139, 149, 193, ... OEIS|id=A063980

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

References

* R. K. Guy, "Unsolved Problems in Number Theory" New York: Springer-Verlag 2004: A2
* G. E. Hardy and M. V. Subbarao, "A modified problem of Pillai and some related questions", "Amer. Math. Monthly" 109 6 (2002): 554 - 559.
*planetmath reference|id=8739|title=Pillai prime