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, $n!\; \backslash equiv\; -1\; \backslash mod\; p$ but $p\; \backslash not\backslash equiv\; 1\; \backslash mod\; n$. 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 studied these numbers. Their infinitude has been proven several times, by Subbarao, Erdős, and Hardy & Subbarao.

==References==

• {{Citation |first=R. K. |last=Guy |title=Unsolved Problems in Number Theory |location=New York |publisher=Springer-Verlag |year=2004 |page=A2 |edition=3rd |isbn=0-387-20860-7 }}.
• {{Citation |first1=G. E. |last1=Hardy |name-list-style=amp |first2=M. V. |last2=Subbarao |title=A modified problem of Pillai and some related questions |journal=American Mathematical Monthly |volume=109 |issue=6 |year=2002 |pages=554–559 |doi=10.2307/2695445 |jstor=2695445 }}.
• https://planetmath.org/pillaiprime, PlanetMath

{{Prime number classes}}

Category:Classes of prime numbers

Category:Eponymous numbers in mathematics

Category:Factorial and binomial topics

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