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| 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, <math>n! \equiv -1 \mod p</math> but <math>p \not\equiv 1 \mod n</math>. The first few Pillai primes are
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| :[[23 (number)|23]], [[29 (number)|29]], [[59 (number)|59]], [[61 (number)|61]], [[67 (number)|67]], [[71 (number)|71]], [[79 (number)|79]], [[83 (number)|83]], [[109 (number)|109]], [[137 (number)|137]], [[139 (number)|139]], [[149 (number)|149]], [[193 (number)|193]], ... {{OEIS|id=A063980}} | |
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| 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.
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| ==References==
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| *{{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 }}.
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| *{{Citation |first=G. E. |last=Hardy |lastauthoramp=yes |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 }}.
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| *{{planetmath reference|id=8739|title=Pillai prime}}
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| {{Prime number classes}}
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| [[Category:Classes of prime numbers]]
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| [[Category:Factorial and binomial topics]]
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| {{numtheory-stub}}
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