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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== | |
| *{{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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Revision as of 22:30, 19 February 2014
Hi there, I am Andrew Berryhill. Office supervising is what she does for a living. For many years she's been living in Kentucky but her spouse wants them to move. The favorite pastime for him and his kids is to perform lacross and he would never give it up.
Feel free to surf to my weblog psychic phone (see here)