Editing Fermat's little theorem (section)

== Pseudoprimes ==
{{main|Pseudoprime}}
If {{mvar|a}} and {{mvar|p}} are coprime numbers such that {{math|''a''<sup>''p''−1</sup> − 1}} is divisible by {{mvar|p}}, then {{mvar|p}} need not be prime. If it is not, then {{mvar|p}} is called a ''(Fermat) pseudoprime'' to base {{mvar|a}}. The first pseudoprime to base 2 was found in 1820 by [[Pierre Frédéric Sarrus]]: 341 = 11&nbsp;×&nbsp;31.<ref>{{Cite OEIS|A128311|Remainder upon division of 2<sup>''n''−1</sup>−1 by ''n''.}}</ref><ref>{{cite journal |first=Frédéric |last=Sarrus |author-link=Pierre Frédéric Sarrus |title=Démonstration de la fausseté du théorème énoncé á la page 320 du IXe volume de ce recueil |trans-title=Demonstration of the falsity of the theorem stated on page 320 of the 9th volume of this collection |journal=Annales de Mathématiques Pures et Appliquées |volume=10 |date=1819–1820 |pages=184–187 |language=fr |url=http://www.numdam.org/item?id=AMPA_1819-1820__10__184_0}}</ref>

A number {{mvar|p}} that is a Fermat pseudoprime to base {{mvar|a}} for every number {{mvar|a}} coprime to {{mvar|p}} is called a [[Carmichael number]]. Alternately, any number {{mvar|p}} satisfying the equality
<math display="block">\gcd\left(p, \sum_{a=1}^{p-1} a^{p-1}\right)=1</math>
is either a prime or a Carmichael number.