Editing Fermat's little theorem (section)

== History ==
[[File:Pierre de Fermat.jpg|thumb|upright=0.75|right|Pierre de Fermat]]
Pierre de Fermat first stated the theorem in a letter dated October 18, 1640, to his friend and confidant [[Frénicle de Bessy]]. His formulation is equivalent to the following:<ref name=Burton />

<blockquote>If {{mvar|p}} is a prime and {{mvar|a}} is any integer not divisible by {{mvar|p}}, then {{math|''a''<sup> ''p'' − 1</sup> − 1}} is divisible by {{mvar|p}}.
</blockquote>

Fermat's original statement was 
<blockquote>{{lang|fr|Tout nombre premier mesure infailliblement une des puissances <math>-1</math> de quelque progression que ce soit, et l'exposant de la dite puissance est sous-multiple du nombre premier donné <math>-1</math>; et, après qu'on a trouvé la première puissance qui satisfait à la question, toutes celles dont les exposants sont multiples de l'exposant de la première satisfont tout de même à la question.}}
</blockquote>
This may be translated, with explanations and formulas added in brackets for easier understanding, as:
<blockquote>
Every prime number [{{mvar|p}}] divides necessarily one of the powers minus one of any [geometric] [[geometric progression|progression]] [{{math|''a'', ''a''<sup>2</sup>, ''a''<sup>3</sup>, …}}] [that is, there exists {{mvar|t}} such that {{mvar|p}} divides {{math|''a<sup>t</sup>'' − 1}}], and the exponent of this power [{{mvar|t}}] divides the given prime minus one [divides {{math|''p'' − 1}}]. After one has found the first power [{{mvar|t}}] that satisfies the question, all those whose exponents are multiples of the exponent of the first one satisfy similarly the question [that is, all multiples of the first {{mvar|t}} have the same property].
</blockquote>

Fermat did not consider the case where {{mvar|a}} is a multiple of {{mvar|p}} nor prove his assertion, only stating:<ref>{{citation|first=Pierre|last=Fermat|title=Oeuvres de Fermat. Tome 2: Correspondance|editor-last1=Tannery|editor-first1=P.|editor-last2=Henry|editor-first2=C.|year=1894|place=Paris|publisher=Gauthier-Villars|url=https://archive.org/stream/oeuvresdefermat02ferm#page/206/mode/2up|pages=206–212}} (in French)</ref>
<blockquote>{{lang|fr|Et cette proposition est généralement vraie en toutes progressions et en tous nombres premiers; de quoi je vous envoierois la démonstration, si je n'appréhendois d'être trop long.}}</blockquote>
<blockquote>(And this proposition is generally true for all series [''sic''] and for all prime numbers; I would send you a demonstration of it, if I did not fear going on for too long.)<ref>{{harvnb|Mahoney|1994|page=295}} for the English translation</ref></blockquote>

[[Euler]] provided the first published proof in 1736, in a paper titled "Theorematum Quorundam ad Numeros Primos Spectantium Demonstratio" (in English: "Demonstration of Certain Theorems Concerning Prime Numbers") in the ''Proceedings'' of the St. Petersburg Academy,<ref>{{cite journal |last1=Euler |first1=Leonhard |title=Theorematum quorundam ad numeros primos spectantium demonstratio |journal=Commentarii Academiae Scientiarum Imperialis Petropolitanae (Memoirs of the Imperial Academy of Sciences in St. Petersburg)|date=1736 |volume=8 |pages=141–146 |url=https://www.biodiversitylibrary.org/item/38573#page/167/mode/1up |trans-title=Proof of certain theorems relating to prime numbers |language=Latin}}</ref><ref>{{harvnb|Ore|1988|page=273}}</ref> but [[Gottfried Leibniz|Leibniz]] had given virtually the same proof in an unpublished manuscript from sometime before 1683.<ref name=Burton />

The term "Fermat's little theorem" was probably first used in print in 1913 in ''Zahlentheorie'' by [[Kurt Hensel]]:<ref>{{cite book |last1=Hensel |first1=Kurt |title=Zahlentheorie |trans-title=Number Theory |date=1913 |publisher=G. J. Göschen |location=Berlin and Leipzig, Germany |page=103 |url=https://books.google.com/books?id=SbhUAAAAYAAJ&pg=PA103 |language=German}}</ref>

<blockquote>{{lang|de|Für jede endliche Gruppe besteht nun ein Fundamentalsatz, welcher der kleine Fermatsche Satz genannt zu werden pflegt, weil ein ganz spezieller Teil desselben zuerst von Fermat bewiesen worden ist.}}</blockquote>

<blockquote>(There is a fundamental theorem holding in every finite group, usually called Fermat's little theorem because Fermat was the first to have proved a very special part of it.)</blockquote>

An early use in English occurs in [[Abraham Adrian Albert|A.A. Albert]]'s ''Modern Higher Algebra'' (1937), which refers to "the so-called 'little' Fermat theorem" on page 206.<ref>{{Harvnb|Albert|2015|p=206}}</ref>

=== Further history ===
{{main|Chinese hypothesis}}
Some mathematicians independently made the related hypothesis (sometimes incorrectly called the Chinese hypothesis) that {{math|2<sup>''p''</sup> ≡ 2 (mod ''p'')}} if and only if {{mvar|p}} is prime. Indeed, the "if" part is true, and it is a special case of Fermat's little theorem. However, the "only if" part is false: For example, {{math|2<sup>341</sup> ≡ 2 (mod 341)}}, but 341&nbsp;=&nbsp;11&nbsp;×&nbsp;31 is a [[pseudoprime]] to base 2. See [[#Pseudoprimes|below]].