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Fermat's little theorem states that if is a prime number, then for any integer , the number is an integer multiple of . In the notation of modular arithmetic, this is expressed as : For example, if = 2 and = 7, 27 = 128, and 128 − 2 = 7 × 18 is an integer multiple of 7. If is not divisible by , Fermat's little theorem is equivalent to the statement that is an integer multiple of , or in symbols : For example, if = 2 and = 7 then 26 = 64 and 64 − 1 = 63 is a multiple of 7. Fermat's little theorem is the basis for the Fermat primality test and is one of the fundamental results of elementary number theory. The theorem is named after Pierre de Fermat, who stated it in 1640. It is called the "little theorem" to distinguish it from Fermat's last theorem. ==History== Pierre de Fermat first stated the theorem in a letter dated October 18, 1640, to his friend and confidant Frénicle de Bessy as the following:〔 If is a prime and is any integer not divisible by , then is divisible by . Fermat did not prove his assertion, only stating: 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. (And this proposition is generally true for all series and for all prime numbers; the proof of which I would send to you, if I did not fear it being too long.)〔 for the English translation〕 Euler provided the first published proof in 1736 in a paper entitled "Theorematum Quorundam ad Numeros Primos Spectantium Demonstratio" in the ''Proceedings'' of the St. Petersburg Academy, but Leibniz had given virtually the same proof in an unpublished manuscript from sometime before 1683.〔 The term "Fermat's Little Theorem" was probably first used in print in 1913 in ''Zahlentheorie'' by Kurt Hensel: Für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." (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.) An early use in English occurs in A.A. Albert, Modern Higher Algebra (1937), which refers to "the so-called "little" Fermat theorem" on page 206. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Fermat's little theorem」の詳細全文を読む スポンサード リンク
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