Fermat's little theorem

Fermat's little theorem states that if p is a prime number, then for any integer a, the number ap − a is an integer multiple of p. In the notation of modular arithmetic, this is expressed asIf p is a prime and a is any integer not divisible by p, then a p − 1 − 1 is divisible by p.Tout nombre premier mesure infailliblement une des puissances – 1 de quelque progression que ce soit, et l'exposant de la dite puissance est sous-multiple du nombre premier donné – 1 ; 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.Every prime number divides necessarily one of the powers minus one of any progression , and the exponent of this power divides the given prime minus one . After one has found the first power that satisfies the question, all those whose exponents are multiples of the exponent of the first one satisfy similarly the question .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; I would send you a demonstration of it, if I did not fear going on for too long.)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.)If p is an odd prime number, and p – 1 = 2s d, with d odd, then for every a prime to p, either ad ≡ 1 mod p, or there exists t such that 0 ≤ t < s and a2td ≡ −1 mod p Fermat's little theorem states that if p is a prime number, then for any integer a, the number ap − a is an integer multiple of p. In the notation of modular arithmetic, this is expressed as For example, if a = 2 and p = 7, then 27 = 128, and 128 − 2 = 126 = 7 × 18 is an integer multiple of 7. If a is not divisible by p, Fermat's little theorem is equivalent to the statement that ap − 1 − 1 is an integer multiple of p, or in symbols: For example, if a = 2 and p = 7, then 26 = 64, and 64 − 1 = 63 = 7 × 9 is thus 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. 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: