- Fermat's little theorem
**Fermat's little theorem**(not to be confused withFermat's last theorem ) states that if "p" is aprime number , then for anyinteger "a", $a^p\; -\; a$ will be evenly divisible by "p". This can be expressed in the notation ofmodular arithmetic as follows::$a^p\; equiv\; a\; pmod\{p\},!$A variant of this theorem is stated in the following form: if "p" is a prime and "a" is an integer

coprime to "p", then $a^\{p-1\}\; -\; 1$ will be evenly divisible by "p". In the notation ofmodular arithmetic ::$a^\{p-1\}\; equiv\; 1\; pmod\{p\},!$Another way to state this is that if "p" is a prime number and "a" is any integer that does not have "p" as a factor, then "a" raised to the "p-1" power will leave a remainder of 1 when divided by "p".

Fermat's little theorem is the basis for the

Fermat primality test .**History**Pierre de Fermat first stated the theorem in a letter datedOctober 18 ,1640 to his friend and confidantFrénicle de Bessy as the following [*http://www.cs.utexas.edu/users/wzhao/fermat2.pdf*] : "p" divides $a^\{p-1\}-1,$ whenever "p" is prime and "a" iscoprime to "p".As usual, 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 progressions and for all prime numbers; the proof of which I would send to you, if I were not afraid to be too long.)

Euler first published a proof in 1736 in a paper entitled "Theorematum Quorundam ad Numeros Primos Spectantium Demonstratio", but Leibniz left virtually the same proof in an unpublished manuscript from sometime before 1683.The term "Fermat's Little Theorem" was first used in 1913 in "Zahlentheorie" by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.)

It was first used in English in an article by

Irving Kaplansky , "Lucas's Tests forMersenne Number s," "American Mathematical Monthly ",**52**(Apr., 1945).**Further history**Chinese

mathematicians independently made the related hypothesis (sometimes called the Chinese Hypothesis) that "p" is a prime if and only if $2^p\; equiv\; 2\; pmod\{p\},$. It is true that if "p" is prime, then $2^p\; equiv\; 2\; pmod\{p\},$. This is a special case of Fermat's little theorem. However, the converse (if $,2^p\; equiv\; 2\; pmod\{p\}$ then "p" is prime) is false. Therefore, the hypothesis, as a whole, is false (for example, $2^\{341\}\; equiv\; 2pmod\{341\},$, but 341=11×31 is apseudoprime . See below.).It is widely stated that the Chinese hypothesis was developed about 2000 years before Fermat's work in the 1600s. Despite the fact that the hypothesis is partially incorrect, it is noteworthy that it may have been known to ancient mathematicians. Some, however, claim that the widely propagated belief that the hypothesis was around so early sprouted from a misunderstanding, and that it was actually developed in 1872. For more on this, see (Ribenboim, 1995).

**Proofs**Fermat gave his theorem without a proof. The first one who gave a proof was

Gottfried Leibniz in a manuscript without a date, where he wrote also that he knew a proof before1683 .See

Proofs of Fermat's little theorem .**Generalizations**A slight generalization of the theorem, which immediately follows from it, is as follows: if "p" is prime and "m" and "n" are "positive" integers with $mequiv\; npmod\{p-1\},$, then $forall\; ainmathbb\{Z\}\; :\; quad\; a^mequiv\; a^npmod\{p\}.$ In this form, the theorem is used to justify the

RSA public key encryption method.Fermat's little theorem is generalized by

Euler's theorem : for any modulus "n" and any integer "a"coprime to "n", we have:$a^\{varphi\; (n)\}\; equiv\; 1\; pmod\{n\}$where φ("n") denotes Euler's φ function counting the integers between 1 and "n" that are coprime to "n". This is indeed a generalization, because if "n" = "p" is a prime number, then φ("p") = "p" − 1.This can be further generalized to Carmichael's theorem.

The theorem has a nice generalization also in

finite field s.**Pseudoprimes**If "a" and "p" are coprime numbers such that $,a^\{p-1\}\; -\; 1$ is divisible by "p", then "p" need not be prime. If it is not, then "p" is called a

pseudoprime to base "a". F. Sarrus in1820 found 341 = 11×31 as one of the first pseudoprimes, to base 2.A number "p" that is a pseudoprime to base "a" for every number "a" coprime to "p" is called a

Carmichael number (e.g. 561).**ee also***Fractions with prime denominators – numbers with behavior relating to Fermat's little theorem

*RSA – How Fermat's little theorem is essential toInternet security**Notes****References***

Paulo Ribenboim (1995). "The New Book of Prime Number Records" (3rd ed.). New York: Springer-Verlag. ISBN 0-387-94457-5.

* [*http://bolyai.port5.com/kisfermat.htm János Bolyai and the pseudoprimes*] (in Hungarian**External links*** [

*http://www.cut-the-knot.org/blue/Fermat.shtml Fermat's Little Theorem*] atcut-the-knot

* [*http://www.cut-the-knot.org/blue/Euler.shtml Euler Function and Theorem*] atcut-the-knot

* [*http://fermatslasttheorem.blogspot.com/2005/08/fermats-little-theorem.html Fermat's Little Theorem and Sophie's Proof*]

* [*http://www.cs.utexas.edu/users/wzhao/fermat2.pdf Text and translation of Fermat's letter to Frenicle*]

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