- Fermat's little theorem
Fermat's little theorem (not to be confused with
Fermat's last theorem) states that if "p" is a prime number, then for any integer"a", will be evenly divisible by "p". This can be expressed in the notation of modular arithmeticas follows::
A variant of this theorem is stated in the following form: if "p" is a prime and "a" is an integer
coprimeto "p", then will be evenly divisible by "p". In the notation of modular arithmetic::
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.
Pierre de Fermatfirst stated the theorem in a letter dated October 18, 1640to his friend and confidant Frénicle de Bessyas the following [http://www.cs.utexas.edu/users/wzhao/fermat2.pdf] : "p" divides whenever "p" is prime and "a" is coprimeto "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.)
Eulerfirst 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"Theorematum Quorundam ad Numeros Primos Spectantium Demonstratio", but Leibniz left virtually the same proof in an unpublished manuscript from sometime before 1683.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
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 for Mersenne Numbers," " American Mathematical Monthly", 52 (Apr., 1945).
mathematiciansindependently made the related hypothesis (sometimes called the Chinese Hypothesis) that "p" is a prime if and only if . It is true that if "p" is prime, then . This is a special case of Fermat's little theorem. However, the converse (if then "p" is prime) is false. Therefore, the hypothesis, as a whole, is false (for example, , but 341=11×31 is a pseudoprime. 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).
Fermat gave his theorem without a proof. The first one who gave a proof was
Gottfried Leibnizin a manuscript without a date, where he wrote also that he knew a proof before 1683.
Proofs of Fermat's little theorem.
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 , then In this form, the theorem is used to justify the
RSA public key encryptionmethod.
Fermat's little theorem is generalized by
Euler's theorem: for any modulus "n" and any integer "a" coprimeto "n", we have: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
If "a" and "p" are coprime numbers such that is divisible by "p", then "p" need not be prime. If it is not, then "p" is called a
pseudoprimeto base "a". F. Sarrus in 1820found 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).
*Fractions with prime denominators – numbers with behavior relating to Fermat's little theorem
RSA– How Fermat's little theorem is essential to Internetsecurity
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
* [http://www.cut-the-knot.org/blue/Fermat.shtml Fermat's Little Theorem] at
* [http://www.cut-the-knot.org/blue/Euler.shtml Euler Function and Theorem] at
* [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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