Fermat's little theorem
For example, if a = 2 and p = 7, 27 = 128, and 128 − 2 = 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 a p − 1 − 1 is an integer multiple of p:
For example, if a = 2 and p = 7, 26 = 64, and 64 − 1 = 63 = 7 × 9.
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 as the following:[dead link] p divides a p−1 − 1 whenever p is prime and a is coprime 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"Theorematum Quorundam ad Numeros Primos Spectantium Demonstratio", but Leibniz left virtually the same proof in an unpublished manuscript from sometime before 1683.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.)
Further history 
Some mathematicians independently made the related hypothesis (sometimes incorrectly called the Chinese Hypothesis) that p is a prime if and only if . This is a special case of Fermat's little theorem. However, the "if" part of this hypothesis is false: for example, , but 341 = 11 × 31 is a pseudoprime. See below.
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 before 1683.
A slight generalization of the theorem, which immediately follows from it, is: if p is prime and m and n are positive integers such that
- then for every integer a we have .
This follows as m is of the form , so
where φ(n) denotes Euler's totient function (which counts 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.
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 in 1820 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). Alternately, any number satisfying the equality
is either a prime or Carmichael number.
The converse of Fermat's little theorem is not generally true, as it fails for Carmichael numbers. However, a slightly stronger form of the theorem is true, and is known as Lehmer's theorem. The theorem is as follows:
If there exists an a such that
and for all prime q dividing p − 1
then p is prime.
See also 
- Fractions with prime denominators–numbers with behavior relating to Fermat's little theorem
- Frobenius endomorphism
- Table of congruences
- Paulo Ribenboim (1995). The New Book of Prime Number Records (3rd ed.). New York: Springer-Verlag. ISBN 0-387-94457-5. pp. 22–25, 49.
- János Bolyai and the pseudoprimes (in Hungarian)
- Necklace Visualization
- Fermat's Little Theorem at cut-the-knot
- Euler Function and Theorem at cut-the-knot
- Fermat's Little Theorem and Sophie's Proof
- Text and translation of Fermat's letter to Frenicle
- Hazewinkel, Michiel, ed. (2001), "Fermat's little theorem", Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4
- Weisstein, Eric W., "Fermat's Little Theorem", MathWorld.
- Weisstein, Eric W., "Fermat's Little Theorem Converse", MathWorld.