For example, if a = 2 and p = 7, then 27 = 128, and 128 − 2 = 126 = 7 × 18 is an integer multiple of 7.
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.
If p is a prime and a is any integer not divisible by p, then a p − 1 − 1 is divisible by p.
Fermat's original statement was
Tout nombre premier mesure infailliblement une des puissances de quelque progression que ce soit, et l'exposant de la dite puissance est sous-multiple du nombre premier donné ; 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.
This may be translated, with explanations and formulas added in brackets for easier understanding, as:
Every prime number [p] divides necessarily one of the powers minus one of any [geometric] progression [a, a2, a3, …] [that is, there exists t such that p divides at – 1], and the exponent of this power [t] divides the given prime minus one [divides p – 1]. After one has found the first power [t] that satisfies the question, all those whose exponents are multiples of the exponent of the first one satisfy similarly the question [that is, all multiples of the first t have the same property].
Fermat did not consider the case where a is a multiple of p nor 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 [sic] and for all prime numbers; I would send you a demonstration of it, if I did not fear going on for too long.)
Euler provided the first published proof in 1736, in a paper titled "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Demonstratio" in the Proceedings of the St. Petersburg Academy, but Leibniz had given virtually the same proof in an unpublished manuscript from sometime before 1683.
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.)
Some mathematicians independently made the related hypothesis (sometimes incorrectly called the Chinese Hypothesis) that 2p ≡ 2 (mod p) if and only if p is prime. Indeed, the "if" part is true, and it is a special case of Fermat's little theorem. However, the "only if" part is false: For example, 2341 ≡ 2 (mod 341), but 341 = 11 × 31 is a pseudoprime to base 2. See below.
Several proofs of Fermat's little theorem are known. It is frequently proved as a corollary of Euler's theorem.
where φ(n) denotes Euler's totient function (which counts the integers from 1 to n that are coprime to n). Fermat's little theorem is indeed a special case, because if n is a prime number, then φ(n) = n − 1.
A corollary of Euler's theorem is: for every positive integer n, if the integer a is coprime with n then
for any integers x and y. This follows from Euler's theorem, since, if , then x = y + kφ(n) for some integer k, and one has
If n is prime, this is also a corollary of Fermat's little theorem. This is widely used in modular arithmetic, because this allows reducing modular exponentiation with large exponents to exponents smaller than n.
retrieving x from the values of y, e and n is easy if one knows φ(n). In fact, the extended Euclidean algorithm allows computing the modular inverse of e modulo φ(n); that is, the integer f such that It follows that
On the other hand, if n = pq is the product of two distinct prime numbers, then φ(n) = (p − 1)(q − 1). In this case, finding f from n and e is as difficult as computing φ(n) (this has not been proven, but no algorithm is known for computing f without knowing φ(n)). Knowing only n, the computation of φ(n) has essentially the same difficulty as the factorization of n, since φ(n) = (p − 1)(q − 1), and conversely, the factors p and q are the (integer) solutions of the equation x2 – (n − φ(n) + 1) x + n = 0.
The basic idea of RSA cryptosystem is thus: if a message x is encrypted as y = xe (mod n), using public values of n and e, then, with the current knowledge, it cannot be decrypted without finding the (secret) factors p and q of n.
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 it is known as Lehmer's theorem. The theorem is as follows:
If there exists an integer a such that
and for all primes q dividing p − 1 one has
then p is prime.
If a and p are coprime numbers such that ap−1 − 1 is divisible by p, then p need not be prime. If it is not, then p is called a (Fermat) pseudoprime to base a. The first pseudoprime to base 2 was found in 1820 by Pierre Frédéric Sarrus: 341 = 11 × 31.
A number p that is a Fermat pseudoprime to base a for every number a coprime to p is called a Carmichael number (for example, 561). Alternately, any number p satisfying the equality
is either a prime or a Carmichael number.
Miller–Rabin primality test
If p is an odd prime and p − 1 = 2sd with s > 0 and d odd > 0, then for every a coprime to p, either ad ≡ 1 (mod p) or there exists r such that 0 ≤ r < s and a2rd ≡ −1 (mod p).
This result may be deduced from Fermat's little theorem by the fact that, if p is an odd prime, then the integers modulo p form a finite field, in which 1 modulo p has exactly two square roots, 1 and −1 modulo p.
Note that ad ≡ 1 (mod p) holds trivially for a ≡ 1 (mod p), because the congruence relation is compatible with exponentiation. And ad = a20d ≡ −1 (mod p) holds trivially for a ≡ −1 (mod p) since d is odd, for the same reason. That is why one usually chooses a random a in the interval 1 < a < p − 1.
The Miller–Rabin test uses this property in the following way: given an odd integer p for which primality has to be tested, write p − 1 = 2sd with s > 0 and d odd > 0, and choose a random a such that 1 < a < p − 1; then compute b = ad mod p; if b is not 1 nor −1, then square it repeatedly modulo p until you get −1 or have squared s − 1 times. If b ≠ 1 and −1 has not been obtained by squaring, then p is a composite and a is a witness for the compositeness of p. Otherwise, p is a strong probable prime to base a; that is, it may be prime or not. If p is composite, the probability that the test declares it a strong probable prime anyway is at most 1⁄4, in which case p is a strong pseudoprime, and a is a strong liar. Therefore after k non-conclusive random tests, the probability that p is composite is at most 4−k, and may thus be made as low as desired by increasing k.
In summary, the test either proves that a number is composite, or asserts that it is prime with a probability of error that may be chosen as low as desired. The test is very simple to implement and computationally more efficient than all known deterministic tests. Therefore, it is generally used before starting a proof of primality.
- Long 1972, pp. 87–88.
- Pettofrezzo & Byrkit 1970, pp. 110–111.
- Burton 2011, p. 514.
- Fermat, Pierre (1894), Tannery, P.; Henry, C. (eds.), Oeuvres de Fermat. Tome 2: Correspondance, Paris: Gauthier-Villars, pp. 206–212 (in French)
- Mahoney 1994, p. 295 for the English translation
- Euler, Leonhard (1736). "Theorematum quorundam ad numeros primos spectantium demonstratio" [Proof of certain theorems relating to prime numbers]. Commentarii Academiae Scientiarum Imperialis Petropolitanae (Memoirs of the Imperial Academy of Sciences in St. Petersburg) (in Latin). 8: 141–146.
- Ore 1988, p. 273
- Hensel, Kurt (1913). Zahlentheorie [Number Theory] (in German). Berlin and Leipzig, Germany: G. J. Göschen. p. 103.
- Albert 2015, p. 206
- Trappe, Wade; Washington, Lawrence C. (2002), Introduction to Cryptography with Coding Theory, Prentice-Hall, p. 78, ISBN 978-0-13-061814-6
- If y is not coprime with n, Euler's theorem does not work, but this case is sufficiently rare for not being considered. In fact, if it occurred by chance, this would provide an easy factorization of n, and thus break the considered instance of RSA.
- Sloane, N. J. A. (ed.). "Sequence A128311 (Remainder upon division of 2n−1−1 by n.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
- Sarrus, Frédéric (1819–1820). "Démonstration de la fausseté du théorème énoncé á la page 320 du IXe volume de ce recueil" [Demonstration of the falsity of the theorem stated on page 320 of the 9th volume of this collection]. Annales de Mathématiques Pures et Appliquées (in French). 10: 184–187.
- Rempe-Gillen, Lasse; Waldecker, Rebecca (2013-12-11). "4.5.1. Lemma (Roots of unity modulo a prime)". Primality Testing for Beginners. American Mathematical Soc. ISBN 9780821898833.
- Albert, A. Adrian (2015) , Modern higher algebra, Cambridge University Press, ISBN 978-1-107-54462-8
- Burton, David M. (2011), The History of Mathematics / An Introduction (7th ed.), McGraw-Hill, ISBN 978-0-07-338315-6
- Long, Calvin T. (1972), Elementary Introduction to Number Theory (2nd ed.), Lexington: D. C. Heath and Company, LCCN 77171950
- Mahoney, Michael Sean (1994), The Mathematical Career of Pierre de Fermat, 1601–1665 (2nd ed.), Princeton University Press, ISBN 978-0-691-03666-3
- Ore, Oystein (1988) , Number Theory and Its History, Dover, ISBN 978-0-486-65620-5
- Pettofrezzo, Anthony J.; Byrkit, Donald R. (1970), Elements of Number Theory, Englewood Cliffs: Prentice Hall, LCCN 71081766
- 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.
- Media related to Fermat's little theorem at Wikimedia Commons
- János Bolyai and the pseudoprimes (in Hungarian)
- Fermat's Little Theorem at cut-the-knot
- Euler Function and Theorem at cut-the-knot
- Fermat's Little Theorem and Sophie's Proof
- "Fermat's little theorem", Encyclopedia of Mathematics, EMS Press, 2001 
- Weisstein, Eric W. "Fermat's Little Theorem". MathWorld.
- Weisstein, Eric W. "Fermat's Little Theorem Converse". MathWorld.