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, Failed to parse (Missing texvc executable; please see math/README to configure.): (a^p - a)

will be evenly divisible by p. This can be expressed in the notation of modular arithmetic as follows:

Failed to parse (Missing texvc executable; please see math/README to configure.): 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 Failed to parse (Missing texvc executable; please see math/README to configure.): (a^{p-1} - 1)

will be evenly divisible by p.  In the notation of modular arithmetic:

Failed to parse (Missing texvc executable; please see math/README to configure.): 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.

Examples

• 4³ − 4 = 60 is divisible by 3.
• 7² − 7 = 42 is divisible by 2.
• (−3)⁷ − (−3) = −(2 184) is divisible by 7.
• 2⁹⁷ − 2 = 158 456 325 028 528 675 187 087 900 670 is divisible by 97.

History

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 [1]: p divides Failed to parse (Missing texvc executable; please see math/README to configure.): 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"Theorematum Quorundam ad Numeros Primos Spectantium Demonstratio", but Leibniz left virtually the same proof in an unpublished manuscript from sometime before 1683.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.)

It was first used in English in an article by Irving Kaplansky, "Lucas's Tests for Mersenne Numbers," 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 Failed to parse (Missing texvc executable; please see math/README to configure.): 2^p \equiv 2 \pmod{p}\, . It is true that if p is prime, then Failed to parse (Missing texvc executable; please see math/README to configure.): 2^p \equiv 2 \pmod{p}\, . This is a special case of Fermat's little theorem. However, the converse (if Failed to parse (Missing texvc executable; please see math/README to configure.): \,2^p \equiv 2 \pmod{p}

then p is prime) is false. Therefore, the hypothesis, as a whole, is false (for example, Failed to parse (Missing texvc executable; please see math/README to configure.): 2^{341} \equiv 2\pmod{341}\,

, 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).

Proofs

Fermat explained 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.

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 Failed to parse (Missing texvc executable; please see math/README to configure.): m\equiv n\pmod{p-1}\, , then Failed to parse (Missing texvc executable; please see math/README to configure.): \forall a\in\mathbb{Z} : \quad a^m\equiv a^n\pmod{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

Failed to parse (Missing texvc executable; please see math/README to configure.): 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 fields.

Pseudoprimes

If a and p are coprime numbers such that Failed to parse (Missing texvc executable; please see math/README to configure.): \,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).

See also

• Fractions with prime denominators – numbers with behavior relating to Fermat's little theorem
• RSA – How Fermat's little theorem is essential to Internet security

References

External links

• 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
• Eric W. Weisstein, Fermat's Little Theorem at MathWorld.ar:مبرهنة فيرما

bg:Малка теорема на Ферма cs:Malá Fermatova věta de:Kleiner fermatscher Satz es:Pequeño teorema de Fermat fi:Fermat'n pieni lause fr:Petit théorème de Fermat he:המשפט הקטן של פרמה hu:Kis Fermat-tétel id:Teorema kecil Fermat it:Piccolo teorema di Fermat ja:フェルマーの小定理 ko:페르마의 소정리 nl:Kleine stelling van Fermat pl:Małe twierdzenie Fermata pt:Teste de primalidade de Fermat ro:Mica teoremă a lui Fermat ru:Малая теорема Ферма sl:Fermatov mali izrek sr:Мала Фермаова теорема sv:Fermats lilla sats ta: ஃபெர்மாவின் சிறிய தேற்றம் th:ทฤษฎีบทเล็กของแฟร์มาต์ tr:Fermat'ın Küçük Teoremizh-classical:費馬小定理