Fermat's little theorem

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For other theorems named after Pierre de Fermat, see Fermat's theorem (disambiguation).

Fermat's little theorem states that if p is a prime number, then for any integer a, the number a p − a is an integer multiple of p. In the notation of modular arithmetic, this is expressed as

a^p \equiv a \pmod p.

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:

a^{p-1} \equiv 1 \pmod 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.

History[edit]

Pierre de Fermat

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 a p−1 − 1 whenever p is prime and a is coprime to p.

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 provided the first published proof in 1736 in a paper entitled "Theorematum Quorundam ad Numeros Primos Spectantium Demonstratio", but Leibniz had given virtually the same proof in an unpublished manuscript from sometime before 1683.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.)

An early use in English occurs in A.A. Albert, Modern Higher Algebra (1937), which refers to "the so-called "little" Fermat theorem" on page 206.

Further history[edit]

Main article: Chinese hypothesis

Some mathematicians independently made the related hypothesis (sometimes incorrectly called the Chinese Hypothesis) that p is a prime if and only if 2^p \equiv 2 \pmod{p}\,. Indeed, the "only if" part is true, and a special case of Fermat's little theorem. However, the "if" part of this hypothesis is false: for example, 2^{341} \equiv 2\pmod{341}\,, but 341 = 11 × 31 is a pseudoprime. See below.

Proofs[edit]

Fermat gave his theorem without a proof. The first one who gave a proof was Gottfried Leibniz in a manuscript without a date, stating that he knew a proof before 1683.

Generalizations[edit]

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

m\equiv n\pmod{p-1}  then for every integer a we have  a^m\equiv a^n\pmod{p}.

This follows as m is of the form b(p-1)+ n, so a^m = a^{b(p-1)}\cdot a^n \equiv \left({a^{p-1}}\right)^b \cdot a^n\equiv 1^b\cdot a^n \equiv a^n \pmod p .

In this form, the theorem is used to justify the RSA public key encryption method.[citation needed]

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 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.

This can be further generalized to Carmichael's theorem, as well as to Lagrange's theorem in group theory.

Fermat's little theorem also has a generalization in finite fields.

Pseudoprimes[edit]

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

\gcd\left(\sum_{a=1}^{p-1} a^{p-1}, p\right)=1

is either a prime or Carmichael number.

Converse[edit]

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

 a^{p-1}\equiv 1\pmod p

and for all primes q dividing p − 1

 a^{(p-1)/q}\not\equiv 1\pmod p

then p is prime.

This theorem forms the basis for the Lucas–Lehmer test, an important primality test.

See also[edit]

Notes[edit]

  1. ^ "Fermat to Frenicle, Thursday, 18 October 1640". Archived from the original on 22 December 2006. 

External links[edit]