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Fermat's little theorem

Fermat's little theorem states that if p is a prime number, then for any integer a,

a^p \equiv a \pmod{p}

This means that if you take some number a, multiply it by itself p times and subtract a, the result is divisible by p (see modular arithmetic). It is often stated in the following equivalent form: if p is a prime and a is an integer coprime to p, then

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

It is called Fermat's little theorem to differentiate it from Fermat's last theorem.

Fermat's little theorem is the basis for the Fermat primality test.

Contents

History

Pierre de Fermat found the theorem around 1636. It appeared in one of his letters, dated October 18 1640 to his confidant Frenicle as the following: p divides ap−1 − 1 whenever p is prime and a is coprime to p.

Chinese mathematicians independently made the related hypothesis (sometimes called the Chinese Hypothesis) that p is a prime if and only if 2p = 2(mod p). It is true that if p is prime, then 2p = 2(mod p) (this is a special case of Fermat's little theorem). However, the converse (if 2p = 2(mod p) then p is prime), and therefore the hypothesis as a whole, is false (e.g. 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 1600's. 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 Wilhelm Leibniz in a manuscipt 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 mn (mod p − 1), then aman (mod p) for all integers a. 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

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 ap−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 is a Carmichael number).

Watch these interesting numbers

The Wiki page recurring decimal includes an article on 'fractions with prime denominators' and some numbers with interesting behaviour, which relates directly to Fermat's little theorem.

References

  • Ribenboim, P. (1995). The New Book of Prime Number Records (3rd ed.). New York: Springer-Verlag. ISBN 0-387-94457-5.
  • JŠnos Bolyai and the pseudoprimes http://bolyai.port5.com/kisfermat.htm (in Hungarian)

External links

  • Fermat's Little Theorem http://www.cut-the-knot.org/blue/Fermat.shtml
  • Euler Function and Theorem http://www.cut-the-knot.org/blue/Euler.shtml



Last updated: 02-16-2005 09:09:03
Last updated: 03-18-2005 11:16:12