Fermat's little theorem
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Fermat's little theorem states that if p is a prime number, then for any integer a,
- <math>a^p \equiv a \pmod{p}\,\!<math>
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
- <math>a^{p-1} \equiv 1 \pmod{p}\,\!<math>
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.
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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 <math>2^p = 2 \pmod{p}<math>. It is true that if p is prime, then <math>2^p = 2 \pmod{p}<math> (this is a special case of Fermat's little theorem). However, the converse (if <math>2^p = 2 \pmod{p}<math> 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 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 m ≡ n (mod p − 1), then am ≡ an (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
- <math>a^{\varphi (n)} \equiv 1 \pmod{n}<math>
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).
See also
- Fractions with prime denominators – numbers with behaviour that relates 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.
- János 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)de:Kleiner fermatscher Satz
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