Fermat's little theorem
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English
Alternative forms
Etymology
Named after French lawyer and amateur mathematician Pierre de Fermat (1601–1665), who stated a version of the theorem in a letter in 1640. Called little to distinguish it from Fermat's Last Theorem.
Proper noun
- (number theory) The theorem that, for any prime number and integer , is an integer multiple of .
- 1999, John Stillwell, Translator's introduction, Peter Gustav Lejeune Dirichlet, Richard Dedekind (supplements), Lectures on Number Theory, , American Mathematical Society, page xi,
- When combined with the historical remarks made by Gauss himself, they give a bird's eye view of number theory from approximately 1640 to 1840 - from Fermat's little theorem to L-functions - the period which produced the problems and ideas which are still at the center of the subject.
- 1999, Siguna Müller, On the Combined Fermat/Lucas Probable Prime Test, Michael Walker (editor), Cryptography and Coding: 7th IMA International Conference, Springer, LNCS 1746, page 222,
- Most of the pseudoprimality tests originate in some sense on Fermat's Little Theorem an−1 ≡ 1 mod n.
- 2007, Thomas Koshy, Elementary Number Theory with Applications, 2nd edition, Elsevier (Academic Press), page 327:
- Incidentally, the special case of Fermat's little theorem for a = 2 was known to the Chinese as early as 500 B.C.
The first proof of Fermat's little theorem was given by Euler in 1736, almost a century after Fermat's announcement.
- 2008, Lawrence C. Washington, Elliptic Curves: Number Theory and Cryptography, 2nd edition, Taylor & Francis (Chapman & Hall/CRC), page 189:
- Fermat's little theorem says that if n is prime and gcd(a,n) = 1, then an−1 ≡ 1 (mod n), so it follows that n must be composite, even though we have not produced a factor.
- 1999, John Stillwell, Translator's introduction, Peter Gustav Lejeune Dirichlet, Richard Dedekind (supplements), Lectures on Number Theory, , American Mathematical Society, page xi,
Synonyms
- (theorem that a p − a is divisible by p): Fermat's theorem
Translations
theorem that ap − a is divisible by p
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See also
Further reading
Euler's theorem on Wikipedia.Wikipedia - Fermat pseudoprime on Wikipedia.Wikipedia
