Fermat's little theorem

Definitions

The theorem that, for any prime number p and integer a, aᵖ-a is an integer multiple of p.

Equivalents

Examples

“1999, John Stillwell, Translator's introduction, Peter Gustav Lejeune Dirichlet, Richard Dedekind (supplements), Lectures on Number⟳ Theory, [1863, P. G. Lejeune Dirichlet, R. Dedekind, Vorlesungen über Zahlentheorie], 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.”

“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.”

“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.”

CEFR level