If p is a prime number and a is any integer not divisible by p, then a raised to the power p minus 1 is congruent to 1 modulo p. It underlies many primality tests and much of elementary number theory, and is a special case of Euler's theorem.
Facts
StatementFermat's little theorem states that if p is a prime number, then for any integer a, the number ap − a is an integer multiple of p. 1 Proof Method Classification
Statement Form Connections
Has Statement Form
In Branch
Named After
Principal Result Of
Source Wikipedia: Pierre de Fermat
Proved By
Sources
1. Fermat's Little Theorem (Wikipedia)
Wikimedia Foundationlead section, first paragraph
Fermat's little theorem states that if p is a prime number, then for any integer a, the number ap − a is an integer multiple of p.
History section, Euler proof sentence
Euler provided the first published proof in 1736
View the Source 2. ProofWiki: Fermat's Little Theorem
ProofWikiProof sectionQuote, Proof section
Proof by induction over $n$.
Wikipedia: Pierre de Fermat
WikipediaPrincipal Result Of: Pierre de Fermat, Article textQuote, Principal Result Of: Pierre de Fermat, Article text
It was while researching perfect numbers that he discovered Fermat's little theorem.
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