Fermat'S Little Theorem

(Theory)

Logo of Fermat's little theorem
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Theory
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What is Fermat's little theorem?

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. In the notation of modular arithmetic, this is expressed as For example, if a = 2 and p = 7, then 27 = 128, and 128 − 2 = 126 = 7 × 18 is an integer multiple of 7. If a is not divisible by p, Fermat's little theorem is equivalent to the statement that ap − 1 − 1 is an integer multiple of p, or in symbols: For example, if a = 2 and p = 7, then 26 = 64, and 64 − 1 = 63 = 7 × 9 is thus a multiple of 7. Fermat's little theorem is the basis for the Fermat primality test and is one of the fundamental results of elementary number theory. The theorem is named after Pierre de Fermat, who stated it in 1640. It is called the "little theorem" to distinguish it from Fermat's last theorem.

Technology Types

definite quantitydiophantine equationequationexperimentinquirymathematical statementmathematical theoremmodular arithmeticprimality testprimeprime numberproblem solvingprocesspropositionstatementtheoremtheorems about prime numbertheorems in group theorytheorems in number theorytheorythinkingtrial

Synonyms

Fermat lesser theoremFermat little theoremFermat simple theoremFermat's lesser theoremFermat's Little Theorem ConverseFermat's simple theoremFermats little theoremFermats little theoromLittle Fermat theorem

Translations

Fermats lilla sats (sv)Kleine stelling van Fermat (nl)Kleiner fermatscher Satz (de)Malá Fermatova věta (cs)Małe twierdzenie Fermata (pl)Malgranda teoremo de Fermat (eo)Pequeño teorema de Fermat (es)Petit teorema de Fermat (ca)Petit théorème de Fermat (fr)Piccolo teorema di Fermat (it)Teorema kecil Fermat (in)Teste de primalidade de Fermat (pt)Μικρό θεώρημα του Φερμά (el)Мала теорема Ферма (uk)Малая теорема Ферма (ru)مبرهنة فيرما الصغرى (ar)페르마의 소정리 (ko)フェルマーの小定理 (ja)费马小定理 (zh)

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