Fermat's little theorem

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For a number not divisible by a prime number , the congruence holds. This theorem was established by P. Fermat (1640). It proves that the order of every element of the multiplicative group of residue classes modulo divides the order of the group. Fermat's little theorem was generalized by L. Euler to the case modulo an arbitrary . Namely, he proved that for every number relatively prime to the given number there is the congruence

where is the Euler function. Another generalization of Fermat's little theorem is the equation , which is valid for all elements of the finite field consisting of elements.


[1] I.M. Vinogradov, "Elements of number theory" , Dover, reprint (1954) (Translated from Russian)



[a1] G.H. Hardy, E.M. Wright, "An introduction to the theory of numbers" , Oxford Univ. Press (1979)
How to Cite This Entry:
Fermat's little theorem. Encyclopedia of Mathematics. URL:
This article was adapted from an original article by S.A. Stepanov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article