# Fermat's little theorem

For a number $a$ not divisible by a prime number $p$, the congruence $a^{p-1}\equiv1\pmod p$ 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 $p$ divides the order of the group. Fermat's little theorem was generalized by L. Euler to the case modulo an arbitrary $m$. Namely, he proved that for every number $a$ relatively prime to the given number $m>1$ there is the congruence

$$a^{\phi(m)}\equiv1\pmod m,$$

where $\phi(m)$ is the Euler function. Another generalization of Fermat's little theorem is the equation $x^q=x$, which is valid for all elements of the finite field $k_q$ consisting of $q$ elements.

#### References

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

#### Comments

#### References

[a1] | G.H. Hardy, E.M. Wright, "An introduction to the theory of numbers" , Oxford Univ. Press (1979) |

**How to Cite This Entry:**

Fermat little theorem.

*Encyclopedia of Mathematics.*URL: http://www.encyclopediaofmath.org/index.php?title=Fermat_little_theorem&oldid=31847