Menu Home My entries Defined browse Select Select in the list MathematicsPhysics Then Select Select in the list Then Select Select in the list Search

# Fermat's little theorem

Edit

 Edit Definition/Summary Fermat's little theorem states that if $p$ is a prime number, then for any integer $a$, $a^{p}-a$ will be divisible by

 Edit Equations $$a^{p-1}\equiv1\pmod p \quad (\text{for\ }a \not\equiv 0 \pmod p)$$ $$a^p\equiv a\pmod p$$

 Edit Scientists Pierre de Fermat (1601?-1665)

 Recent forum threads on Fermat's little theorem

 Edit Breakdown Mathematics > Number Theory >> Congruences

 Edit See Also WikipediaMathworld

 Edit Images

 Edit Extended explanation Fermat's Little Theorem If p is a prime number and a an integer, then $$a^p\equiv a\ (p)$$ In order to prove Fermat's Little theorem, we will start by proving a superficially slightly weaker result, which is also referred to as Fermat's Little Theorem, on occasion. The two results imply each other, however. Theorem Let a and p be coprime, then $$a^{p-1}-1 \equiv 0\ (p).$$ Proof Start by listing the first p-1 positive multiples of a: $$a, 2a, 3a, \ldots, (p -1)a$$ Suppose that $ra$ and $sa$ are the same modulo $p$, with [itex]0

Commentary

 matt grime @ 10:45 AM Jun8-08 Done the tex'ing. Removed (unneccessary and unused) reference to there being a minimal d with some property. Made it clearer what each 'proof' was proving.

 Gokul43201 @ 12:37 PM May24-08 Definition amended

 Gokul43201 @ 05:30 PM May22-08 Error in definition - fixed now, but it's still only the weaker form.

 Edit log Date User Name Edit type Mar13-13 Unregistered definition Mar13-13 Unregistered definition