Math Is Fun Forum

Jai Ganesh

Administrator

Registered: 2005-06-28

Posts: 48,384

Fermat's Little Theorem

Can anybody post the proof to Fermat's Little Theorem (The Theorm on prime numbers)?

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Ricky

Moderator

Registered: 2005-12-04

Posts: 3,791

Re: Fermat's Little Theorem

There are two basic proofs for this, a number theoretic proof and a group theory proof.  This is the group theory proof:

Note that the integers modulo p form a group under multiplication (with 0 removed).  By Lagrange's theorem, for any element a in a group G, a^|G| = e, the identity.  Specifically, if a is a non-zero integer modulo p, then a^(p-1) = 1, as p-1 is the order of this group under multiplication.

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"In the real world, this would be a problem.  But in mathematics, we can just define a place where this problem doesn't exist.  So we'll go ahead and do that now..."