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## #1 2007-12-09 18:50:08

ganesh
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Registered: 2005-06-28
Posts: 21,730

### Fermat's Little Theorem

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

It is no good to try to stop knowledge from going forward. Ignorance is never better than knowledge - Enrico Fermi.

Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.

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## #2 2007-12-11 14:33:30

Ricky
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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.

"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..."

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