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**ganesh****Moderator**- Registered: 2005-06-28
- Posts: 21,307

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

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

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**Ricky****Moderator**- Registered: 2005-12-04
- Posts: 3,791

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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