# All theorems that should be considered before proving Fermat's Little theorem?

Hello fellow mathematicians. I was allowed to choose any topic that we learned about in our proof-writing/number theory class. And I decided to go with Fermat's little theorem because I find it really interesting and its applications in cryptography are also cool.

The goal of this assignment is to basically write a 3 page paper proving the theorem but it must be extremely explicit. Meaning I need to provide and prove theorems, lemmas, etc, before actually proving Fermat's Little theorem.

For now all I have considered is proving unique multiplicative inverses (modulo). But that is really all that I can think of currently. Is there anything else I should consider showing to a reader who may not be the most mathematically adept?

My goal with this question is to build a flowchart of how my paper should be laid out. Your help and guidance would be super appreciated!

• I think, the best approach is induction over the base $a$. If we expand $(a+1)^p$ , we can show that modulo $p$ only two terms remain. This way we can show $a^p\equiv a\mod p$ and the theorem easily follows. Nov 24 '20 at 10:42
• Are you familiar with Lagrange theorem in Group theory? There is a nice way to derive Fermat's little theorem using unitary groups by applying Lagrange theorem. Nov 24 '20 at 11:46

My favorite is the group theory proof, where by considering the multiplicative group $$\Bbb F_p^*$$ of the field, we know by Lagrange's theorem that any element has order dividing the order of that group, $$p-1$$.

Or you can do induction on the base, and invoke the freshman's binomial, more from the point of view of number theory, I guess.

(Both of these nice proofs were mentioned in the comments.)

Perhaps it is worth mentioning that Euler's theorem is a generalization of it.

Here is a roadmap for a nice proof that $$a^{p-1} \equiv 1 \bmod p$$ when $$a \not\equiv 0 \bmod p$$.

• The map $$x \mapsto ax$$ is injective in the residue classes mod $$p$$

• The map $$x \mapsto ax$$ is bijective in the residue classes mod $$p$$

• $$1\cdot 2 \cdots (p-1) \equiv (a\cdot1)(a\cdot2)\cdots(a\cdot(p-1))=a^{p-1}(1\cdot 2 \cdots (p-1)) \bmod p$$

It's essentially the proof of Lagrange's theorem for finite abelian groups.