Theorem 14: Fermat’s Little Theorem

Firstly, apologies for the long gap.  Very far from being Theorem of the Week, I know.  Here’s another theorem for now, and I’ll do what I can to revert to a weekly post.

So, to this week’s theorem.  I have previously promised to write about Fermat‘s Little Theorem, and I think it’s time to keep that promise.  In that post (Theorem 10, about Lagrange’s theorem in group theory), I introduced the theorem, so I’m going to state it straightaway.  If you haven’t seen the statement before, I suggest you look back at that post to see an example.

Theorem (Fermat’s Little Theorem) Let $latex p$ be a prime, and let $latex a$ be an integer not divisible by $latex p$.  Then $latex a^{p-1} \equiv 1\mod{p}$.

If you aren’t comfortable with the notation of modular arithmetic, you might like to phrase the conclusion of the theorem as saying that $latex…