Something dynamics can tell us about number theory

Now for some number theory

We have seen for all integers n > 1, and for all positive integers p, fnp has np fixed points.

So if p is a prime number, we have seen that each of these fixed points of fnp belongs to a p-cycle or is a fixed point.

We have seen that fn has n fixed points.

Consequently, np - n of the fixed points of fnp belong to p-cycles of f.

Each p-cycle consists of p points, so these np - n points separate themselves into p-cycles.

That is, for all prime numbers p and for all integers n > 1,

p divides np - n.

This is Fermat's Little Theorem.

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