Fermat generalization, not sure if it's true or how to prove it I have this rule in my notebook, but I don't remember when I took it:
$$a^{p^n}$$ is congruent with a(p) (a modulo p), where p is a prime number, a is an integer and n is a natural number.
Or, in other words, that: 
$$p | a^{p^n} - a$$ 
I can't find this rule in Internet. It's supposed to be a generalization of Fermat Little Theorem.
Is this true? If so, how can be derivated from Fermat Little Theorem?

Fermat's (little) theorem is: If $p$ doesn't divide $a$ then $a^{p-1} \equiv 1$ mod $p.$ When  both sides multiplied by $a$ it becomes $a^p \equiv a$ mod $p$ and now one can allow $a$ divisible by $p.$ At this point apply an inductive argument, next case being $(a^p)^p \equiv a^p,$ then use inductive hypothesis to finish. Higher powers go similarly, each going down to the previous power.
