I'm just starting with integer division and congruence in an algebra course and I have this problem:
Let $a$ be an odd integer. Prove that $\forall n \in \Bbb N$:
$$2^{n+2}\ |\ a^{2^n} - 1$$
I've tried a few things but got nowhere. For instance, if I try by induction, I end up with this:
First I check $P(1):$
$$2^3\ |\ a^2-1$$ Being an odd number: $$8\ |\ (2k+1)^2-1$$ $$8\ |\ 4k^2+4k$$ But how can I continue? Or can you think of another way of proving it? Thanks a lot.