If $p^q - 1$ is a prime, then $p=2$ and $q$ is a prime

Question

I was working my way through some number theoretic proofs and being a newbie am stuck on this problem :

If $p$ and $q$ are positive integers ($\mathbb{Z}^+$) such that $q \gt 1$ and $(p^q - 1)$ is a prime, then $p$ is $2$ and $q$ is a prime.

My Question :

I am unable to make any concrete progress . Even a decent hint would be acceptable so that I can build on that ...

Answer 1

Since if $p$ is odd then $p^q-1$ is even and larger then two you must have $p=2$. Now, if $q=ab$ then $p^q-1$ is divided by $p^a-1$. (show it). Form here $q$ must be prime.