Considering the equation $p^{k+1}-1=(p-1)q^n$, where $p$ and $q$ are primes, $k$ and $n$ are integers such that $k>1$ and $n>0$, is it true that $p?
Thanks in advance.
Edit: it can be shown that $p\lt q$ iff $n\lt k$. I think that whenever $p$ is not a divisor of $k$, then one has $n=\varphi(k)/l_{k}(p)$, where $\varphi$ is Euler's totient function and $l_{k}(p)$ is the order of $p$ in $(\mathbb{Z}/k\mathbb{Z})^{\times}$. I'm not sure this kind of question really fits the "elementary number theory" tag, but I'd be glad if someone could prove this.