Find all primes $p$ so that $p+1$ is a perfect power.

Question

Inspired by this question, in a more general setting when can we say that $p+1 = k^n$ ?

If we take a sneak peek to the answers to the linked question.. this would be the same as to try and factor $k^n-1$ which is nice to do if $n=2$. But how can we do that for other $n$? Does there perhaps exist a generalization to the conjugate rule..?

Answer 1

If $k\neq 2$ then $p=k^n -1$ is composite number since it is equal to $(k-1)\cdot\sum_{0\leq j\leq n-1} k^j .$ If $k=2$ then we obtain Mersenne numbers.

Answer 2

$$p=k^n-1=(k-1)(k^{n-1}+k^{n-2}+\dots+k+1)$$and this will give a non-trivial factorisation unless $k=2$ (or $n=1$), so you get the Mersenne primes.