$p^n+1=k^3$ where $p$ is a prime and $k$ is a positive integer.

Question

My problem:

Find all the prime numbers $p$ for which there exists a positive integer $n$ such that $p^n+1$ is a cube of a positive integer.

Answer 1

$p^n+1=k^3\Rightarrow p^n=(k^3-1)=(k-1)(k^2+k+1)\Rightarrow (k-1)=1$. (As an exercise, prove to yourself that $k-1\ne p^m$)

$k=2\Rightarrow p=7,\ n=1$