## Prime Power

#### Problem

Given that $p$ is prime it can be seen that $p + 1$ is a perfect cube when $p$ = 7. What is most surprising is that this is the only value of $p$ for which $p + 1$ is a perfect cube.

Prove that there only ever exists one prime value $p$ for which $p + 1$ is a perfect power and determine the condition for this perfect power to exist.

Problem ID: 290 (22 Sep 2006) Difficulty: 3 Star