TITLE: Take a cube, add one, when is this a prime?
QUESTION [5 upvotes]: Take 
$$
p = n^3 + 1
$$
I just realised, checking some numbers, that this only occurs for $n=1$. I was wondering if there is some general proof for this?

REPLY [5 votes]: You can make your understanding stronger by the fact for any odd number $k > 1$ and $n > 1$, the number $p = n^k + 1$ is never a prime because it is divisible by $n+1$. The cubes or $k = 3$ is a special case of this.