7-11 polynomial

For what positive integers $n$ is the polynomial

$$n^{11} + n^7 + 1$$

a prime number?

Scroll down for a solution.

Note that if $w \neq 1$ is a cube root of unity, then 

$$w^{11} +w^7 + 1 = w^2 + w + 1 = 0$$

Thus $x^{11} +x^7 + 1$ is divisible by $x^2 + x + 1$ and for $n \gt 1$, $n^{11} + n^7  + 1 > n^2 + n + 1$.

Thus $n=1$ is the only number for which $n^{11} +  n^7 + 1$ is a prime.