For what positive integers $n$ is the polynomial
$$n^{11} + n^7 + 1$$
a prime number?
$$n^{11} + n^7 + 1$$
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.
