Thrice Composite... When is it Prime?

Notice that if $\omega^3 = 1$ , then $\omega^5 + \omega^4 + 1 = \omega^2 + \omega + 1 = 0$ , and hence follows $n^2 + n + 1|n^5 + n^4 + 1$ : Indeed , if $\omega$ is a root of both of the polynomials then one divides the other. Being $1^2 + 1 + 1=1^5 + 1^4 + 1 = 3$ , the other factor will be $1$ for $n=1$ , but neither of the factors will be $1$ henceforward. So, $n=1$ is the only solution.