Let $\Phi_n(x)$ be the $n$ th cyclotomic polynomial. I've checked the values of $\Phi_n(2)$ for some small $n\geq 2$ and noticed that there is always a divisor of $\Phi_n(2)$ in of the form of $kn+1$ (regardless of $\Phi_n(2)$ being a prime or not) with the only exception of $n=6$. Does anyone know any counterexample to this or a proof for its validity?