Let $\ \phi_m(x)\ $ denote the $\ m\ $-th cyclotomic polynomial.
For $\ n\ge 2\ $ define $\ t:=\phi_n(n)\ $
Must all the prime factors $\ p\ $ of $\ \phi_t(n)\ $ satisfy $\ p\equiv 1\mod t\ $ ?
It is easy to show that $\ t\ $ must be odd. It is known that a prime factor $\ p\ $ of $\phi_t(n)$ must either saitisfy $\ t\mid p-1\ $ or it must be the largest prime factor of $t$. I tried to rule out $p\mid t$ by writing $\ t\ $ as $\ t=p^k\cdot u\ $ with $\ p\nmid u\ $ and I worked out that $\ p\mid n^u-1\ $. Here is where I am stuck.