Let $n\ge 2$ be an integer and $p\ge 7$ be a prime number.
Is there a prime of the form $\phi_n(n^p)$ , where $\phi_n$ denotes the $n^\text{th}$ cyclotomic polynomial ?
We must have $p\mid n$ because if $p\nmid n$ , we have $$\phi_n(n^p)=\phi_n(n)\cdot \phi_{np}(n)$$
Hence, the problem boils down to find a prime $p\ge 7$ and an integer $s\ge 1$ , such that $$\phi_{sp^2}(sp)$$ is prime. According to my calculations such a prime must have more than $10\ 000$ digits. Can we rule out somehow such a prime, or do I have to continue the search ?