Can $$f(n):=\Phi_n(\Phi_n(n))$$ where $n$ is a positive integer, be a prime number , where $\Phi_n(x)$ denotes the $n-th$ cyclotomic polynomial ?
- If $n$ is a prime power $p^k$ ($p$ prime , $k$ positive integer), then we can show $p\mid f(n)$ , hence we cannot have a prime number.
- I checked upto $n=389$ without finding a prime number.
- Small factors are not forced : I know no prime factor of $f(45)$ (I invite everyone to search for one) , the smallest has very probably more than $30$ digits.
- I could not find forced algebraic or aurifeuillan factors.
Can we show that there is no prime of this form or do we just have to continue the search ?