Let $p$ be a rational prime and $n$ be a positive integer.

It can be easily deduced from Zsigmondy's theorem that $p^n+1$ has a prime divisor greater than $2n$ except when $(p,n)=(2,3)$ or $(2^k-1,1)$ for some positive integer $k$. Hence we know that there exists an odd prime divisor of $p^n+1$ greater than $n$ if and only if $(p,n)\neq(2,3)$ or $(2^k-1,1)$ for any positive integer $k$.

**Question:**

**(1)**. For which $(p,n)$ does there exist at least two odd prime divisors of $p^n+1$ coprime to $n$?

**(2)**. For which $(p,n)$ does there exist at least two odd prime divisors of $p^n+1$ greater than $n$?