Dear all,

I have a problem relating to primitive prime divisors. (For a sequence $(A_n)_{n\ge1}$, a primitive prime divisor of a term $A_n$ is a prime $\ell$ that divides $A_n$, but does not divide $A_i$ for all $i<n$.)

Assume that $p$ and $q$ are distinct odd primes and $n\geq 3$ a fixed positive integer, so that $p^n-1$ and $q^n-1$ have at least one primitive prime divisor (by the Bang-Zsigmondy theorem).

Question 1: When do $p^n-1$ and $q^n-1$ have a common primitive prime divisor?

Question 2: When do $p^n-1$ and $p^n-q^n$ have a common primitive prime divisor (which will be a divisor, although not necessarily primitive, of $q^n-1$)?

If you have some source, please tell me.

Thanks a lot in advance.