It might be well-known (and sorry if it is), but a quick search did not return the answer.


Consider prime numbers $p \neq q$.


Are $\displaystyle \frac{p^q-1}{p-1}$ and $\displaystyle \frac{q^p-1}{q-1}$ relatively prime?



Prime numbers that lead to relatively prime
The answer is no. As the Wikipedia article in my comment states, the counterexample $p = 17, q = 3313$ was found by Stephens in 1971, but the stronger question of whether one can ever divide the other is a famous open problem because its solution would greatly simplify a step in the proof of the Feit-Thompson theorem.


