# Prime numbers that lead to relatively prime

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?

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–  Qiaochu Yuan Mar 26 '10 at 2:12
I've deleted my comment, which was incorrect and apologize to Portland for my flippancy. –  Harry Gindi Mar 26 '10 at 2:29
@QY, turn the comment into an answer and let Portland accept it. –  Mariano Suárez-Alvarez Mar 26 '10 at 3:24
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.
I've just check all pairs $p,q\leq 1000$th prime out of curiosity... –  Mariano Suárez-Alvarez Mar 26 '10 at 4:00