It might be wellknown (and sorry if it is), but a quick search did not return the answer.
Consider prime numbers $p \neq q$.
Are $\displaystyle \frac{p^q1}{p1}$ and $\displaystyle \frac{q^p1}{q1}$ relatively prime?
It might be wellknown (and sorry if it is), but a quick search did not return the answer. Consider prime numbers $p \neq q$. Are $\displaystyle \frac{p^q1}{p1}$ and $\displaystyle \frac{q^p1}{q1}$ 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 FeitThompson theorem. 

