Few weeks ago I wondered about if the following conjecture is in the literature or well if it is possible to find a counterexample. I evoke a generalization of a well-known conjecture, I mean the Feit–Thompson conjecture from Wikipedia.
An integer $n>1$ is square-free or squarefree if has no repeated prime factors. For example $n=15$ is squarefree, while than $12$ has repeated prime factors.
Conjecture. There are no distinct square-free integers $p$ and $q$ (both greater than $1$) with $\gcd(p,q)=1$ such that $\frac{p^q-1}{p-1}$ divides $\frac{q^p-1}{q-1}$.
Question. What work can be done about the veracity of this conjecture? Since Feit–Thompson conjecture seems a particular case of this, then I'm asking if previous conjecture is in the literature, or well if you've some approach or heuristic about the Conjecture or well if it is possible to find a counterexample. Many thanks.
I wrote a Pari/GP program and I've tested the Conjecture, let's say for the segments of integers $2\leq p,q\leq 2000$. I hope that there were not mistakes and this post is interesting.