For fixed prime $p$ and $q$ is it true that the following equation has at most one pair $(m,n)$ of solution? $$\frac{p^n-1}{p-1}=\frac{q^m-1}{q-1}$$ where $m,n>1$.
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$\begingroup$ See also mathoverflow.net/questions/451767/111-11-base-p-111-11-base-q/… $\endgroup$– Gerry MyersonCommented Jul 31, 2023 at 4:33
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Yes for fixed $p,q$ it is know there is at most one solution. Even without the condition $p,q$ prime.
This was proved by He and Togbé, On the number of solutions of Goormaghtigh equation for given x and y (Indagationes Math, 2008)