TITLE: Goormaghtigh equation
QUESTION [5 upvotes]: 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 $m,n>1$.

REPLY [7 votes]: 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)