After reading up on Catalan's Conjecture, a related equation piqued my interest:
Let $a>c>1$ and $b,d>2$, where $a,b,c,d \in \mathbb{Z}$. I am looking for integer solutions to $$ \frac{a^b-1}{a-1}=\frac{c^d-1}{c-1}. $$ By some trial and error (educated guessing), I found the solution $$\begin{align*} \frac{5^3-1}{5-1}&=\frac{2^5-1}{2-1}\\ \frac{125-1}{4}&=\frac{31}{1}\\ \frac{125-1}{4}&=\frac{31}{1}\\ 31&=31 \end{align*}$$
Is there a technique to find the whole solution set? I have not been successful in finding any patterns whatsoever.
Thanks to everyone who commented! I'll go ahead and answer my own question. I found a wikipedia page on this exact problem, where it is labelled the Goormaghtigh conjecture, and is listed under the wiki page of unsolved problems in number theory. So it seems that if anyone knew how to solve this equation that would be mighty impressive.
Also, the paper that John Omielan linked in the comments seems to be the best resource on this problem.