I posted this question on SE, and was told I should repost it here.

The Goormaghtigh conjecture explores the Diophantine equation of the form $$ \frac{a^b-1}{a-1}=\frac{c^d-1}{c-1}, $$ where $a>c>1$ and $b,d>2$, where $a,b,c,d \in \mathbb{Z}$.

A Weiferich Prime is a prime $p$ such that $p^2 \mid 2^{p-1}-1$. There are only two of these primes known: $1093$ and $3511$. This self-published "note" explains how they can be rewritten as $$ \begin{align*} 1093-1&=4\cdot \frac{16^3-1}{16-1}\\ 3511-1&=6\cdot \frac{8^4-1}{8-1}, \end{align*} $$ which look awfully similar to that of the Goormaghtigh conjecture.

There are also only two know solutions of the Goormaghtigh conjecture: $31$, and $8191$. Is there any known connection between these two open problems? If there is not, oh well, it was worth a question, but if there is, how interesting might that be!

Perhaps there is something more to this related question?