I posted this question on [SE][1], and was told I should repost it here.
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The [Goormaghtigh conjecture][2] 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][3] 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][4]" 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.

Both of these questions have only two solutions: the Goormaghtigh conjecture has $31$ and $8191$ as the only known solutions.

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][5]?


  [1]: https://math.stackexchange.com/questions/4427135/is-there-a-known-connection-between-wieferich-primes-and-the-goormaghtigh-conjec
  [2]: https://en.wikipedia.org/wiki/Goormaghtigh_conjecture
  [3]: https://en.wikipedia.org/wiki/Wieferich_prime
  [4]: https://johnblythedobson.org/mathematics/Wieferich_primes.html
  [5]: https://math.stackexchange.com/questions/3052133/is-this-variant-of-goormaghtighs-conjecture-known