On the Wikipedia page of Goormaghtigh 's conjecture, there is the following claim:
The Goormaghtigh conjecture may be expressed as saying that $31$ ($111$ in base $5$, $11111$ in base $2$) and $8191$ ($111$ in base $90$, $1111111111111$ in base $2$) are the only two numbers that are repunits with at least 3 digits in two different bases.
I don't see how this follows from the original conjecture, i.e., that the only non-trivial integer solutions of
$$ {\frac {x^{m}-1}{x-1}}={\frac {y^{n}-1}{y-1}} $$
satisfying ${\displaystyle x>y>1}$ and ${\displaystyle n,m>2}$ are
$$ {\displaystyle {\frac {5^{3}-1}{5-1}}={\frac {2^{5}-1}{2-1}}=31} $$
and
$$ {\displaystyle {\frac {90^{3}-1}{90-1}}={\frac {2^{13}-1}{2-1}}=8191.} $$
Can someone show a proof that these two statements are equivalent?
This follows from the fact that $\frac{x^m - 1}{x-1} = 1 + x + x^2 + \cdots + x^{m-1}$ is the sum of a geometric series, and hence in base $x$, $\frac{x^m - 1}{x-1} = 111\ldots1$ is a repunit with $m$ digits.