Goormaghtigh's conjecture states 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
$$ {\displaystyle {\frac {90^{3}-1}{90-1}}={\frac {2^{13}-1}{2-1}}=8191.} $$ Now let $a$ and $b$ denote positive integers such that $a\neq b$ and $a,b>1$. Then there are many non-trivial integer solutions of
$$ a{\frac {x^{m}-1}{x-1}}=b{\frac {y^{n}-1}{y-1}}, $$
for comprime $x$ and $y$, albeit with the relaxed condition that $m,n>1$. For instance,
$$ 84493{\frac {118352^{2}-1}{118352-1}}=39451{\frac {253478^{2}-1}{253478-1}}=10000000029. $$
Is such a generalisation to Goormaghtigh's conjecture already known and studied in literature somewhere?
There will be infinitely many since $$a{\frac {x^{m}-1}{x-1}}=b{\frac {y^{n}-1}{y-1}}\implies\frac ab=\frac{(x-1)(y^n-1)}{(y-1)(x^m-1)}$$ is always a rational number for $a,b,x,y\in\Bbb Q$.