I've asked two years ago a post on Mathematics Stack Exchange, were provided two excellent answers. I'm asking on MathOverflow in the hope that some professor can to expand/improve (if it is possible) these results answering my question. The post has the same title and identifier *3757149* on Mathematics Stack Exchange.

I don't know if the following diophantine equation (problem) is in the literature. We consider the diophantine equation $$x^{m-1}(x+1)=y^{n-1}(y+1)\tag{1}$$

over integers $x\geq 2$ and $y\geq 2$ with $x>y$, and over integers $m\geq 2$ and $n\geq 2$. These are four integral variables $x,y,m$ and $n$. The solutions that I know for the problem $(1)$ are two, the solution $(x,y;m,n)=(3,2;2,3)$ and $(98,21;2,3)$.

Question 1.Do you know if this problem is in the literature? Alternatively, if this problem isn't in the literature can you find more solutions?Many thanks.

If the equation or problem $(1)$ is in the literature please refer it answering this question as a reference request, and I try to search and read the statements for new solutions from the literature. In other case compute more solutions or add upto what uppers limits you got evidence that there aren't more solutions.

I would like to know **what work can be done** with the purpose to know if the problem $(1)$ have finitely many solutions $(x,y;m,n)$.

Question 2.Are there finitely many solutions $(x,y;m,n)$ of stated problem $(1)$? I mean what relevant reasonings or heuristics you can to deduce with the purpose to study if the problem have finitely many solutions.Many thanks

If this second question is in the literature, please refer the literature answering this question as a reference request, and I try to search and read the statements from the literature.