<s>[INST] On the diophantine equation $x^{m-1}(x+1)=y^{n-1}(y+1)$ with $x>y$, over integers greater or equal than two 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?

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.

Question 2. I would like to know what work can be done with the purpose to know if the problem $(1)$ have finitely many solutions $(x,y;m,n)$. I mean what relevant reasonings or heuristics you can to deduce with the purpose to study if the problem have finitely solutions.

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.
 [/INST] My approach is to restrict the exponents to $m = 2, n = 3$, because both of your solutions have these values, and due to $m - 1 = 1$, this could makes the question different, as the LHS becomes simply a product of two consecutive numbers.
This approach of mine might not be very useful, as I thought I can come up with something that proves that there are infinitely many solutions for this type of equality, but at the end you see it might be the other way round, your two solutions might be the only ones.
I will use the identity $a^b - 1 = (a - 1)(a^{b - 1} + a^{b - 2} + ... + a^2 + a + 1)$ in my approach. $\tag{*}$
We have $x(x + 1) = y^2(y + 1) = y^3 + y^2$
and to be able to use $(*)$, subtract $2$ from both sides:
$x(x + 1) - 2 = y^3 + y^2 - 2$
$x^2 + x - 2 = y^3 - 1 + y^2 - 1 = (y - 1)(y^2 + y + 1) + (y - 1)(y + 1)$
Now I would like to have an expression on both sides so that I can piecewise make them equal:
$x^2 + x - 2 = (y - 1)(y^2 + 2y + 1) + y - 1$
$x^2 + x - 2 = (y - 1)(y + 1)^2 + (y + 1) - 2$
$x^2 + x = (y - 1)(y + 1)^2 + (y + 1)$
Now we had on both sides pairwise equal terms of $x = (y + 1)$, if $(y - 1) = 1$. So one solution is $y = 2, x = (y + 1) = 3$. This is what you got. But this is not giving your other solution. Something is needed to make it more general.
My idea is that as we already have an expression on the RHS that has the right exponents, we need to get rid of the coefficient $(y - 1)$. Therefore we change it to
$x^2 + x = (y - 1)(y + 1)^2 + (y + 1) = (y + 1)^2 + (y + 1) + (y - 2)(y + 1)^2$
and so we must have $(y - 2)(y + 1)^2 = 0$ to obtain $x = (y + 1)$. But this only give $y = -1$ besides $y = 2$, so nothing useful.
So let us change the delta between $x$ and $y$, and instead of having $x = (y + 1)$, let us change to $x = (y + \alpha)$.
We need the following:
$(y + \alpha) = (y + 1) + (\alpha - 1) \implies (y + 1) = (y + \alpha) - (\alpha - 1)$
$(y + \alpha)^2 = y^2 + 2\alpha y + \alpha^2 = y^2 + 2y + 1 + (2\alpha - 2)y + (\alpha^2 - 1) = (y + 1)^2 + (2\alpha - 2)y + (\alpha^2 - 1) \implies (y + 1)^2 = (y + \alpha)^2 - (2\alpha y - 2y + \alpha^2 - 1)$
Now we have
$x^2 + x = (y + 1)^2 + (y + 1) + (y - 2)(y + 1)^2 = (y + \alpha)^2 + (y + \alpha) - (2\alpha y - 2y + \alpha^2 - 1) - (\alpha - 1) + (y - 2)((y + \alpha)^2 - (2\alpha y - 2y + \alpha^2 - 1))$
and so we must have $ - (2\alpha y - 2y + \alpha^2 - 1) - (\alpha - 1) + (y - 2)((y + \alpha)^2 - (2\alpha y - 2y + \alpha^2 - 1)) = 0$ to obtain $x = (y + \alpha)$.
After a lot of transformations one gets a quadratic for $\alpha$:
$\alpha_{1, 2} = \alpha^2 + (2y + 1)\alpha + y - y^3$
from which $\alpha$ based on the quadratic formula is
$\alpha_{1, 2} = \frac{-(2y + 1) \pm \sqrt{4y^3 + 4y^2 + 1}}{2}$
I thought I can come up with something useful for the expression under the square root, but I could not determine for what values of $y$ it gives a square number. When it does, $\alpha$ is an integer, as the number under the square root is odd, therefore its square root is odd, and $(2y + 1)$ is odd, so the $\pm$ of these are even. For $y = 2$ it gives $\alpha = 1$ so $x = 3$, which is your first solution and my previous one, and for $y = 21$ it gives $\alpha = 77$ so $x = 98$ which is your other solution. I brute-forced the expression under the square root up to one million, but there are no other solutions.
Of course even if it is provable that there are no other solutions for this expression to be a square, other solutions might exist for the original equality, as it only related to the piecewise equality logic applied.
</s>