Is there a natural number $k>1$ such that there are infinitely many pairs $(x,y)$ of natural numbers such that $\frac{x^2-1}{y^2-1}$ is a power of $k$?
In particular, is $k=2$ or $k=10$ good in this sense? (are there finitely or infinitely many solutions (in natural numbers) to equations $\frac{x^2-1}{y^2-1}=2^n$? or $\frac{x^2-1}{y^2-1}=10^n$?)
In case we put $x=y^2$, we would obtain equation $y^2+1=k^n$, which has been considered during investigations on Catalan conjecture.
Plugging in the fraction $\dfrac{x^2-1}{y^2-1}$ values $x=2^{2 n + 1} - 1;\;y=2^n$ we get $$\dfrac{\left(2^{2 n+1}-1\right)^2-1}{2^{2 n}-1}=2^{2n+2}$$ so there are infinite pairs $(x,\;y)$ which give $2^k$
I think there are also infinite solution to $10^k$ but I still can't find a closed form
Hope this helps