Prove that if $$\frac{x^2+y^2}{1+xy}$$ is a positive integer, where $(x,y) \in(0,\infty)$ then it is always a square number.
My work:
I decided to find out all values of $x,y$ for which the expression given above evolves to an integer. With trial-and-error I found out some values and surprisingly for that values the expression given above was actually a square number. But I could not determine if there are infinite solutions to this or not. If that were true, then I am not able to find a generalized solution.
Anyways, I know this is not a good way to prove something. Any help is greatly appreciated. Do we need modular arithmetic here$?$ I also tried to assume the value of the expression as $k^2$ but in vain.
