Find all positive integers which are representable uniquely as $\frac{x^2+y}{xy+1}$ with $x,y$ positive integers.

$$\textbf{Question:}$$ Find all positive integers,which are representable uniquely as

$$\frac{x^2+y}{xy+1}$$

where x and y are positive integers.

I think this question maybe has something to do with vieta jumping. I also found that for such x,y to exist $$y < x^2$$ must hold.

• See here. Russian Mathematical Olympiad $2001$. – Dietrich Burde May 31 at 18:58