# What is the algebraic intuition behind Vieta jumping in IMO1988 Problem 6?

Let $a$ and $b$ be positive integers and $k=\frac{a^2+b^2}{1+ab}$. Show that if $k$ is an integer then $k$ is a perfect square.

The usual way to show this involves a technique called Vieta jumping. See Wikipedia or this MSE post.

I can follow the Vieta jumping proof, but it seems a bit strained to me. You play around with equations that magically work out at the end. I don’t see how anyone could have come up with that problem using that proof.

Is there a natural or canonical way to see the answer to the problem, maybe using (abstract) algebra or more powerful tools? In addition, how can someone come up with a problem like this?