so I've set myself this question as a challenge and because of that do not want to read any solutions on the internet, so it would be appreciated if no one spoiled the answer, please just tell me if I'm incorrect
the question goes: $$\frac{a^2+b^2}{ab+1}$$ prove that the expression above is equal to the square of an integer if $a$ and $b$ are positive integers.
Now, I've been able to prove that when $b = a^3$, the expression will equal $a^2$, but am I mistaken to believe that the expression has many different sets of numbers that will satisfy the question?
In that case, is this a proper solution or is there another one that can somehow describe all sets at once?
I haven't looked at any solutions on the internet as I don't want to ruin the challenge for myself.
(Possibly too big) hint: Fix an integer $k$, for which there is a solution to $\frac{a^2+b^2}{ab+1}=k$. Let $a$ be unknown and see that there are two solutions with the same $b$ and $k$.
This forms a single link in a chain of solutions. Study this chain, and you have solved the problem.