Prove that if a²+b² is a multiple of ab+1, for positive integer a and b, then (a²+b²)/(ab+1) is a perfect square.
Let (a, b) be any solution pair with a>b. Let s = (aČ+bČ)/(ab+1). Then another solution can be derived by creating solution pair (s*a-b, a).
Plugging in the trivial solution of (a, 0) with a>1 and repeatedly applying the transform above creates the following sequence:
For different choices of a, I believe this algorithm will generate every solution.