RO-TST 1988 Problem 13

Let $a$ be a positive integer. The sequence $\{x_n\}_{n\geq 1}$ is defined by $x_1=1$ , $x_2=a$ and $x_{n+2} = ax_{n+1} + x_n$ for all $n\geq 1$ . Prove that $(y,x)$ is a solution of the equation $$|y^2 - axy - x^2 | = 1$$ if and only if there exists a rank $k$ such that $(y,x)=(x_{k+1},x_k)$ .

Serban Buzeteanu