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
Problem
Source: Romanian IMO Team Selection Test TST 1988, problem 13
Tags: quadratics, algebra proposed, algebra