VMO 2012 Problem 6

Consider two odd natural numbers $a$ and $b$ where $a$ is a divisor of $b^2+2$ and $b$ is a divisor of $a^2+2.$ Prove that $a$ and $b$ are the terms of the series of natural numbers $\langle v_n\rangle$ defined by $$v_1 = v_2 = 1; v_n = 4v_ {n-1}-v_{n-2} \ \ \text{for} \ n\geq 3.$$