Question 4
Let $a$, $b$ be two odd natural numbers. Suppose that $a$ is a divisor of $b^2 + 2$ and $b$ is a divisor of $a^2 + 2$. Prove that $a$ and $b$ belong to the sequence $(v_n)$ defined by
$$
v_1 = v_2 = 1 \quad \text{and} \quad v_n = 4v_{n-1} - v_{n-2} \quad \forall n \ge 3.
$$
$$
v_1 = v_2 = 1 \quad \text{and} \quad v_n = 4v_{n-1} - v_{n-2} \quad \forall n \ge 3.
$$