Question 180
Given are two coprime positive integers $a, b$ with $b$ odd and $a \gt 2$. The sequence $(x_n)$ is defined by $x_0 = 2$, $x_1 = a$ and $x_{n+2} = a x_{n+1} + b x_n$ for $n \ge 0$. Prove that
a) If $a$ is even then there do not exist positive integers $m, n, p$ such that $\frac{x_m}{x_n x_p}$ is a positive integer.
b) If $a$ is odd then there do not exist positive integers $m, n, p$ such that $mnp$ is even and $\frac{x_m}{x_n x_p}$ is a perfect square.
a) If $a$ is even then there do not exist positive integers $m, n, p$ such that $\frac{x_m}{x_n x_p}$ is a positive integer.
b) If $a$ is odd then there do not exist positive integers $m, n, p$ such that $mnp$ is even and $\frac{x_m}{x_n x_p}$ is a perfect square.