Question 4
Let $a,b$ be relatively prime positive integers with $b$ odd and $a\gt 2$. Define the sequence by $x_0=2$, $x_1=a$, and $x_{n+2}=ax_{n+1}+bx_n$ for every positive integer $n$.
(a) If $a$ is even, prove that there are no positive integers $m,n,p$ for which $x_m/(x_nx_p)$ is an integer.
(b) If $a$ is odd, prove that there are no positive integers $m,n,p$ such that $mnp$ is even and $x_m/(x_nx_p)$ is a perfect square.
(a) If $a$ is even, prove that there are no positive integers $m,n,p$ for which $x_m/(x_nx_p)$ is an integer.
(b) If $a$ is odd, prove that there are no positive integers $m,n,p$ such that $mnp$ is even and $x_m/(x_nx_p)$ is a perfect square.