RO-TST 2004 Problem 7

Let $a,b,c$ be 3 integers, $b$ odd, and define the sequence $\{x_n\}_{n\geq 0}$ by $x_0=4$ , $x_1=0$ , $x_2=2c$ , $x_3=3b$ and for all positive integers $n$ we have $$x_{n+3} = ax_{n-1}+bx_n + cx_{n+1} .$$ Prove that for all positive integers $m$ , and for all primes $p$ the number $x_{p^m}$ is divisible by $p$ .