MathNet 2008-part-01 — 165 problems (EN–VI) Question 137

The sequence $\{x_n\}$ is defined by $x_1 = 2$, $x_2 = 12$, $x_{n+2} = 6x_{n+1} - x_n$, $n = 1, 2, \dots$. Let $p$ be an odd prime number. Let $q$ be a prime number such that $q \mid x_p$. Prove that if $q \neq 2$, then $q \ge 2p - 1$.