MK-MO 2021 Problem 5

Let $(x_{n})_{n=1}^{+\infty}$ be a sequence defined recursively with $x_{n+1} = x_{n}(x_{n}-2)$ and $x_{1} = \frac{7}{2}$ . Let $x_{2021} = \frac{a}{b}$ , where $a,b \in \mathbb{N}$ are coprime. Show that if $p$ is a prime divisor of $a$ , then either $3|p-1$ or $p=3$ .

Authored by Nikola Velov