Question 34
Consider an infinite sequence $a_0, a_1, a_2, \dots$ of integers that satisfies $a_0 \ge 2$ and $a_{n+1} \in \{2a_n - 1, 2a_n + 1\}$ for all indices $n \ge 0$. Prove that any such infinite sequence contains infinitely many composite numbers. (Martin Melicher, Josef Tkadlec)