MathNet 2018-part-03 — 181 problems (EN–VI) Question 165

An integer $a \ge 1$ is called Aegean if none of the numbers $a^{n+2} + 3a^n + 1$ with $n \ge 1$ is prime. Prove that there are at least 500 Aegean integers in the set $\{1, 2, \ldots, 2018\}$.