MathNet 2024-part-05 — 168 problems (EN–VI) Question 10

For every $n = 1, 2, 3, \ldots$, define $a_n = 3^{3^n-1} + 2$. Prove that there are infinitely many prime numbers $p$ for which there exists a natural number $n$ such that $p$ is a divisor of $a_n$.