MathNet 2025-part-01 — 187 problems (EN–VI) Question 185

The sequence $(a_n)$ of positive integers is defined by $a_1 = 1$ and $a_n = (a_{n-1}+1)^2$ for $n \ge 2$. Let $p$ be an odd prime. Prove that $a_{2p} - a_p$ has at least $p$ pairwise distinct prime divisors.