MK-MO 2022 Problem 3

The sequence $(a_n)_{n \ge 1}^\infty$ is given by: $a_1=2$ and $a_{n+1}=a_n^2+a_n$ for all $n \ge 1$ . For an integer $m \ge 2$ , $L(m)$ denotes the greatest prime divisor of $m$ . Prove that there exists some $k$ , for which $L(a_k) > 1000^{1000}$ .

Proposed by Nikola Velov