NMC 2018 Problem 2

A sequence of primes $p_1, p_2, \dots$ is given by two initial primes $p_1$ and $p_2$ , and $p_{n+2}$ being the greatest prime divisor of $p_n + p_{n+1} + 2018$ for all $n \ge 1$ . Prove that the sequence only contains finitely many primes for all possible values of $p_1$ and $p_2$ .