RU-ALLRUSSIAN 2018 Problem 91

Suppose $a_1,a_2, \dots$ is an infinite strictly increasing sequence of positive integers and $p_1, p_2, \dots$ is a sequence of distinct primes such that $p_n \mid a_n$ for all $n \ge 1$ . It turned out that $a_n-a_k=p_n-p_k$ for all $n,k \ge 1$ . Prove that the sequence $(a_n)_n$ consists only of prime numbers.