MathNet 2016-part-06 — 184 problems (EN–VI) Question 121

Let $p_1, p_2, \dots$ be the sequence of positive integers defined by $p_1 = 2$, and for all positive integers $n$, $p_{n+1}$ is defined to be the least prime number dividing $n p_1^{1!} p_2^{2!} \cdots p_n^{n!} + 1$.
Prove that all prime numbers appear in this sequence.