Do all primes occur as a factor of $p_{k}-2$ for some k?

Question

Can we prove that for any prime p, sufficiently large n, and $$A_n=\prod_{k=2}^n (p_k-2)$$ that $p|A_n$? I checked through $p=p_{50}.$

Answer 1

Yes: given an odd prime $p$, there is a prime $q$ of the form $2+kp$.

In fact, there are infinitely many primes of that form. That's a direct consequence of Dirichlet's theorem on arithmetic progressions.