About all primes $p$ such that $s_p=1+\prod_{k\leq p,k\text{ prime}}k$ is prime

Question

Inspired by Euclid's proof that there are infinitely many prime numbers, I started looking at numbers of the form $$s_p=1+\prod_{k\leq p,k\text{ prime}}k$$ where $p$ is a prime number. I couldn't find any literature that tries to characterize the set of all prime numbers $p$ such that $s_p$ is prime as well. With some brute force computations, we see that if $p=2,3,5,7,11$ then $s_p$ is prime as well but $s_{13}$ is not. My question is, is the set $$S=\{p\in\mathbb{Z}_{\geq 2}\text{ prime}:s_p\text{ prime}\}$$ infinite? Or finite? Can one prove anything about the cardinality of $S$? Does $S$ admit any (obvious) structures? I wouldn't have any idea on how to start, so I'd appreciate any suggestions/ideas/solutions!

Answer 1

A computer check gives values $2, 3, 5, 7, 11, 31, 379, 1019, 1021,...$. Putting this into the Online Encyclopaedia of Integer Sequences gives this sequence, and there are several links from that page with more information. In particular there is a scan of a journal article from 1987 in which it is conjectured that there are infinitely many such numbers. I doubt this will be proved anytime soon; intuitively it feels significantly harder than the unsolved conjecture that there are infinitely many Sophie Germain primes, for example.