TITLE: Does the algorithm of the Greeks produce all prime numbers?
QUESTION [34 upvotes]: Let ${\cal P}$ be the set of prime numbers.  Define a subset ${\cal P}'=\{p_1,p_2,p_3,\cdots\}$ of ${\cal P}$ by setting $p_1=2$ and defining $p_{n+1}$ to be the smallest element of ${\cal P}$ dividing $1+p_1\cdots p_n$. Is there any obstruction to ${\cal P}'={\cal P}$ ?

REPLY [32 votes]: According to Booker - A variant of the Euclid-Mullin sequence containing every prime, as of 2016, this question remains open.

One of the central questions in this area was posed by Mullin [6] in 1963: Does the Euclid–Mullin sequence contain every prime number? Despite a compelling heuristic argument of Shanks [9] that the answer is yes, even the broader question of whether there is any Euclid sequence containing every prime number remains open.

The OEIS contains a decent amount of information. For example, the primes up to 37 do appear within the first 50 terms of the sequence.