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 ${\cal P}'={\cal P}$ ?



Does the algorithm of the Greeks produce all prime numbers?
According to [Booker - A variant of the Euclid-Mullin sequence containing every prime](https://arxiv.org/abs/1605.08929), as of 2016, this question remains open.



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> 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.
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The [OEIS](https://oeis.org/A000945) contains a decent amount of information. For example, the primes up to 37 do appear within the first 50 terms of the sequence.


