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I was learning Euclid's theorem. If we repeat his construction (properly modified to give only primes) then will we skip any primes? Formally:

$p_1 = 2$, and $p_{n + 1} = \text{smallest} \; q \in \mathbb{N} - \{1\} \; \text{s.t} \quad q \; | \;(p_1\cdot ... \cdot p_n + 1)$

In other words, $p_{n + 1}$ is the smallest number that divides the number made by Euclid's construction on $\{p_1, ..., p_n\}$.

Call the above Euclid primes (starting at $2$). The first few numbers in this sequence are: $2,3,7,43,13,53,5,...$ Note how it skips $5$, but then comes back to it. Are there any primes that will not appear in this sequence? If so, can we pick a different $p_1$ to avoid this? Can we predict the skipped numbers based on $p_1$?

Related question

Is there an infinite number of primes constructed as in Euclid's proof?

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There is some information here: – Matthew Conroy Jan 8 '13 at 18:14
up vote 5 down vote accepted

As the references in the answer to this question say, this is an open problem; the sequence $(p_n)$ is known as the Euclid–Mullin sequence.

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