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A generalized Euclidean number is the product of a finite set of primes, plus one.

A generalized Euclidean prime is a generalized Euclidean number that is prime.

Are there infinitely many generalized Euclidean primes?

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    I'd venture to guess yes, but I have no idea as to how one would show this. – Rushabh Mehta Nov 12 '19 at 22:32
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    See this article. – Keith Backman Nov 12 '19 at 22:39
  • Just some notes: let $S$ a finite set of primes and $p(S)=1+\prod S$. If $p(S)$ is prime then $2\in S$. When you use the word 'set' you're are saying that the primes in the product are distinct. And the "official" definition of Euclidean number is the product of the first $n$ primes plus $1$. And $p(S)$ is prime if and only if $\phi(p(S))=\prod S$. – ajotatxe Nov 12 '19 at 22:49
  • Possible duplicate of this. The answer seems to be: Yes. There are infinitely many such numbers. Looks like the linchpin is the Siegel–Walfisz theorem. Note the connection here is that $\prod S$ is squarefree because $S$ is a set (No duplicates) of primes. – Mason Nov 12 '19 at 22:53
  • @Mason I have to admit it. I don't understand a single word of the answer. – ajotatxe Nov 12 '19 at 22:59
  • @ajotatxe. Ah. This may be so. This question may be a duplicate of question without an amazing answer. But in this case we should probably still mark this as a duplicate and aim to answer the original in a more compelling way. I am not so sure but I think the $\mu$ in the other post may refer to mobius inversion. Sorry if this is a redherring. – Mason Nov 12 '19 at 23:03
  • @Mason The $\mu(n)$ function at https://math.stackexchange.com/q/3258045 is the Möbius function (see https://en.wikipedia.org/wiki/M%C3%B6bius_function) which only takes on non-zero values at square-free integers and hence the connection between $\mu(p-1)$ and the other question. – Steven Clark Nov 13 '19 at 03:44
  • A proof that there are infinitely many primes that are equal to the sum of a square-free number plus 1 does not imply there are an infinite number of Euclid numbers that are primes, but I'm not sure a generalized Euclidean number is the same thing as a Euclid number. $E_n-1$ is a square-free number (see https://en.wikipedia.org/wiki/Euclid_number), but there are infinitely many square-free numbers that are not of the form $E_n-1$. – Steven Clark Nov 13 '19 at 03:44

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