How can we determine whether $\sqrt {2 \sqrt {3 \sqrt {4 \ldots}}}$ is rational or not?
Is it transcendental or algebraic?
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18$\begingroup$ You should link to previous related discussion, e.g. here and here $\endgroup$– Zev ChonolesJun 21, 2015 at 17:41
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8$\begingroup$ this very same question has a live bounty by another user on MSE, is it appropriate to just copy it here? math.stackexchange.com/questions/898405/… $\endgroup$– Carlo BeenakkerJun 21, 2015 at 21:43
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4$\begingroup$ Its good to know that someone on the internet has discussed my question a year ago. But they didnt solve it $\endgroup$– badmfJun 21, 2015 at 21:44
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13$\begingroup$ I'll just re-post the essence of my comment: There's no sense of posting the same question on two communities which share a large intersection at the same time. The fact that you are an MSE user, and you didn't include links to those posts makes me think that asking this question isn't exactly a scientific interest to you. $\endgroup$– Asaf Karagila ♦Jun 21, 2015 at 22:27
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9$\begingroup$ The mathematical question is interesting, which is why so many people have voted it up. However, it should be stated that this seems to be an open question, and there should be links to the other places the question was recently asked. The way this question was presented with no context was poor, and that's why there are so many down votes. $\endgroup$– Douglas ZareJun 22, 2015 at 6:34
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1 Answer
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So that readers don't have to chase down all the hyperlinks, I'll summarize by saying that your number is the square $\sigma^2$ of the Somos quadratic recurrence constant $\sigma$. As the OEIS entry makes clear, numerous papers have been published about $\sigma$, but they do not comment explicitly on its irrationality, even those papers whose sole purpose is to explain how to quickly compute $\sigma$ to high precision. It is a plausible inference that the irrationality of $\sigma$ (and of $\sigma^2$) is an open problem.
answered Jun 22, 2015 at 14:15