Does the set $\{\tan n, n\in\mathbb{Z}\}$ contain any integers, and if so, how many and how can we determine these values?
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Yes: $0$ for $n=0$. Only. – Przemysław Scherwentke Apr 13 '18 at 00:50
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@PrzemysławScherwentke How do we prove there are no other zeros? – Sully Chen Apr 13 '18 at 00:53
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$\pi$ is irrational and zeroes are only in $k\pi$.. – Przemysław Scherwentke Apr 13 '18 at 00:57
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@PrzemysławScherwentke There exist non-zero integers :). – Erick Wong Apr 13 '18 at 01:01
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1Have a look a look at this previous Question, which seems to rule out such integers besides zero by application of the Lindemann-Weierstrass theorem. – hardmath Apr 13 '18 at 01:02
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I think this should follow from the transcendentality of $e^i$, which is a consequence of Lindemann's theorem. – Erick Wong Apr 13 '18 at 01:02
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@ErickWong I know. That's why I only commented. – Przemysław Scherwentke Apr 13 '18 at 01:04
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@PrzemysławScherwentke Ah, yes I just noticed the comment was only asking about existence of other zeros. – Erick Wong Apr 13 '18 at 01:04