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Is there a characterisation for which $x\in\mathbb{R}$ the value $\arctan(x)$ is a rational multiple of $\pi$?

Or reformulated: What is the "structure" of the subset $A\subseteq\mathbb{R}$ which fulfils $$ \arctan(x) \in \pi\mathbb{Q} \Leftrightarrow x\in A$$ for all $x\in\mathbb{R}$?

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I can't see that one can say anything much more than $x=\tan q\pi$ with $q$ rational. –  Robin Chapman Aug 21 '10 at 12:04
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All the elements of $A$ are real algebraic numbers, with all their Galois conjugates real as well. Other than that, of course, we can define polynomials $P_n$ and $Q_n$ for every $n\in\mathbb N$ such that $\tan\left(nx\right)=\frac{P_n\left(\tan x\right)}{Q_n\left(\tan x_\right)}$, and then (if we take these polynomials coprime) your set $A$ will be the union of the sets of roots of all $P_n$. –  darij grinberg Aug 21 '10 at 13:06
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The malformed equation should mean $\tan\left(nx\right)=\dfrac{P_n\left(\tan x\right)}{Q_n\left(\tan x\right)}$. –  darij grinberg Aug 21 '10 at 13:06
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A reference of possible interest, even though it only deals with the case where $x$ is rational: ma.utexas.edu/users/jack/gausspi.pdf –  Doug Chatham Aug 22 '10 at 20:18
    
The article @Doug linked to is also at oberlin.edu/faculty/jcalcut/gausspi.pdf ; see also oberlin.edu/faculty/jcalcut/arctan.pdf . –  J. M. Dec 18 '11 at 10:47
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1 Answer

A partial answer was provided in response to my MSE question, "ArcTan(2) a rational multiple of $\pi$?"

There Thomas Andrews showed that $\arctan(x)$ is not a rational multiple of $\pi$ for any $x$ rational, except for $-1,0,1$. More specifically:

$\arctan(x)$ is a rational multiple of $\pi$ if and only if the complex number $1+xi$ has the property that $(1+xi)^n$ is a real number for some positive integer $n$. This is not possible if $x$ is a rational, $|x|\neq 1$, because $(q+pi)^n$ cannot be real for any $n$ if $(q,p)=1$ and $|qp|> 1$. So $\arctan(\frac{p}q)$ cannot be a rational multiple of $\pi$.

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