Some time ago I encountered in my work the following equation
$ cos(\frac{2 \pi k}{l} )= B$
The problem consists in finding for a given irrational number $B$, a pair of integers $(k,l)$ satisfying the written equation.
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$\begingroup$ This is, of course, not a diophantine equation. Also: $A$ serves no purpose here, right? $\endgroup$– Steven LandsburgCommented Sep 24, 2013 at 22:29
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$\begingroup$ Yes you are right. I firstly shouldn't have called it "diophantine" . It is also true that $A$ has no purpose. $\endgroup$– user38651Commented Sep 24, 2013 at 22:42
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1 Answer
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This is about to be closed as "unclear what you're asking", perhaps I misunderstood the question, but it seems you ask for a characterization of the rational angles $\alpha=2\pi k/l$ that produce an irrational cosine.
This question is addressed by Jörg Jahnel in these notes, see in particular section 6: $\cos\alpha$ is an algebraic number of degree $d>1$ if and only if the Euler function $\phi(l)=2d$. For example, a quintic irrationality is obtained only for angles $\alpha/2\pi=1/22$ or $5/22$.
answered Sep 24, 2013 at 21:52
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$\begingroup$ Even though you understood what I wanted. Thanks $\endgroup$ Commented Sep 24, 2013 at 22:42
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1$\begingroup$ @user38651 Was that really your question? One possible interpretation of what you wrote was that you were given $B$ somehow and and wanted to know if the $k,l$ existed. For most $B$ it doesn't. $\endgroup$ Commented Sep 25, 2013 at 0:48