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Consider the equation below:

$$\cos\dfrac{\pi}{m}=2\cos\dfrac{\pi}{r}\cos\dfrac{\pi}{n},$$ where $m,n$ and $r$ are non-zero integers.

Equality holds when $m=2$ and $r=2$ (or $n=2$), and also when $m=n$ and $r=3$ (alternatively $m=r$ and $n=3$).

I would like to know any general conditions (if there are) between $m,n$ and $r$ for equality to hold.

TMM
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Hesky Cee
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  • Related : http://math.stackexchange.com/questions/537275/relationship-among-a-b-c-d-for-cos-a-cos-b-cos-c-cos-d – lab bhattacharjee Jan 27 '14 at 16:35
  • @labbhattacharjee: yes they are related but not same. I think there are no other solutions. However, I still don't have a proof to it. – Hesky Cee Jan 27 '14 at 18:27

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After my discussion with Jim, I have just confirmed that those are the only solutions. The simple proof follows from the fact that if $x\geq 4$, then $2\cos\dfrac{\pi}{x}\geq \sqrt{2}$. Hence since $2\cos\dfrac{\pi}{m}< {2}$, the result follows by multiplying boths sides of the equation in the problem by $2$ since $(2\cos\dfrac{\pi}{n})(2\cos\dfrac{\pi}{r})\geq {2}$ for $r,n\geq 4.$

I should also add that I am only interested in non-units $m,n,r\in \mathbb{Z}$. If units are allowed, the only other soultions are: $m=1 ,n=-1$ (or $n=3$) and $r=3$ (or $r=-1$) AND $m=-1 ,n=1$ (or $n=-3$) and $r=-3$ (or $r=1$).

Hesky Cee
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