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AN INTERESTING RELATION

  I just copy here an answer of mine of this post, just in case OP decided to delete it.  Link: https://math.stackexchange.com/q/5005043/1231540 First we state two results:  - **Proposition 1:** Let $n \ge 3$. The minimal polynomial of $\cos(2\pi/n)$ - denoted by $\psi_n$ - has degree $\phi(n)/2.$ Here $\phi$ is Euler totient function.  - **Proposition 2:** The roots of $\psi_n$ are $\cos(2k\pi/n)$ where $\gcd(k,n)=1$ and $0 \le k \le \lfloor \frac{n}{2}\rfloor$. The proofs of the above two propositions can be found in this paper: https://www.jstor.org/stable/2324301 ***Proof of the first identity:*** It can be observed that $4\sin^2(\pi/n) = 2-4\cos(2\pi/n)$, so the polynomial $\Phi_n$ and $\psi_n$ have the same degree, which is $\phi(n)/2$. In particular, $\Phi_{2m}$ has degree $\phi(2m)/2$.  If $m$ is odd, we clearly have $\phi(2m) = \phi(m)$, which means $\Phi_{2m}$ and $\Phi_m$ have the same degrees. By definition, $\Phi_{2m}(x)$ is irreducible and has...
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GREAT COMMENT ON LEBESGUE NUMBER LEMMA

 The sources is here: https://math.stackexchange.com/q/105337/1231540

I just rewrite everything in case I lost track of the link.

Question: the Lebesgue number lemma is stated as follows: 

For every open cover  $\mathcal{U}$ of a compact metric space $X$, there exists a number $\delta$ such  that every open ball of radius $\delta$ is contained in some element of  $\mathcal{U}$. The number $\delta$ is called the Lebesgue number of open cover $\mathcal{U}$. 


Answer: The intuition answer is the following: for every point $x_0$ inside an open set $U \in \mathcal{U}$, we can draw a ball centered at $x$ with radius $r$ small enough such that $B(x,r) \subset U$.  Since $r$ depends on $x,U$, it is natural to ask for the largest possible value of $r$. We define

\[r(x)= \sup \left\lbrace r(x,U): x \in X, U \in \mathcal{U} \right\rbrace\]

Intuitive, for $x_n$ is closed enough to $x$, they will like in the same open set $U$, and thus $r(x_n)$ will tends to $r(x)$. This implies $r$ is a continuous function on $X$. Since $X$ is compact, the function can attain a minima, and this minimum value is called Lesbegue number!


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