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$$\pi $$π -Formulas and Gray code
Ricerche di Matematica ( IF 1.2 ) Pub Date : 2018-10-11 , DOI: 10.1007/s11587-018-0426-4
Pierluigi Vellucci , Alberto Maria Bersani

In previous papers we introduced a class of polynomials which follow the same recursive formula as the Lucas–Lehmer numbers, studying the distribution of their zeros and remarking that this distribution follows a sequence related to the binary Gray code. It allowed us to give an order for all the zeros of every polynomial \(L_n\). In this paper, the zeros, expressed in terms of nested radicals, are used to obtain two formulas for \(\pi \): the first can be seen as a generalization of the known formula$$\begin{aligned} \pi =\lim _{n\rightarrow \infty } 2^{n+1}\cdot \sqrt{2-\underbrace{\sqrt{2+\sqrt{2+\sqrt{2+\cdots +\sqrt{2}}}}}_{n}}, \end{aligned}$$related to the smallest positive zero of \(L_n\); the second is an exact formula for \(\pi \) achieved thanks to some identities valid for \(L_n\).

中文翻译:

$$ \ pi $$π-公式和格雷码

在先前的论文中,我们介绍了一类与Lucas-Lehmer数相同的递归公式的多项式,研究了它们的零分布,并指出该分布遵循与二进制格雷码相关的序列。它使我们能够为每个多项式\(L_n \)的所有零给出一个阶。在本文中,用嵌套根表示的零用于获得\(\ pi \)的两个公式:第一个可以看作是已知公式$$ \ begin {aligned} \ pi =的推广。\ lim _ {n \ rightarrow \ infty} 2 ^ {n + 1} \ cdot \ sqrt {2- \ underbrace {\ sqrt {2+ \ sqrt {2+ \ sqrt {2+ \ cdots + \ sqrt {2} }}}} _ {n}},\ end {aligned} $$\(L_n \)的最小正零有关;第二个是精确的公式\(\ pi \)由于对\(L_n \)有效的身份而得以实现。
更新日期:2018-10-11
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