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\(\pi \)-Formulas and Gray code

  • Pierluigi Vellucci
  • Alberto Maria Bersani
Article
  • 6 Downloads

Abstract

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 \(L_n\).

Keywords

\(\pi \) Formulas Gray code Continued roots Nested square roots Zeros of Chebyshev polynomials 

Mathematics Subject Classification

40A05 11Y60 

Notes

Acknowledgements

We thank the Editor-in-Chief and the anonymous reviewers for their careful reading of the paper and for their excellent and constructive comments and suggestions.

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Copyright information

© Università degli Studi di Napoli "Federico II" 2018

Authors and Affiliations

  1. 1.Department of EconomicsUniversity of Roma TRERomeItaly
  2. 2.Department of Mechanical and Aerospace EngineeringSapienza UniversityRomeItaly

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