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| Home > Articles > Published articles > Three essays on Machin's type formulas |
(Universitat Autònoma de Barcelona. Departament de Matemàtiques) (Universidad de La Rioja. Departamento de Matemáticas y Computación)| Date: | 2023 |
| Abstract: | We study three questions related to Machin's type formulas. The first one gives all two terms Machin formulas where both arctangent functions are evaluated 2-integers, that is values of the form b/2a for some integers a and b. These formulas are computationally useful because multiplication or division by a power of two is a very fast operation for most computers. The second one presents a method for finding infinitely many formulas with N terms. In the particular case N=2 the method is quite useful. It recovers most known formulas, gives some new ones, and allows to prove, in an easy way, that there are two terms Machin formulas with Lehmer measure as small as desired. Finally, we correct an oversight from previous result and give all Machin's type formulas with two terms involving arctangents of powers of the golden section. |
| Grants: | Agencia Estatal de Investigación PID2019-104658GB-I00 Ministerio de Ciencia e Innovación CEX2020-001084-M Agència de Gestió d'Ajuts Universitaris i de Recerca 2021/SGR-113 Agencia Estatal de Investigación PID2021-124332NB-C22 |
| Rights: | Aquest document està subjecte a una llicència d'ús Creative Commons. Es permet la reproducció total o parcial, la distribució, la comunicació pública de l'obra i la creació d'obres derivades, fins i tot amb finalitats comercials, sempre i quan es reconegui l'autoria de l'obra original.  |
| Language: | Anglès |
| Document: | Article ; recerca ; Versió publicada |
| Subject: | Machin's type formulas ; Lehmer measure ; Computation of π ; Golden section |
| Published in: | Indagationes Mathematicae, Vol. 34, Issue 6 (November 2023) , p. 1373-1396, ISSN 0019-3577 |
