dc.contributor.author
Gasull, A.
dc.contributor.author
Luca, F.
dc.contributor.author
Varona, J.L.
dc.date.accessioned
2023-08-29T11:48:11Z
dc.date.accessioned
2024-09-19T14:35:30Z
dc.date.available
2023-08-29T11:48:11Z
dc.date.available
2024-09-19T14:35:30Z
dc.date.issued
2023-07-22
dc.identifier.uri
http://hdl.handle.net/2072/536867
dc.description.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. © 2023 The Author(s)
eng
dc.description.sponsorship
The first author is partially supported by the Ministerio de Ciencia e Innovación ( PID2019-104658GB-I00 grant), by the grant Severo Ochoa and María de Maeztu Program for Centers and Units of Excellence in R&D ( CEX2020-001084-M ) and also by the Agència de Gestió d’Ajuts Universitaris i de Recerca ( 2021 SGR 113 grant). The second author worked on this paper during a visit at the Max Planck Institute for Software Science in Saarbrücken, Germany , in Spring of 2022. He thanks the people of this Institute for hospitality and support. This author was also partially supported by Project 2022-064-NUM-GANDA from the CoEMaSS at Wits. The third author is partially supported by the Ministerio de Ciencia e Innovación ( PID2021-124332NB-C22 grant).
dc.format.extent
24 p.
cat
dc.publisher
Elsevier B.V.
cat
dc.relation.ispartof
Indagationes Mathematicae
cat
dc.rights
L'accés als continguts d'aquest document queda condicionat a l'acceptació de les condicions d'ús establertes per la següent llicència Creative Commons: https://creativecommons.org/licenses/by/4.0/
dc.source
RECERCAT (Dipòsit de la Recerca de Catalunya)
dc.subject.other
Computation of π; Golden section; Lehmer measure; Machin's type formulas
cat
dc.title
Three essays on Machin's type formulas
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dc.type
info:eu-repo/semantics/article
cat
dc.type
info:eu-repo/semantics/publishedVersion
cat
dc.identifier.doi
10.1016/j.indag.2023.07.002
cat
dc.rights.accessLevel
info:eu-repo/semantics/openAccess
