{"title":"Three essays on Machin’s type formulas","authors":"Armengol Gasull , Florian Luca , Juan L. Varona","doi":"10.1016/j.indag.2023.07.002","DOIUrl":null,"url":null,"abstract":"<div><p>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 <span><math><mrow><mi>b</mi><mo>/</mo><msup><mrow><mn>2</mn></mrow><mrow><mi>a</mi></mrow></msup></mrow></math></span> for some integers <span><math><mi>a</mi></math></span> and <span><math><mi>b</mi></math></span>. 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 <span><math><mi>N</mi></math></span> terms. In the particular case <span><math><mrow><mi>N</mi><mo>=</mo><mn>2</mn></mrow></math></span> 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.</p></div>","PeriodicalId":56126,"journal":{"name":"Indagationes Mathematicae-New Series","volume":null,"pages":null},"PeriodicalIF":0.5000,"publicationDate":"2023-07-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"1","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Indagationes Mathematicae-New Series","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0019357723000666","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 1
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 for some integers and . 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 terms. In the particular case 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.
期刊介绍:
Indagationes Mathematicae is a peer-reviewed international journal for the Mathematical Sciences of the Royal Dutch Mathematical Society. The journal aims at the publication of original mathematical research papers of high quality and of interest to a large segment of the mathematics community. The journal also welcomes the submission of review papers of high quality.