Finding and Excluding b-ary Machin-Type Individual Digit Formulae

Author:

Borwein Jonathan M.,Borwein David,Galway William F.

Abstract

AbstractConstants with formulae of the form treated by D. Bailey, P. Borwein, and S. Plouffe (BBP formulae to a given base b) have interesting computational properties, such as allowing single digits in their base b expansion to be independently computed, and there are hints that they should be normal numbers, i.e., that their base b digits are randomly distributed. We study a formally limited subset of BBP formulae, which we call Machin-type BBP formulae, for which it is relatively easy to determine whether or not a given constant κ has a Machin-type BBP formula. In particular, given b ∈ ℕ, b > 2, b not a proper power, a b-ary Machin-type BBP arctangent formula for κ is a formula of the form κ = Σmam arctan(–bm), am ∈ ℚ, while when b = 2, we also allow terms of the form am arctan(1/(1 – 2m)). Of particular interest, we show that π has no Machin-type BBP arctangent formula when b ≠ 2. To the best of our knowledge, when there is no Machin-type BBP formula for a constant then no BBP formula of any form is known for that constant.

Publisher

Canadian Mathematical Society

Subject

General Mathematics

Cited by 8 articles. 订阅此论文施引文献 订阅此论文施引文献,注册后可以免费订阅5篇论文的施引文献,订阅后可以查看论文全部施引文献

1. Jonathan Borwein: Renaissance Mathematician;The American Mathematical Monthly;2021-10-21

2. On the genesis of BBP formulas;Acta Arithmetica;2021

3. On Leaving as Little to Chance as Possible;Mario Bunge: A Centenary Festschrift;2019

4. A Novel Approach to the Discovery of Binary BBP-Type Formulas for Polylogarithm Constants;Integers;2012-01

5. Experimental mathematics and computational statistics;Wiley Interdisciplinary Reviews: Computational Statistics;2009-07

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