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Finding and Excluding b-ary Machin-Type BBP Formulae

Borwein, Jonathan M. and Galway, William F. and Borwein, David (2004) Finding and Excluding b-ary Machin-Type BBP Formulae. Canadian J. Math, 56 . pp. 897-925.

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      Abstract

      Constants 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, 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 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 $κ =\sum_m a_m arctan(−b^{−m}), a_m ∈ Q,$ while when b = 2 we also allow terms of the form $a_m arctan(1/(1 − 2^m)).$ Of particular interest, we show that π has no Machin-type BBP arctangent formula when 4b\neq 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.

      Item Type: Article
      Additional Information: pubdom FALSE
      Uncontrolled Keywords: BBP formulae, Machin-type formulae, arctangents, logarithms, normality, Mersenne primes, Bang's theorem, Zaigmondy's theorem, primitive prime factors, p-adic analysis
      Subjects: 33-xx Special functions > 33Bxx Elementary classical functions
      11-xx Number theory > 11Kxx Probabilistic theory: distribution modulo $1$; metric theory of algorithms
      11-xx Number theory > 11Axx Elementary number theory
      11-xx Number theory > 11Yxx Computational number theory
      Faculty: UNSPECIFIED
      Depositing User: Users 1 not found.
      Date Deposited: 30 Oct 2003
      Last Modified: 12 Jan 2015 15:14
      URI: https://docserver.carma.newcastle.edu.au/id/eprint/47

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