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

  • Jonathan M. Borwein (a1), David Borwein (a2) and William F. Galway (a3)
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, 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 = Σ m am arctan(–b m ), am ∈ ℚ, while when b = 2, we also allow terms of the form am arctan(1/(1 – 2 m )). 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.

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References
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Canadian Journal of Mathematics
  • ISSN: 0008-414X
  • EISSN: 1496-4279
  • URL: /core/journals/canadian-journal-of-mathematics
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