- Title
- Finding and excluding b-ary machin-type individual digit formulae
- Creator
- Borwein, Jonathan M.; Borwein, David; Galway, William F.
- Relation
- Canadian Journal of Mathematics Vol. 56, Issue 5, p. 897-925
- Publisher Link
- http://dx.doi.org/10.4153/CJM-2004-041-2
- Publisher
- University of Toronto Press
- Resource Type
- journal article
- Date
- 2004
- Description
- 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 = ∑m am arctan(−b−m), 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.
- Subject
- BBP formulae; machin type formulae; arctangents; logarithms; normality; Mersenne primes; Bang's theorem
- Identifier
- http://hdl.handle.net/1959.13/1039922
- Identifier
- uon:13725
- Identifier
- ISSN:0008-414X
- Language
- eng
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