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Finding and Excluding b-ary Machin-Type Individual Digit Formulae
Published online by Cambridge University Press: 20 November 2018
Abstract
Constants with formulae of the form treated by D. Bailey, P. Borwein, and S. Plouffe ($b$ 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
$K$ has a Machin-type BBP formula. In particular, given
$b\,\in \,\mathbb{N},\,b\,>\,2,\,b$ not a proper power, a
$b$-ary Machin-type BBP arctangent formula for
$K$ is a formula of the form
$k\,=\,{{\Sigma }_{m}}\,{{a}_{m}}\,\arctan \,(-{{b}^{-m}}),\,{{a}_{m}}\,\in \,\mathbb{Q}$
, while when
$b\,=\,2$, we also allow terms of the form
${{a}_{m}}\,\arctan \,(1/1\,-\,{{2}^{m}}))$
. Of particular interest, we show that
$\pi$ has no Machin-type BBP arctangent formula when
$b\,\ne \,2$. 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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- Copyright © Canadian Mathematical Society 2004
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