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

  Published:2004-10-01
 Printed: Oct 2004
  • Jonathan M. Borwein
  • David Borwein
  • William F. Galway
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Abstract

Constants with formulae of the form treated by D.~Bailey, P.~Borwein, and S.~Plouffe (\emph{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 \emph{normal} numbers, {\em i.e.,} that their base $b$ digits are randomly distributed. We study a formally limited subset of BBP formulae, which we call \emph{Machin-type BBP formulae}, for which it is relatively easy to determine whether or not a given constant $\kappa$ 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 $\kappa$ is a formula of the form $\kappa = \sum_{m} a_m \arctan(-b^{-m})$, $a_m \in \mathbb{Q}$, 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 $b \neq 2$. 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.
Keywords: BBP formulae, Machin-type formulae, arctangents, logarithms, normality, Mersenne primes, Bang's theorem, Zsigmondy's theorem, primitive prime factors, $p$-adic analysis BBP formulae, Machin-type formulae, arctangents, logarithms, normality, Mersenne primes, Bang's theorem, Zsigmondy's theorem, primitive prime factors, $p$-adic analysis
MSC Classifications: 11Y99, 11A51, 11Y50, 11K36, 33B10 show english descriptions None of the above, but in this section
Factorization; primality
Computer solution of Diophantine equations
Well-distributed sequences and other variations
Exponential and trigonometric functions
11Y99 - None of the above, but in this section
11A51 - Factorization; primality
11Y50 - Computer solution of Diophantine equations
11K36 - Well-distributed sequences and other variations
33B10 - Exponential and trigonometric functions
 

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