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Finding and Excluding $b$ary MachinType Individual Digit Formulae


Published:20041001
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{Machintype BBP formulae}, for which it
is relatively easy to determine whether or not a given constant
$\kappa$ has a Machintype BBP formula. In particular, given $b \in
\mathbb{N}$, $b>2$, $b$ not a proper power, a $b$ary Machintype
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 $a_m \arctan(1/(12^m))$. Of
particular interest, we show that $\pi$ has no Machintype BBP
arctangent formula when $b \neq 2$. To the best of our knowledge,
when there is no Machintype BBP formula for a constant then no BBP
formula of any form is known for that constant.
Keywords: 
BBP formulae, Machintype formulae, arctangents, logarithms, normality, Mersenne primes, Bang's theorem, Zsigmondy's theorem, primitive prime factors, $p$adic analysis
BBP formulae, Machintype formulae, arctangents, logarithms, normality, Mersenne primes, Bang's theorem, Zsigmondy's theorem, primitive prime factors, $p$adic analysis
