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1. CJM 2004 (vol 56 pp. 897)
Finding and Excluding $b$-ary Machin-Type Individual Digit Formulae 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 Categories:11Y99, 11A51, 11Y50, 11K36, 33B10 |