Finding and Excluding <i>b</i>-ary Machin-Type Individual Digit Formulae

Abstract

Abstract 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 &gt; 2, b not a proper power, a b -ary Machin-type BBP arctangent formula for κ is a formula of the form κ = Σ m a m arctan(– b – m ), a m ∈ 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 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.

Locations

• Canadian Journal of Mathematics - View - PDF