FINDING AND EXCLUDING b-ARY MACHIN-TYPE BBP FORMULAE

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, 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 relatively easy to determine whether or not a given constant κ has a Machin-type BBP formula. In particular, given b ∈ N, b > 2, b not a proper power, a b-ary Machin-type BBP arctangent formula for κ is a formula of the form κ = ∑ m am arctan(−b −m), am ∈ Q, while when b = 2 we also allow terms of the form am arctan(1/(1 − 2)). Of particular interest, we show that π has no Machin-type BBP arctangent formula when b 6= 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.