Bailey–Borwein–Plouffe formula

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The Bailey–Borwein–Plouffe formula (BPP formula) was discovered in 1995 by Simon Plouffe and was named after the authors of the paper in which the formula was published, David H. Bailey, Peter Borwein, and Simon Plouffe. Before that paper, it had been published by Plouffe on his own site.

     

π  =    ∞ ∑   k  = 0    4 8 k + 1 − 2 8 k + 4 − 1 8 k + 5 − 1 8 k + 6 16  − k  =

∞ ∑   k  = 0    120 k  2 + 151 k + 47 512 k  4 + 1024 k  3 + 712 k  2 + 194 k + 15 16  − k  =    ∞ ∑   k  = 0    ak 16  − k,

which allows to compute a given base-16 digit of

π

without calculating the preceding digits.

Note that we have (see Bailey and Crandall conjecture)

     

ak  − 1  =  4 8 (k − 1) + 1 − 2 8 (k − 1) + 4 − 1 8 (k − 1) + 5 − 1 8 (k − 1) + 6  =  120 k  2 − 89 k + 16 512 k  4 − 1024 k  3 + 712 k  2 − 206 k + 21  .

A062964:

π

(referred to as "Pi" in the OEIS) in hexadecimal.

{3, 2, 4, 3, 15, 6, 10, 8, 8, 8, 5, 10, 3, 0, 8, 13, 3, 1, 3, 1, 9, 8, 10, 2, 14, 0, 3, 7, 0, 7, 3, 4, 4, 10, 4, 0, 9, 3, 8, 2, 2, 2, 9, 9, 15, 3, 1, 13, 0, 0, 8, 2, 14, 15, 10, 9, 8, 14, 12, 4, 14, 6, 12, ...}

Bailey and Crandall conjecture

David H. Bailey and Richard Crandall conjecture that the terms of this sequence, apart from the first (which is the integer part), are given by the formula

⌊  16 ( x (n)  −  ⌊  x (n)⌋  )⌋

, where

x (n)

is determined by the recurrence equation      

x (0)  =  0; x (n)  =  16 x (n − 1) + an  − 1  =  16 x (n − 1) + 120 n 2 − 89 n + 16 512 n 4 − 1024 n 3 + 712 n 2 − 206 n + 21 , n ≥ 1.

They have numerically verified the conjecture for the first 100,000 terms of the sequence.

External links

• Finding the Nth digit of a number a without needing to know the previous one thanks to the BBP formula