A class of digit extraction BBP-type formulas in general binary bases

Kunle Adegoke, Jaume Oliver Lafont and Olawanle Layeni
Notes on Number Theory and Discrete Mathematics, ISSN 1310-5132
Volume 17, 2011, Number 4, Pages 18—32
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Authors and affiliations

Kunle Adegoke
Department of Physics, Obafemi Awolowo University
Ile-Ife, Nigeria

Jaume Oliver Lafont
Conselleria d’Educacio i Cultura, Govern de les Illes Balears
Palma, Spain

Olawanle Layeni
Department of Mathematics, Obafemi Awolowo University
Ile-Ife, Nigeria

Abstract

BBP-type formulas are usually discovered experimentally, one at a time and in specific bases, through computer searches. In this paper, however, we derive explicit digit extraction BBPtype formulas in general binary bases b = 212p, for p ∈ Z+ and mod (p, 2) = 1. As particular examples, new binary formulas are presented for π√3, π√3log2, √3Cl2(π/3) and a couple of other polylogaritm constants. A variant of the formula for π√3log2 derived in this paper has been known for over ten years but was hitherto unproved. Binary BBP-type formulas for the logarithms of an infinite set of primes and binary BBP-type representations for the arctangents of an infinite set of rational numbers are also presented. Finally, new binary BBP-type zero relations are established.

Keywords

  • BBP-type formulas
  • General binary bases

AMS Classification

  • 11Y60
  • 30B99

References

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  2. Bailey, D. H. A compendium of BBP-type formulas for mathematical constants, http://crd.lbl.gov/~dhbailey/dhbpapers/bbp-formulas.pdf February 2011.
  3. Bailey, D. H., R. E. Crandall. On the random character of fundamental constant expansions. Experimental Mathematics, Vol. 10, 2001, No. 2, 175–190.
  4. Borwein, J. M., D. Borwein, W. F. Galway. Finding and excluding b-ary Machintype BBP formulae. http://docserver.carma.newcastle.edu.au/47/1/
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  5. Chamberland, M. Binary BBP-formulae for logarithms and generalized GaussianMersenne primes. Journal of Integer Sequences, Vol. 6, 2003.
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  7. Lafont, J. O. Degree 1 BBP-Type Zero Relations. Private communication, available from the author, January 2011.

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Cite this paper

APA

Adegoke, K., Lafont, J. O. & Layeni, O. (2011). A class of digit extraction BBP-type formulas in general binary bases, Notes on Number Theory and Discrete Mathematics, 17(4), 18-32.

Chicago

Adegoke, Kunle, Jaume Oliver Lafont, and Olawanle Layeni. “A Class of Digit Extraction BBP-type Formulas in General Binary Bases.” Notes on Number Theory and Discrete Mathematics 17, no. 4 (2011): 18-32.

MLA

Adegoke, Kunle, Jaume Oliver Lafont, and Olawanle Layeni. “A Class of Digit Extraction BBP-type Formulas in General Binary Bases.” Notes on Number Theory and Discrete Mathematics 17.4 (2011): 18-32. Print.

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