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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## Details

### 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* = 2^{12p}, for *p* ∈ Z^{+} and mod (*p*, 2) = 1. As particular examples, new binary formulas are presented for *π*√3, *π*√3log2, √3Cl_{2}(*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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- Bailey, D. H. A compendium of BBP-type formulas for mathematical constants, http://crd.lbl.gov/~dhbailey/dhbpapers/bbp-formulas.pdf February 2011.
- Bailey, D. H., R. E. Crandall. On the random character of fundamental constant expansions. Experimental Mathematics, Vol. 10, 2001, No. 2, 175–190.
- Borwein, J. M., D. Borwein, W. F. Galway. Finding and excluding b-ary Machintype BBP formulae. http://docserver.carma.newcastle.edu.au/47/1/

machin.pdf, 2002. - Chamberland, M. Binary BBP-formulae for logarithms and generalized Gaussian—Mersenne primes. Journal of Integer Sequences, Vol. 6, 2003.
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- Lafont, J. O. Degree 1 BBP-Type Zero Relations. Private communication, available from the author, January 2011.

## Related papers

## Cite this paper

APAAdegoke, 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.

ChicagoAdegoke, 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.

MLAAdegoke, 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.