BBP-type formulas, in general bases, for arctangents of real numbers

Kunle Adegoke and Olawanle Layeni
Notes on Number Theory and Discrete Mathematics, ISSN 1310-5132
Volume 19, 2013, Number 3, Pages 33—54
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Authors and affiliations

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

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


BBP-type formulas are usually discovered experimentally, one at a time and in specific bases, through computer searches using PSLQ or other integer relations finding algorithms. In this paper, however, we give a systematic analytical derivation of numerous new explicit digit extraction BBP-type formulas for the arctangents of real numbers in general bases. Our method has the clear advantage that the proofs of the formulas are contained in the derivations, whereas in the experimental approach, proofs of discovered formulas have to be found separately. The high point of this paper is perhaps the discovery, for the first time, of a BBP-type formula for \pi \sqrt{5}.


  • BBP-type formulas
  • Polyalgorithm constants
  • Digit extraction formulas

AMS Classification

  • 11Y60
  • 30B99


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Cite this paper

Adegoke, K., & Layeni, O. (2013). BBP-type formulas, in general bases, for arctangents of real numbers. Notes on Number Theory and Discrete Mathematics, 19(3), 33-54.

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