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On the genesis of BBP formulas

Volume 198 / 2021

Daniel Barsky, Vicente Muñoz, Ricardo Pérez-Marco Acta Arithmetica 198 (2021), 401-426 MSC: 11K16, 11J99. DOI: 10.4064/aa200619-28-9 Published online: 8 February 2021


We present a general procedure to generate infinitely many BBP and BBP-like formulas for the simplest transcendental numbers. This provides some insight into and a better understanding of their nature. In particular, we can derive the main known BBP formulas for $\pi $. We can understand why many of these formulas are rearrangements of each other. We also understand better where some null BBP formulas representing $0$ come from. We also explain what is the observed relation between some BBP formulas for $\log 2$ and $\pi $, which are obtained by taking the real and imaginary parts of a general complex BBP formula. Our methods are elementary, but motivated by transalgebraic considerations, and offer a new way to obtain and to search for many new BBP formulas and, conjecturally, to better understand transalgebraic relations between transcendental constants.


  • Daniel Barsky7 rue La Condamine
    75017 Paris, France
  • Vicente MuñozDepartamento de Algebra, Geometría y Topología
    Universidad de Málaga
    Campus de Teatinos, s/n
    29071 Málaga, Spain
  • Ricardo Pérez-MarcoCNRS, IMJ-PRG, Univ. de Paris
    Bât. Sophie Germain, Case 7012
    75205 Paris Cedex 13, France

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