Acta Mathematica Hungarica

, Volume 157, Issue 2, pp 327–348 | Cite as

Settling some sum suppositions

  • C. Vignat
  • T. WakhareEmail author


We solve multiple conjectures by Byszewski and Ulas about the base b sum-of-digits function. In order to do this, we develop general results about summations over the sum-of-digits function. As a corollary, we describe an unexpected new result about the Prouhet–Tarry–Escott problem. In some cases, this allows us to partition fewer than bN values into b sets \({\{S_1,\ldots,S_b\}}\) such that
$$\sum_{s\in S_1}s^k = \sum_{s\in S_2}s^k = \cdots = \sum_{s\in S_b}s^k $$
for \({0\leq k \leq N-1}\). The classical construction can only partition bN values such that the first N powers agree. Our results are amenable to a computational search, which may discover new, smaller solutions to this classical problem.

Key words and phrases

Prouhet–Tarry–Escott problem digit sum 

Mathematics Subject Classification

primary 11A63 secondary 05A18 


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  1. 1.
    J.-P. Allouche and J. Shallit, Automatic Sequences: Theory, Applications, Generalizations, Cambridge University Press (New York, 2003).Google Scholar
  2. 2.
    J.-P. Allouche and J. Shallit, The ubiquitous Prouhet–Thue–Morse sequence, in: Sequences and their Applications, C. Ding, T. Helleseth, H. Niederreiter (Eds.), Proceedings of SETA, 98, Springer Verlag (1999), pp. 1-16.Google Scholar
  3. 3.
    Byszewski J., Ulas M.: Some identities involving the Prouhet–Thue–Morse sequence and its relatives. Acta Math. Hungar., 127, 438–456 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Dorwart H.L., Brown O.E.: The Tarry–Escott problem. Amer. Math. Monthly, 44, 613–626 (1937)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Gawron M., Miska P., Ulas M.: Arithmetic properties of coefficients of power series expansion of \({\prod_{n=0}^{\infty}( 1-x^{2^n} )^t}\) (with an Appendix by Andrzej Schinzel). Monatsh. Math., 185, 307–360 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    C. Vignat and T. Wakhare, Finite generating functions for the sum of digits sequence, Ramanujan J., to appear.Google Scholar

Copyright information

© Akadémiai Kiadó, Budapest, Hungary 2018

Authors and Affiliations

  1. 1.Tulane UniversityNew OrleansUSA
  2. 2.Université Paris SudParisFrance
  3. 3.University of MarylandCollege ParkUSA

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