Waring’s Theorem for Binary Powers

  • Daniel M. KaneEmail author
  • Carlo SannaEmail author
  • Jeffrey ShallitEmail author


A natural number is a binary k’ th power if its binary representation consists of k consecutive identical blocks. We prove, using tools from combinatorics, linear algebra, and number theory, an analogue of Waring’s theorem for sums of binary k’th powers. More precisely, we show that for each integer k> 2, there exists an effectively computable natural number n such that every sufficiently large multiple of Ek:=gcd(2k - 1,k) is the sum of at most n binary k’th powers. (The hypothesis of being a multiple of Ek cannot be omitted, since we show that the gcd of the binary k’th powers is Ek.) Furthermore, we show that n = 2O(k3). Analogous results hold for arbitrary integer bases b>2.

Mathematics Subject Classification (2010)

11B13 68R15 


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We are grateful to Igor Pak for introducing the first and third authors to each other. We thank the referees for their careful reading of the paper.


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Copyright information

© János Bolyai Mathematical Society and Springer-Verlag 2019

Authors and Affiliations

  1. 1.MathematicsUniversity of California, San DiegoLa JollaUSA
  2. 2.Dipartimento di Matematica “Giuseppe Peano”Università degli Studi di TorinoTorinoItaly
  3. 3.School of Computer ScienceUniversity of WaterlooWaterlooCanada

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