Waring’s Theorem for Binary Powers

Abstract

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.

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Acknowledgments

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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Correspondence to Daniel M. Kane or Carlo Sanna or Jeffrey Shallit.

Additional information

D. M. Kane was supported by NSF Award CCF-1553288 (CAREER) and a Sloan Research Fellowship.

C. Sanna is a member of the INdAM group GNSAGA.

J. Shallit was supported by NSERC Discovery Grants #105829/2013 and 2018-04118.

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Kane, D.M., Sanna, C. & Shallit, J. Waring’s Theorem for Binary Powers. Combinatorica 39, 1335–1350 (2019). https://doi.org/10.1007/s00493-019-3933-3

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Mathematics Subject Classification (2010)

  • 11B13
  • 68R15