Advertisement

Waring’s Theorem for Binary Powers

  • Daniel M. KaneEmail author
  • Carlo SannaEmail author
  • Jeffrey ShallitEmail author
Article
First Online:
  • 15 Downloads

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.

Mathematics Subject Classification (2010)

11B13 68R15 
This is a preview of subscription content, log in to check access.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Notes

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.

References

  1. [1]
    W. D. BANKS: Every natural number is the sum of forty-nine palindromes, INTEGERS - Electronic J. Combinat. Number Theory, 16 (2016), Paper #A3Google Scholar
  2. [2]
    A. BRIDY, R. J. LEMKE-OLIVER, A. SHALLIT and J. SHALLIT: The generalized Nagell-Ljunggren problem: powers with repetitive representations, Experimental Math. 0 (2018), 1–12.CrossRefGoogle Scholar
  3. [3]
    R. D. CARMICHAEL: On the numerical factors of certain arithmetic forms, Amer. Math. Monthly 16 (1909), 153–159.MathSciNetCrossRefGoogle Scholar
  4. [4]
    J. CILLERUELO, F. LUCA and L. BAXTER: Every positive integer is a sum of three palindromes, Math. Comp. 87 (2018), 3023–3055.MathSciNetCrossRefGoogle Scholar
  5. [5]
    P. CUBRE and J. ROUSE: Divisibility properties of the Fibonacci entry point, Proc. Amer. Math. Soc. 142 (11) (2014), 3771–3785.MathSciNetCrossRefGoogle Scholar
  6. [6]
    E. GROSSWALD: Representations of Integers as Sums of Squares, Springer-Verlag, 1985.CrossRefGoogle Scholar
  7. [7]
    D. HILBERT: Beweis fu¨r die Darstellbarkeit der ganzen Zahlen durch eine feste Anzahlnter Potenzen (Waringsches Problem), Math. Annalen 67 (1909), 281–300.MathSciNetCrossRefGoogle Scholar
  8. [8]
    D. E. KNUTH: The Art of Computer Programming, Vol. 1, Fundamental Algorithms, Third Edition, Addison-Wesley, 1997.zbMATHGoogle Scholar
  9. [9]
    D. E. KNUTH: The Art of Computer Programming, Vol. 2, Seminumerical Algorithms, Second Edition, Addison-Wesley, 1981.zbMATHGoogle Scholar
  10. [10]
    J. L. LAGRANGE: Demonstration d’un theoreme d’arithmetique, Nouv. Mem. Acad. Roy. Sc. de Berlin (1770), 123–133. Also in Oeuvres de Lagrange, 3 (1869), 189–201.Google Scholar
  11. [11]
    P. LEONETTI and C. SANNA: On the greatest common divisor of n and the nth Fibonacci number, Rocky Mountain J. Math., 48 (2018), 1191–1199.MathSciNetCrossRefGoogle Scholar
  12. [12]
    P. MADHUSUDAN, D. NOWOTKA, A. RAJASEKARAN and J. SHALLIT: Lagrange’s theorem for binary squares, in I. Potapov, P. Spirakis, and J. Worrell, eds., 43rd International Symposium on Mathematical Foundations of Computer Science (MFCS 2018), Leibniz International Proceedings in Informatics (LIPIcs) 117 (2018), 18:1- 18:14.MathSciNetGoogle Scholar
  13. [13]
    C. J. MORENO and S. S. WAGSTAFF, Jr.: Sums of Squares of Integers, Chapman and Hall/CRC, 2005.CrossRefGoogle Scholar
  14. [14]
    M. B. NATHANSON: Additive Number Theory: The Classical Bases, Springer, 1996.CrossRefGoogle Scholar
  15. [15]
    M. B. NATHANSON: Elementary Methods in Number Theory, Springer, 2000.zbMATHGoogle Scholar
  16. [16]
    G. PÓLYA and G. SZEGŐ: Problems and Theorems in Analysis II, Springer-Verlag, 1976.CrossRefGoogle Scholar
  17. [17]
    J. L. RAMÍREZ-ALFONSÍN: The Diophantine Frobenius Problem, Oxford University Press, 2006.zbMATHGoogle Scholar
  18. [18]
    C. SANNA, J. SHALLIT and S. ZHANG: Largest entry in the inverse of a Vandermonde matrix, manuscript in preparation, February 2019.Google Scholar
  19. [19]
    C. SANNA and E. TRON: The density of numbers n having a prescribed G.C.D. with the nth Fibonacci number, Indag. Math. 29 (2018), 972–980.MathSciNetCrossRefGoogle Scholar
  20. [20]
    N. J. A. SLOANE et al.: The On-Line Encyclopedia of Integer Sequences, 2017. Available at https://doi.org/oeis.org.zbMATHGoogle Scholar
  21. [21]
    C. SMALL: Waring’s problem. Math. Mag. 50 (1977), 12–16.MathSciNetCrossRefGoogle Scholar
  22. [22]
    R. C. VAUGHAN and T. WOOLEY: Waring’s problem: a survey, in: M. A. Bennett, B. C. Berndt, N. Boston, H. G. Diamond, A. J. Hildebrand, and W. Philipp, editors, Number Theory for the Millennium. III, 301–340. A. K. Peters, 2002.Google Scholar
  23. [23]
    K. WIERTELAK: On the density of some sets of primes p, for which nordpa, Funct. Approx. Comment. Math. 28 (2000), 237–241.MathSciNetCrossRefGoogle Scholar

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

Personalised recommendations

Cite article

Buy options