Annals of Combinatorics

, Volume 17, Issue 1, pp 15–26 | Cite as

Congruence Properties of Binary Partition Functions

  • Katherine Anders
  • Melissa Dennison
  • Jennifer Weber Lansing
  • Bruce Reznick


Let \({\mathcal{A}}\) be a finite subset of \({\mathbb{N}}\) containing 0, and let f (n) denote the number of ways to write n in the form \({\sum \varepsilon _{j}2^{j}}\) , where \({\varepsilon _{j} \epsilon \mathcal{A}}\) . We show that there exists a computable \({T = T (\mathcal{A})}\) so that the sequence (f (n) mod 2) is periodic with period T. Variations and generalizations of this problem are also discussed.

Mathematics Subject Classification

11A63 11P81 11B34 11B50 


partitions digital representations Stern sequence 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Anders, K., Weber, J.: The parity of generalized binary representations. REGS report (2010)Google Scholar
  2. 2.
    Berlekamp, E.: Algebraic Coding Theory. Aegean Park Press, Laguna Hills, CA (1984)Google Scholar
  3. 3.
    Brent R.P., Zimmermann P.: The great trinomial hunt. Notices Amer. Math. Soc. 58(2), 233–239 (2011)MathSciNetMATHGoogle Scholar
  4. 4.
    Churchhouse R.F.: Congruence properties of the binary partition function. Proc. Cambridge Philos. Soc. 66(2), 371–376 (1969)MathSciNetMATHCrossRefGoogle Scholar
  5. 5.
    Cooper J.N., Eichhorn D., O’Bryant K.: Reciprocals of binary series. Int. J. Number Theory 2(4), 499–522 (2006)MathSciNetMATHCrossRefGoogle Scholar
  6. 6.
    Dennison, M.A.: A Sequence Related to the Stern Sequence. Ph.D. dissertation, University of Illinois at Urbana-Champaign, Urbana (2010)Google Scholar
  7. 7.
    Klosinski L.F., Alexanderson G.L., Hillman A.P.: The William Lowell Putnam Mathematical Competition. Amer. Math. Monthly 91(8), 487–495 (1984)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Lehmer D.H.: On Stern’s diatomic series. Amer. Math. Monthly 36(2), 59–67 (1929)MathSciNetMATHCrossRefGoogle Scholar
  9. 9.
    Lidl R., Niederreiter H.: Finite Fields, 2nd Edition. Cambridge University Press, Cambridge (1997)Google Scholar
  10. 10.
    Reznick, B.: Some binary partition functions. In: Berndt, B.C. et al. (eds.) Analytic Number Theory, pp. 451–477, Birkhäuser Boston, Boston, MA (1990)Google Scholar
  11. 11.
    Reznick, B.: Regularity properties of the Stern enumeration of the rationals. J. Integer Seq. 11(4), Article 08.4.1 (2008)Google Scholar
  12. 12.
    Rødseth Ø.J., Sellers J.A.: On m-ary partition function congruences: a fresh look at a past problem. J. Number Theory 87(2), 270–281 (2001)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Stern M.A.: Ueber eine zahlentheoretische Funktion. J. Reine Angew. Math. 55, 193–220 (1858)MATHCrossRefGoogle Scholar

Copyright information

© Springer Basel 2013

Authors and Affiliations

  • Katherine Anders
    • 1
  • Melissa Dennison
    • 2
  • Jennifer Weber Lansing
    • 1
  • Bruce Reznick
    • 1
  1. 1.Department of MathematicsUniversity of Illinois at Urbana-ChampaignUrbanaUSA
  2. 2.Department of Mathematics and Computer ScienceBaldwin-Wallace CollegeBereaUSA

Personalised recommendations