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Periodica Mathematica Hungarica

, Volume 79, Issue 2, pp 227–235 | Cite as

Proof of a conjecture on a congruence modulo 243 for overpartitions

  • Xiaoqian Huang
  • Olivia X. M. YaoEmail author
Article
  • 67 Downloads

Abstract

Let \(\bar{p}(n)\) denote the number of overpartitions of n. Recently, numerous congruences modulo powers of 2, 3 and 5 were established regarding \(\bar{p}(n)\). In particular, Xia discovered several infinite families of congruences modulo 9 and 27 for \(\bar{p}(n)\). Moreover, Xia conjectured that for \(n\ge 0\), \(\bar{p}(96n+76) \equiv 0\ (\mathrm{mod}\ 243)\). In this paper, we confirm this conjecture by using theta function identities and the (pk)-parametrization of theta functions.

Keywords

Overpartitions Congruences Theta functions 

Mathematics Subject Classification

11P83 05A17 

Notes

Acknowledgements

The authors would like to thank the anonymous referee for valuable corrections and comments. This work was supported by Jiangsu National Funds for Distinguished Young Scientists (Grant No. BK20180044) and Chinese Postdoctoral Science Foundation (Grant No. 2018T110444).

References

  1. 1.
    A. Alaca, S. Alaca, K.S. Williams, On the two-dimensional theta functions of the Borweins. Acta Arith. 124, 177–195 (2006)MathSciNetCrossRefGoogle Scholar
  2. 2.
    S. Alaca, K.S. Williams, The number of representations of a positive integer by certain octonary quadratic forms. Funct. Approx. Comment. Math. 43, 45–54 (2010)MathSciNetCrossRefGoogle Scholar
  3. 3.
    N.D. Baruah, K.K. Ojah, Analogues of Ramanujan’s partition identities and congruences arising from his theta functions and modular equations. Ramanujan J. 28, 385–407 (2012)MathSciNetCrossRefGoogle Scholar
  4. 4.
    B.C. Berndt, Ramanujan’s Notebooks, Part III (Springer, New York, 1991)CrossRefGoogle Scholar
  5. 5.
    S. Corteel, J. Lovejoy, Overpartitions. Trans. Am. Math. Soc. 356, 1623–1635 (2004)MathSciNetCrossRefGoogle Scholar
  6. 6.
    M.D. Hirschhorn, F. Garvan, J. Borwein, Cubic analogs of the Jacobian cubic theta function \(\theta (z, q)\). Can. J. Math. 45, 673–694 (1993)CrossRefGoogle Scholar
  7. 7.
    M.D. Hirschhorn, J.A. Sellers, Arithmetic relations for overpartitions. J. Comb. Math. Comb. Comput. 53, 65–73 (2005)MathSciNetzbMATHGoogle Scholar
  8. 8.
    K.S. Williams, Fourier series of a class of eta quotients. Int. J. Number Theory 8, 993–1004 (2012)MathSciNetCrossRefGoogle Scholar
  9. 9.
    E.X.W. Xia, Congruences modulo 9 and 27 for overpartitions. Ramanujan J. 42, 301–323 (2017)MathSciNetCrossRefGoogle Scholar
  10. 10.
    E.X.W. Xia, O.X.M. Yao, New Ramanujan-like congruences modulo powers of 2 and 3 for overpartitions. J. Number Theory 133, 1932–1949 (2013)MathSciNetCrossRefGoogle Scholar
  11. 11.
    E.X.W. Xia, O.X.M. Yao, Analogues of Ramanujan’s partition identities. Ramanujan J. 31, 373–396 (2013)MathSciNetCrossRefGoogle Scholar
  12. 12.
    E.X.W. Xia, Y. Zhang, Proofs of some conjectures of Sun on the relations between sums of squares and sums of triangular numbers. Int. J. Number Theory 15, 189–212 (2019)MathSciNetCrossRefGoogle Scholar
  13. 13.
    O.X.M. Yao, Congruences modulo 64 and 1024 for overpartitions. Ramanujan J. 46, 1–18 (2018)MathSciNetCrossRefGoogle Scholar

Copyright information

© Akadémiai Kiadó, Budapest, Hungary 2019

Authors and Affiliations

  1. 1.Department of MathematicsJiangsu UniversityZhenjiangPeople’s Republic of China

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