Skip to main content
SpringerLink
Log in

Congruences and recursions for the cubic partition

  • Published:
The Ramanujan Journal Aims and scope Submit manuscript

Abstract

Let \(p_2(n)\) denote the number of cubic partitions. In this paper, we shall present two new congruences modulo 11 for \(p_2(n).\) We also provide an elementary alternative proof of a congruence established by Chan. Furthermore, we will establish a recursion for \(p_2(n),\) which is a special case of a broader class of recursions.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Ahmed, Z., Baruah, N.D., Dastidar, M.G.: New congruences modulo 5 for the number of 2-color partitions. J. Number Theory 157, 184–198 (2015)

    Article  MATH  MathSciNet  Google Scholar 

  2. Andrews, G.E., Deutsch, E.: A note on a method of Erdös and the Stanley–Elder theorems. Integers 16, Paper No. A24 (2016)

  3. Berndt, R.C.: Ramanujan’s Notebooks. Part III. Springer, New York (1991)

    Book  MATH  Google Scholar 

  4. Chan, H.-C.: Ramanujan’s cubic continued fraction and an analog of his “most beautiful identity”. Int. J. Number Theory 6(3), 673–680 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  5. Chen, W.Y.C., Lin, B.L.S.: Congruences for the number of cubic partitions derived from modular forms. Preprint. arXiv:0910.1263

  6. Chern, S.: New congruences for 2-color partitions. J. Number Theory 163, 474–481 (2016)

    Article  MATH  MathSciNet  Google Scholar 

  7. Erdös, P.: On an elementary proof of some asymptotic formulas in the theory of partitions. Ann. Math. (2) 43, 437–450 (1942)

    Article  MATH  MathSciNet  Google Scholar 

  8. Ford, W.B.: Two theorems on the partitions of numbers. Am. Math. Mon. 38(4), 183–184 (1931)

    Article  MATH  MathSciNet  Google Scholar 

  9. Radu, S.: An algorithmic approach to Ramanujan’s congruences. Ramanujan J. 20(2), 215–251 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  10. Radu, S., Sellers, J.A.: Congruence properties modulo 5 and 7 for the pod function. Int. J. Number Theory 7(8), 2249–2259 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  11. Ramanujan, S.: Collected Papers of Srinivasa Ramanujan. AMS Chelsea Publishing, Providence (2000)

    MATH  Google Scholar 

  12. Xiong, X.H.: The number of cubic partitions modulo powers of 5 (Chinese). Sci. Sin. Math. 41(1), 1–15 (2011)

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. School of Mathematical Sciences, Zhejiang University, Hangzhou, 310027, China

    Shane Chern

  2. Department of Mathematical Sciences, Pondicherry University, R. V. Nagar, Kalapet, Pondicherry, 605014, India

    Manosij Ghosh Dastidar

Authors
  1. Shane Chern
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Manosij Ghosh Dastidar
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Manosij Ghosh Dastidar.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Chern, S., Dastidar, M.G. Congruences and recursions for the cubic partition. Ramanujan J 44, 559–566 (2017). https://doi.org/10.1007/s11139-016-9852-7

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11139-016-9852-7

Keywords

Mathematics Subject Classification

Search

Navigation