Congruences for Partition Quadruples with t-Cores

  • M. S. Mahadeva NaikaEmail author
  • S. Shivaprasada Nayaka


Let Ct(n) denote the number of partition quadruples of n with t-cores for t = 3,5,7,25. We establish some Ramanujan type congruences modulo 5, 7, 8 for Ct(n). For example, n ≥ 0, we have
$$ \begin{array}{@{}rcl@{}} C_{5}(5n+4)&\equiv& 0\pmod{5},\\ C_{7}(7n+6)&\equiv& 0\pmod{7},\\ C_{3}(16n+14)&\equiv& 0\pmod{8}. \end{array} $$


Congruences Partition quadruples t-core partition 

Mathematics Subject Classification (2010)

11P81 11P83 



  1. 1.
    Baruah, N.D., Nath, K.: Infinite families of arithmetic identities for 4-cores. Bull. Aust. Math. Soc. 87(2), 304–315 (2013)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Baruah, N.D., Nath, K.: Infinite families of arithmetic identities and congruences for bipartitions with 3-cores. J. Number Theory 149, 92–104 (2015)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Berndt, B.C.: Ramanujan’s Notebooks Part, vol. III. Springer, New York (1991)CrossRefGoogle Scholar
  4. 4.
    Hirschhorn, M.D., Garvan, F., Borwein, J.: Cubic analogs of the Jacobian cubic theta function 𝜃(z,q). Canad. J. Math. 45, 673–694 (1993)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Hirschhorn, M.D., Sellers, J.A.: Some amazing facts about 4-cores. J. Number Theory 60, 51–69 (1996)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Hirschhorn, M.D., Sellers, J.A.: Two congruences involving 4-cores. Electron J. Combin. 3(2), R 10 (1996)MathSciNetzbMATHGoogle Scholar
  7. 7.
    Hirschhorn, M.D., Sellers, J.A.: A congruence modulo 3 for partitions into distinct non-multiples of four. J. Integer Sequen 17, Article 14.9.6 (2014)Google Scholar
  8. 8.
    Lin, B.L.S.: Some results on bipartitions with 3-core. J. Number Theory 139, 41–52 (2014)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Ramanujan, S.: Collected papers. Cambridge University Press, Cambridge (2000). reprinted by Chelsea, New York (1962), reprinted by the American mathematical society, Providence RIzbMATHGoogle Scholar
  10. 10.
    Wang, L.: Arithmetic identities and congruences for partition triples with 3-cores. Int. J. Number Theory 12, 995 (2016)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Xia, E.X.W., Yao, O.X.M.: Analogues of Ramanujan’s partition identities. Ramanujan J. 31, 373–396 (2013)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Xia, E.X.W., Yao, O.X.M.: New Ramanujan-like congruences modulo powers of 2 and 3 for overpartitions. J. Number Theory 133, 1932–1949 (2013)MathSciNetCrossRefGoogle Scholar

Copyright information

© Institute of Mathematics, Vietnam Academy of Science and Technology (VAST) and Springer Nature Singapore Pte Ltd. 2019

Authors and Affiliations

  • M. S. Mahadeva Naika
    • 1
    Email author
  • S. Shivaprasada Nayaka
    • 1
  1. 1.Department of MathematicsBangalore UniversityBengaluruIndia

Personalised recommendations