Acta Mathematica Vietnamica

, Volume 42, Issue 3, pp 551–561 | Cite as

Congruences of − Regular Partition Triples for ∈{2, 3, 4, 5}



For any positive integer , let B (n) denotes the number of -regular partition triples of a positive integer n. By employing q −series identities, we prove infinite family of arithmetic identities and congruences modulo 4 for B 2(n), modulo 2 and 9 for B 3(n), modulo 2 for B 4(n) and modulo 2 and 5 for B 5(n).


-regular partition Partition triples Partition congruence q −series identities Ramanujan’s theta-functions 

Mathematics Subject Classification (2010)

11P83 05A15 05A17 



The first author (N. Saikia) is thankful to Council of Scientific and Industrial Research of India for partially supporting the research work under the Research Scheme No. 25(0241)/15/EMR-II (F. No. 25(5498)/15).

The authors thank anonymous referee for his/her valuable suggestions and comments.


  1. 1.
    Baruah, N. D., Ahmed, Z.: Congruences modulo p 2 and p 3 for k dots bracelet partitons with k = m p s. J. Number Theory 151(5), 129–146 (2015)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Baruah, N. D., Ahmed, Z.: New congruences for −regular partitions for {5, 6, 7, 49}. Ramanujan J. doi: 10.1007/s11139-015-9752-2 (2016)
  3. 3.
    Baruah, N. D., Das, K.: Parity results for 7-regular and 23-regular partitions. Int. J. Number Theory II, 2221–2238 (2015)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Baruah, N. D., Sarmah, B. K.: Identities and congruences for the general partition and Ramanujan’s Tau functions. Indian J. Pure Appl. Math. 44(5), 643–671 (2013)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Berndt, B. C.: Ramanujan’s notebooks part, vol. III. Springer, New York (1991)Google Scholar
  6. 6.
    Carlson, R., Webb, J. J.: Infinite families of congruences for k-regular partitions. Ramanujan J. 33, 329–337 (2014)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Cui, S. P., Gu, N. S. S.: Arithmetic properties of - regular partitions. Adv. Appl. Math. 51, 507–523 (2013)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Dandurand, B., Penniston, D.: -divisibility of -regular partition functions. Ramanujan J. 19, 63–70 (2009)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Hirschhorn, M. D., Sellers, J. A.: Elementary proofs of parity results for 5-regular partitions. Bull. Aust. Math. Soc. 81, 58–63 (2010)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Lin, B.L.S.: An infinite family of congruences modulo 3 for 13-regular bipartitions. Ramanujan J. doi: 10.1007/s11139-014-9610-7 (2014)
  11. 11.
    Penniston, D.: Arithmetic of -regular partition functions. Int. J. Number Theory 4, 295–302 (2008)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Wang, L.: Arithmetic properties of overpartition triples arXiv:1410.7898v2 [math.NT] 12 (2015)
  13. 13.
    Wang, L.: Arithmetic identities and congruences for partition triples with 3-cores. Int. J Number Theory. doi: 10.1142/S1793042116500627 (2015)
  14. 14.
    Webb, J.J.: Arithmetic of the 13-regular partition function modulo 3. Ramanujan J. 25, 49–56 (2011)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Xia, E. X. W., Yao, O. X. M.: Parity results for 9-regular partitions. Ramanujan J. 34, 109–117 (2014)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Xia, E.X.W., Yao, O.X.M.: A proof of Keith’s conjecture for 9-regular partitions modulo 3. Int. J. Number Theory 10, 669–674 (2014)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Xia, E.X.W., Yao, O.X.M.: Analogues of Ramanujan’s partition identities. Ramanujan J. 31, 373–396 (2013)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Institute of Mathematics, Vietnam Academy of Science and Technology (VAST) and Springer Science+Business Media Singapore 2017

Authors and Affiliations

  1. 1.Department of MathematicsRajiv Gandhi UniversityDoimukhIndia

Personalised recommendations