Arithmetic properties of 7-regular partitions

  • Liuquan Wang


Let \(b_{\ell }(n)\) denote the number of \(\ell \)-regular partitions of n. By employing the modular equation of seventh order, we establish the following congruence for \(b_{7}(n)\) modulo powers of 7: for \(n\ge 0\) and \(j\ge 1\),
$$\begin{aligned} b_{7}\left( 7^{2j-1}n+\frac{3\cdot 7^{2j}-1}{4}\right) \equiv 0 \pmod {7^j}. \end{aligned}$$
We also find some infinite families of congruences modulo 2 and 7 satisfied by \(b_{7}(n)\).


Partitions Congruences 7-Regular partitions Modular equation 

Mathematics Subject Classification

Primary 05A17 Secondary 11P83 



The author thanks the referee for his/her careful reading of the manuscript and helpful suggestions which improved the presentation of this work.


  1. 1.
    Ahlgren, S., Lovejoy, J.: The arithmetic of partitions into distinct parts. Mathematika 48, 203–211 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Andrews, G.E., Hirschhorn, M.D., Sellers, J.A.: Arithmetic properties of partitions with even parts distinct. Ramanujan J. 23, 169–181 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Atkin, A.O.L.: Proof of a conjecture of Ramanujan. Glasg. Math. J. 8, 14–32 (1967)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Calkin, N., Drake, N., James, K., Law, S., Lee, P., Penniston, D., Radder, J.: Divisibility properties of the 5-regular and 13-regular partition functions. Integers 8, A60 (2008)MathSciNetzbMATHGoogle Scholar
  5. 5.
    Chan, H.H., Toh, P.C.: New analogues of Ramanujan’s partition identities. J. Number Theory 130, 1898–1913 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Chen, S.C.: On the number of partitions with distinct even parts. Discret. Math. 311, 940–943 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Cui, S.P., Gu, N.S.S.: Arithmetic properties of the \(\ell \)-regular partitions. Adv. Appl. Math. 51, 507–523 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Cui, S.P., Gu, N.S.S.: Congruences for 9-regular partitions modulo 3. Ramanujan J. 38, 503–512 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Dandurand, B., Penniston, D.: \(\ell \)-Divisibility of \(\ell \)-regular partition functions. Ramanujan J. 19, 63–70 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Furcy, D., Penniston, D.: Congruences for \(\ell \)-regular partition functions modulo 3. Ramanujan J. 27, 101–108 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Garvan, F.G.: A simple proof of Watson’s partition congruences for powers of 7. J. Aust. Math. Soc. 36, 316–334 (1984)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Gordon, B., Ono, K.: Divisibility of certain partition functions by powers of primes. Ramanujan J. 1, 25–34 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Hirschhorn, M.D., Hunt, D.C.: A simple proof of the Ramanujan conjecture for powers of 5. J. Reine Angew. Math. 326, 1–17 (1981)MathSciNetzbMATHGoogle Scholar
  14. 14.
    Hirschhorn, M.D., Sellers, J.A.: Elementary proofs of parity results for 5-regular partitions. Bull. Aust. Math. Soc. 81, 58–63 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Koblitz, N.: Introduction to Elliptic Curves and Modular Forms. Springer, New York (1984)CrossRefzbMATHGoogle Scholar
  16. 16.
    Lovejoy, J.: The divisibility and distribution of partitions into distinct parts. Adv. Math. 158, 253–263 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Lovejoy, J.: The number of partitions into distinct parts modulo powers of 5. Bull. Lond. Math. Soc. 35, 41–46 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Lovejoy, J., Penniston, D.: 3-Regular partitions and a modular \(K3\) surface. Contemp. Math. 291, 177–182 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Newman, M.: The coefficients of certain infinite products. Proc. Am. Math. Soc. 4, 435–439 (1953)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Ono, K., Penniston, D.: The 2-adic behavior of the number of partitions into distinct parts. J. Comb. Theory A 92, 138–157 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Penniston, D.: Arithmetic of \(\ell \)-regular partition functions. Int. J. Number Theory 4, 295–302 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Ramanujan, S.: The Lost Notebook and Other Unpublished Paper. Narosa, New Delhi (1998)Google Scholar
  23. 23.
    Serre, J.-P.: Sur la lacunarité des puissances de \(\eta \). Glasgow Math. J. 27, 203–221 (1985)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Wang, L.: Congruences for 5-regular partitions modulo powers of 5. Ramanujan J. 44, 343–358 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Wang, L.: Congruences modulo powers of 11 for some partition functions. Proc. Am. Math. Soc.
  26. 26.
    Webb, J.J.: Arithmetic of the 13-regular partition function modulo 3. Ramanujan J. 25, 49–56 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Xia, E.X.W.: Congruences for some \(\ell \)-regular partitions modulo \(\ell \). J. Number Theory 152, 105–117 (2015)MathSciNetCrossRefGoogle Scholar
  28. 28.
    Xia, E.X.W., Yao, O.X.M.: Parity results for 9-regular partitions. Ramanujan J. 34, 109–117 (2014)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.School of Mathematics and StatisticsWuhan UniversityWuhanChina

Personalised recommendations