Search Advanced

AIMS Mathematics

2025, Volume 10, Issue 5: 11444-11464. doi: 10.3934/math.2025521
Previous Article Next Article
Research article

A Ramanujan-type supercongruence related to harmonic numbers

  • 1.
    Department of Applied Mathematics, Guangzhou Huashang College, Guangzhou, 511300, China
  • 2.
    School of Mathematics and Physics, Nanjing Institute of Technology, Nanjing, 211167, China
  • Published: 20 May 2025

  • MSC : Primary: 11A07, 11B75; Secondary: 05A10, 11B65

  • We prove that, for all primes $ p > 5 $, positive integers $ r $ and $ m $ with $ p\nmid m $ and $ p\nmid \binom{2mp^{r-1}}{mp^{r-1}} $, there holds

    $ \begin{align*} &\frac{1}{m^4p^{4r}{\binom{2mp^{r-1}}{mp^{r-1}}}^3}\bigg(\sum\limits_{k = 0}^{mp^r-1}(21k+8){\binom{2k}{k}}^3-p\sum\limits_{k = 0}^{mp^{r-1}-1}(21k+8){\binom{2k}{k}}^3\bigg)\\ &\quad \equiv-6\frac{H_{p-1}}{p^{2}}\ ({\rm{mod}}\ {p^{2}}), \end{align*} $

    where $H_n=\sum_{j=1}^{n}\frac{1}{j}$ ($n=1, 2, \cdots$) are the usual harmonic numbers. This partially confirms a conjecture by Z.-W. Sun.

    Citation: Wenbin Zhang, Yong Zhang, Jiachen Wu. A Ramanujan-type supercongruence related to harmonic numbers[J]. AIMS Mathematics, 2025, 10(5): 11444-11464. doi: 10.3934/math.2025521

    Related Papers:

  • We prove that, for all primes $ p > 5 $, positive integers $ r $ and $ m $ with $ p\nmid m $ and $ p\nmid \binom{2mp^{r-1}}{mp^{r-1}} $, there holds

    $ \begin{align*} &\frac{1}{m^4p^{4r}{\binom{2mp^{r-1}}{mp^{r-1}}}^3}\bigg(\sum\limits_{k = 0}^{mp^r-1}(21k+8){\binom{2k}{k}}^3-p\sum\limits_{k = 0}^{mp^{r-1}-1}(21k+8){\binom{2k}{k}}^3\bigg)\\ &\quad \equiv-6\frac{H_{p-1}}{p^{2}}\ ({\rm{mod}}\ {p^{2}}), \end{align*} $

    where $H_n=\sum_{j=1}^{n}\frac{1}{j}$ ($n=1, 2, \cdots$) are the usual harmonic numbers. This partially confirms a conjecture by Z.-W. Sun.



    加载中


    [1] A. O. L. Atkin, H. P. F. Swinnerton-Dyer, Modular forms on noncongruence subgroups, in: Combinatorics, Proc. Sympos. Pure Math. 19, Amer. Math. Soc., Providence, RI, 1971, 1–25. http://dx.doi.org/10.1016/B978-0-12-775850-3.50017-0
    [2] F. Beukers, Some congruences for the Apéry numbers, J. Number Theory, 21 (1985), 141–155. https://doi.org/10.1016/0022-314X(85)90047-2 doi: 10.1016/0022-314X(85)90047-2
    [3] I. Gessel, Some congruences for generalized Euler numbers, Canad. J. Math., 35 (1983), 687–709. https://doi.org/10.4153/CJM-1983-039-5 doi: 10.4153/CJM-1983-039-5
    [4] J. W. L. Glaisher, Congruences relating to the sums of products of the first $n$ numbers and to other sums of products, Quart. J. Math., 31 (1900), 1–35.
    [5] J. W. L. Glaisher, On the residues of the sums of products of the first $p-1$ numbers, and their powers, to modulus $p^2$ or $p^3$, Quart. J. Math., 31 (1900), 321–353.
    [6] A. Granville, Arithmetic properties of binomial coefficients, I: binomial coefficients modulo prime powers, in Organic mathematics (Burnaby, BC, 1995), CMS Conf. Proc., Amer. Math. Soc., Providence, RI, 20 (1997), 253–275.
    [7] J. Guillera, Some binomial series obtained by the WZ-method, Adv. in Appl. Math., 29 (2002), 599–603. https://doi.org/10.1016/S0196-8858(02)00034-9 doi: 10.1016/S0196-8858(02)00034-9
    [8] J. Guillera, Hypergeometric identities for $10$ extended Ramanujan-type series, Ramanujan J., 15 (2008), 219–234. https://doi.org/10.1007/s11139-007-9074-0 doi: 10.1007/s11139-007-9074-0
    [9] J. Guillera, W. Zudilin, "Divergent" Ramanujan-type supercongruences, Proc. Amer. Math. Soc., 140 (2012), 765–777. https://doi.org/10.1090/S0002-9939-2011-10950-X doi: 10.1090/S0002-9939-2011-10950-X
    [10] V. J. W. Guo, Some variations of a "divergent" Ramanujan-type $q$-supercongruence, J. Difference Equ. Appl., 27 (2021), 376–388. https://doi.org/10.1080/10236198.2021.1900140 doi: 10.1080/10236198.2021.1900140
    [11] V. J. W. Guo, L. Li, $q$-Supercongruences from squares of basic hypergeometric series, Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. RACSAM, 117 (2023), Art. 26. https://doi.org/10.1007/s13398-022-01364-9 doi: 10.1007/s13398-022-01364-9
    [12] V. J. W. Guo, M. J. Schlosser, New $q$-analogues of Van Hamme's (E.2) supercongruence and of a supercongruence by Swisher, Results Math., 78 (2023), Art. 105. https://doi.org/10.1007/s00025-023-01885-8 doi: 10.1007/s00025-023-01885-8
    [13] V. J. W. Guo, S. D. Wang, Factors of certain sums involving central $q$-binomial coefficients, Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. RACSAM, 116 (2022), Art. 46. https://doi.org/10.1007/s13398-021-01192-3 doi: 10.1007/s13398-021-01192-3
    [14] H. H. He, X. X. Wang, Some congruences that extend Van Hamme's (D.2) supercongruence, J. Math. Anal. Appl., 527 (2022), Art. 127344. https://doi.org/10.1016/j.jmaa.2023.127344 doi: 10.1016/j.jmaa.2023.127344
    [15] C. Helou, G. Terjanian, On Wolstenholme's theorem and its converse, J. Number Theory, 128 (2008), 475–499. https://doi.org/10.1016/j.jnt.2007.06.008 doi: 10.1016/j.jnt.2007.06.008
    [16] J. C. Liu, Ramanujan-type supercongruences involving Almkvist-Zudilin numbers, Results Math., 77 (2022), Art. 67. https://doi.org/10.1007/s00025-022-01607-6 doi: 10.1007/s00025-022-01607-6
    [17] R. Meštrović, An extension of a congruence by Tauraso, ISRN Combinatorics, (2013), Article ID 363724. https://doi.org/10.1155/2013/363724
    [18] R. Osburn, B. Sahu, A. Straub, Supercongruences for sporadic sequences, P. Edinburgh Math. Soc., 59 (2016), 503–518. https://doi.org/10.1017/S0013091515000255 doi: 10.1017/S0013091515000255
    [19] M. Petkovšek, H. S. Wilf, D. Zeilberger, $A = B$, A. K. Peters, Wellesley, 1996.
    [20] S. Ramanujan, Modular equations and approximations to $\pi$, Quart. J. Math. Oxford Ser., 45 (1914), 350–372, reprinted in: G.H. Hardy, P.V. Sechu Aiyar, B.M. Wilson (Eds.), Collected Papers of Srinivasa Ramanujan, Cambridge University Press, Cambridge, 1927; Chelsea Publ., New York, 1962, pp. 23–39. https://doi.org/10.1007/978-1-4757-4217-6-29
    [21] H. F. Song, C. Wang, Some $q$-supercongruences from squares of basic hypergeometric series, Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. RACSAM, 118 (2024), Art. 36. https://doi.org/10.1007/s13398-023-01534-3 doi: 10.1007/s13398-023-01534-3
    [22] A. Straub, Multivariate Apéry numbers and supercongruences of rational functions, Algebra Number Theory, 8 (2014), 1985–2008. https://doi.org/10.2140/ant.2014.8.1985 doi: 10.2140/ant.2014.8.1985
    [23] Z. H. Sun, Congruences concerning Bernoulli numbers and Bernoulli polynomials, Discrete Appl. Math., 105 (2000), 193–223. https://doi.org/10.1016/S0166-218X(00)00184-0 doi: 10.1016/S0166-218X(00)00184-0
    [24] Z. W. Sun, A new series for $\pi^3$ and related congruences, Internat. J. Math., 26 (2015), 1550055. https://doi.org/10.1142/S0129167X1550055X doi: 10.1142/S0129167X1550055X
    [25] Z. W. Sun, Open conjectures on congruences, Nanjing Univ. J. Math. Biquarterly, 36 (2019), 1–99.
    [26] Z. W. Sun, Super congruences and Euler numbers, Sci. China Math., 54 (2011), 2509–2535. https://doi.org/10.1007/s11425-011-4302-x doi: 10.1007/s11425-011-4302-x
    [27] N. Tang, A new $q$-supercongruence modulo the fourth power of a cyclotomic polynomial, Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. RACSAM, 117 (2023), Art. 101. https://doi.org/10.1007/s13398-023-01432-8 doi: 10.1007/s13398-023-01432-8
    [28] L. Van Hamme, Some conjectures concerning partial sums of generalized hypergeometric series, in: $p$-Adic Functional Analysis, Nijmegen, 1996, in: Lect. Notes Pure Appl. Math., vol. 192, Dekker, New York, 1997, pp. 223–236.
    [29] C. Wei, Generalizations of the (F.2) supercongruence of Van Hamme, Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. RACSAM, 118 (2024), Art. 33. https://doi.org/10.1007/s13398-023-01532-5 doi: 10.1007/s13398-023-01532-5
    [30] W. Xia, Proof of some conjectural congruences of Z. W. Sun, J. Combin. Number Theory, 11 (2019), 1–11.
    [31] Y. H. Youssri, W. M. Abd-Elhameed, Harmonic numbers operational matrix for solving fifth-order two point boundary value problems, Tbil. Math. J., 11 (2018), 17–33.
    [32] D. Zeilberger, Closed form (pun intended!), Contemp. Math., 143 (1993), 579–607.
    [33] Y. Zhang, Some conjectural supercongruences related to Bernoulli and Euler numbers, Rocky Mountain J. Math., 52 (2022), 1105–1126. https://doi.org/10.1216/rmj.2022.52.1105 doi: 10.1216/rmj.2022.52.1105
    [34] J. Zhao, Wolstenholme type theorem for multiple harmonic sums, Int. J. Number Theory, 4 (2008), 73–106. https://doi.org/10.1142/S1793042108001146 doi: 10.1142/S1793042108001146
    [35] W. Zudilin, Ramanujan-type supercongruences, J. Number Theory, 129 (2009), 1848–1857. https://doi.org/10.1016/j.jnt.2009.01.013 doi: 10.1016/j.jnt.2009.01.013
  • Reader Comments
  • © 2025 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)
通讯作者: 陈斌, bchen63@163.com
  • 1. 

    沈阳化工大学材料科学与工程学院 沈阳 110142

  1. 本站搜索
  2. 百度学术搜索
  3. 万方数据库搜索
  4. CNKI搜索
AIMS Mathematics
1.8 3.1

Metrics

Article views(629) PDF downloads(32) Cited by(0)

Preview PDF
Download XML
Export Citation
Article outline

Figures and Tables

Tables(1)

Other Articles By Authors

Related pages

Tools

/

DownLoad:  Full-Size Img  PowerPoint
Return
Return

Catalog