Congruences for the Apéry numbers modulo p3

Zhi-Hong Sun
Notes on Number Theory and Discrete Mathematics
Print ISSN 1310–5132, Online ISSN 2367–8275
Volume 31, 2025, Number 3, Pages 646–657
DOI: 10.7546/nntdm.2025.31.3.646-657
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Zhi-Hong Sun
School of Mathematics and Statistics, Huaiyin Normal University
Huaian, Jiangsu 223300, P. R. China

Abstract

Let \{A'_n\} be the Apéry numbers given by A'_n=\sum_{k=0}^n\binom nk^2\binom{n+k}k. For any prime p\equiv 3\pmod 4 we show that

    \[A'_{\frac{p-1}2}\equiv \frac{p^2}{3\binom{(p-3)/2}{(p-3)/4}^2}\pmod {p^3}.\]

Let \{t_n\} be given by t_0=1, t_1=5 and t_{n+1}=(8n^2+12n+5)t_n-4n^2(2n+1)^2t_{n-1}\ (n\ge 1). We also establish the congruences for t_p\pmod {p^3},\ t_{p-1}\pmod {p^2} and t_{\frac{p-1}2}\pmod {p^2}, where p is an odd prime.

Keywords

  • Apéry number
  • Congruence
  • Combinatorial identity
  • Binary quadratic form
  • Euler number

2020 Mathematics Subject Classification

  • 11A07
  • 05A10
  • 05A19
  • 11B68
  • 11E25

References

  1. Ahlgren, S. (1999). Gaussian hypergeometric series and combinatorial congruences. In: Symbolic Computation, Number Theory, Special Functions, Physics and Combinatorics (Garvan, F. G., & Ismail, M. E. H. (eds).). Developments in Mathematics, vol 4. Springer, Boston, MA, 1–12.
  2. Apéry, R. (1979). Irrationalité de ζ(2) et ζ(3). Astérisque, 61, 11–13.
  3. Beukers, F. (1985). Some congruences for the Apery numbers. Journal of Number Theory, 21(2), 141–155.
  4. Beukers, F. (1987). Another congruence for the Apéry numbers. Journal of Number Theory, 25(2), 201–210.
  5. Chen, W. Y. C., Hou, Q.-H., & Mu, Y.-P. (2006). A telescoping method for double summations. Journal of Computational and Applied Mathematics, 196(2), 553–566.
  6. Gould, H. W. (1972). Combinatorial Identities. A Standardized Set of Tables Listing 500 Binomial Coefficient Summations. West Virginia University, Morgantown, WV.
  7. Ishikawa, T. (1990). Super congruence for the Apéry numbers. Nagoya Mathematical Journal, 118, 195–202.
  8. Sloane, N. J. A. The On-Line Encyclopedia of Integer Sequences. Available online at: https://oeis.org/.
  9. Sun, Z.-H. (2000). Congruences concerning Bernoulli numbers and Bernoulli polynomials. Discrete Applied Mathematics, 105(1–3), 193–223.
  10. Sun, Z.-H. (2011). Congruences concerning Legendre polynomials. Proceedings of the American Mathematical Society, 139(6), 1915–1929.
  11. Sun, Z.-H. (2017). Some further properties of even and odd sequences. International Journal of Number Theory, 13(6), 1419–1442.
  12. Sun, Z.-H. (2018). Super congruences for two Apéry-like sequences. Journal of Difference Equations and Applications, 24(10), 1685–1713.
  13. Sun, Z.-H. (2020). Congruences involving binomial coefficients and Apéry-like numbers. Publicationes Mathematicae Debrecen, 96(3–4), 315–346.
  14. Sun, Z.-H. (2022). Supercongruences involving Apéry-like numbers and binomial  coefficients. AIMS Mathematics, 7(2), 2729–2781.
  15. Sun, Z.-H. (2025). Binomial Coefficients, Recurrence Sequences and Congruences. Science Press, Beijing.
  16. Tauraso, R. (2010). Congruences involving alternating multiple harmonic sums. Electronic Journal of Combinatorics, 17, Article ID R16.
  17. Tauraso, R. (2018). Supercongruences related to 3F2(1) involving harmonic numbers. International Journal of Number Theory, 14(4), 1093–1109.
  18. Tauraso, R. (2020). A supercongruence involving cubes of Catalan numbers. Integers, 20, Article ID A44.
  19. Van Hamme, L. (1987). Proof of a conjecture of Beukers on Apéry numbers. Proceedings of the Conference on p-adic Analysis (De Grande-De Kimpe, N., & van Hamme, L. (eds.)). Houthalen, 189–195, Vrije Univ. Brussel, Brussels, 1986.

Manuscript history

  • Received: 3 May 2025
  • Revised: 3 September 2025
  • Accepted: 14 September 2025
  • Online First: 25 September 2025

Copyright information

Ⓒ 2025 by the Author.
This is an Open Access paper distributed under the terms and conditions of the Creative Commons Attribution 4.0 International License (CC BY 4.0).

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Cite this paper

Sun, Z.-H. (2025). Congruences for the Apéry numbers modulo p3. Notes on Number Theory and Discrete Mathematics, 31(3), 646-657, DOI: 10.7546/nntdm.2025.31.3.646-657.

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