Congruences for harmonic sums

Yining Yang and Peng Yang
Notes on Number Theory and Discrete Mathematics
Print ISSN 1310–5132, Online ISSN 2367–8275
Volume 29, 2023, Number 1, Pages 137–146
DOI: 10.7546/nntdm.2023.29.1.137-146
Full paper (PDF, 237 Kb)

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Authors and affiliations

Yining Yang
School of Science, University of Science and Technology Liaoning
Anshan, Liaoning 114051, P. R. China

Peng Yang
School of Science, University of Science and Technology Liaoning
Anshan, Liaoning 114051, P. R. China

Abstract

Zhao found a curious congruence modulo p on harmonic sums. Xia and Cai generalized his congruence to a supercongruence modulo p^2. In this paper, we improve the harmonic sums

    \[ H_{p}(n)=\sum\limits_{\substack{l_{1}+l_{2}+\cdots+l_{n}=p\\ l_{1}, l_{2}, \ldots , l_{n}>0}} \frac{1}{l_{1} l_{2} \cdots l_{n}} \]

to supercongruences modulo p^3 and p^4 for odd and even where prime p>8 and 3 \leq n \leq p-6.

Keywords

  • Harmonic sums
  • Bernoulli numbers
  • Congruence

2020 Mathematics Subject Classification

  • 11A25

References

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  2. Ireland, K., & Rosen, M. (1990). A Classical Introduction to Modern Number Theory. Springer, New York.
  3. Meštrović, R. (2013). Some Wolstenholme type congruences. Mathematics for Applications, 2, 35–42.
  4. Shen, Z. (2011). Some congruences involving multiple harmonic sums. Journal of Zhejiang International Studies University, 5, 95–99.
  5. Sun, Z. (2000). Congruences concerning Bernoulli numbers and Bernoulli polynomials. Discrete Applied Mathematics, 105(1), 193–223.
  6. Xia, B., & Cai, T. (2010). Bernoulli numbers and congruences for harmonic sums. International Journal of Number Theory, 6(4), 849–855.
  7. Zhao, J. (2006). Bernoulli numbers, Wolstenholme’s theorem, and 𝑝5 variations of Lucas’ theorem. Journal of Number Theory, 123(1), 18–26.
  8. Zhou, X., & Cai, T. (2007). A generalization of a curious congruence on harmonic sums. Proceedings of the American Mathematical Society, 135(5), 1329–1333.

Manuscript history

  • Received: 23 November 2022
  • Revised: 9 March 2023
  • Accepted: 20 March 2023
  • Online First: 21 March 2023

Copyright information

Ⓒ 2023 by the Authors.
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

Yang, Y., & Yang, P. (2023). Congruences for harmonic sums. Notes on Number Theory and Discrete Mathematics, 29(1), 137-146, DOI: 10.7546/nntdm.2023.29.1.137-146.

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