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On Two Congruences Involving Apéry and Franel Numbers

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Abstract

In this paper, we mainly prove a congruence conjecture of Z.-W. Sun involving Franel numbers: For any prime \(p>3\), we have

$$\begin{aligned} \sum _{k=0}^{p-1}(-1)^kf_k\equiv \left( \frac{p}{3}\right) +\frac{2p^2}{3}B_{p-2}\left( \frac{1}{3}\right) \pmod {p^3}, \end{aligned}$$

where \(B_n(x)\) is the n-th Bernoulli polynomial.

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Acknowledgements

The author would like to thank the anonymous referee for helpful comments. This work is funded by the National Natural Science Foundation of China (12001288) and the Startup Foundation for Introducing Talent of Nanjing University of Information Science and Technology (2019r062), and it is partially supported by the National Natural Science Foundation of China (12071208).

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  1. Department of Mathematics, Nanjing University of Information Science and Technology, Nanjing, 210044, People’s Republic of China

    Guo-Shuai Mao

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Mao, GS. On Two Congruences Involving Apéry and Franel Numbers. Results Math 75, 159 (2020). https://doi.org/10.1007/s00025-020-01291-4

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  • DOI: https://doi.org/10.1007/s00025-020-01291-4

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