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On a Supercongruence Conjecture of Z.-W. Sun

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Abstract

In this paper, the author partly proves a supercongruence conjectured by Z.-W. Sun in 2013. Let p be an odd prime and let a ∈ ℤ+. Then, if p ≡ 1 (mod 3)

$$\sum\limits_{k = 0}^{\left\lfloor {{5 \over 6}{p^a}} \right\rfloor } {{{\left( {\matrix{{2k} \cr k \cr } } \right)} \over {{{16}^k}}} \equiv \left( {{3 \over {{p^a}}}} \right)\,\,\left( {\bmod \,{p^2}} \right)} $$

is obtained, where (÷) is the Jacobi symbol.

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Acknowledgement

The author would like to thank the anonymous referees for helpful comments.

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Authors and Affiliations

  1. Department of Mathematics, Nanjing University of Information Science and Technology, Nanjing, 210044, China

    Guo-shuai Mao

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  1. Guo-shuai Mao
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Correspondence to Guo-shuai Mao.

Additional information

This work was supported by the National Natural Science Foundation of China (Nos. 12001288, 12071208).

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Mao, Gs. On a Supercongruence Conjecture of Z.-W. Sun. Chin. Ann. Math. Ser. B 43, 417–424 (2022). https://doi.org/10.1007/s11401-022-0332-7

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  • DOI: https://doi.org/10.1007/s11401-022-0332-7

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