The Ramanujan Journal

, Volume 48, Issue 2, pp 233–244 | Cite as

Proof of some congruence conjectures of Guo and Liu

  • Guo-Shuai MaoEmail author


Let n and r be positive integers. Define the numbers \(S_{n}^{(r)}\) by \(S_{n}^{(r)}=\sum _{k=0}^n\left( {\begin{array}{c}n\\ k\end{array}}\right) ^2\left( {\begin{array}{c}2k\\ k\end{array}}\right) (2k+1)^r.\) In this paper we prove some conjectures of Guo and Liu which extend some conjectures of Sun (Two new kinds of numbers and related divisibility results, 2014), such as: There exist integers \(a_{2r-1}\) and \(b_r\), independent of n, such that
$$\begin{aligned} a_{2r-1}\sum _{k=0}^{n-1}S_k^{(2r-1)}\equiv 0 \, (\mathrm{mod}\, n^2) \quad \text{ and } \quad b_r\sum _{k=0}^{n-1}kS_k^{(r)}\equiv 0\, (\mathrm{mod}\, n^2). \end{aligned}$$
By the Zeilberger algorithm, we find that for all \(0\le j<n\),
$$\begin{aligned} (2j+1)\left( {\begin{array}{c}2j\\ j\end{array}}\right) \sum _{k=j}^{n-1}(2k-j+1)\left( {\begin{array}{c}k\\ j\end{array}}\right) ^2\equiv 0\, (\mathrm{mod}\, n^2). \end{aligned}$$


Central binomial coefficients Congruences Bernoulli numbers Zeilberger algorithm 

Mathematics Subject Classification

11B65 11B68 05A10 11A07 



The author would like to thank Prof. Z.-W. Sun, Prof. Hao Pan and the referees for helpful comments.


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Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of MathematicsNanjing UniversityNanjingPeople’s Republic of China

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