Proof of Some Congruence Conjectures of Z.-H. Sun

Abstract

In this paper, we mainly prove some conjectural congruences of Z.-H. Sun involving Almkvist–Zudilin numbers

b_n=\sum _{k=0}^{\lfloor \frac{n}{3}\rfloor }\left(\begin{array}{c}2k\\ k\end{array}\right)\left(\begin{array}{c}3k\\ k\end{array}\right)\left(\begin{array}{c}n\\ 3k\end{array}\right)\left(\begin{array}{c}n+k\\ k\end{array}\right){\left(-3\right)}^{n-3k}.

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$${b_n} = \sum\limits_{k = 0}^{\left\lfloor {{n \over 3}} \right\rfloor} {\left({\matrix{{2k} \cr k \cr}} \right)\left({\matrix{{3k} \cr k \cr}} \right)} \left({\matrix{n \cr {3k} \cr}} \right)\left({\matrix{{n + k} \cr k \cr}} \right){\left({- 3} \right)^{n - 3k}}.$$\end{document}

Let p > 3 be a prime. If p ≡ 3 (mod 4), then

Keywords

Congruences / Apéry-like numbers / harmonic numbers / Legendre symbol / 11A07 / 11B65 / 11B68

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Peking University