Proof of Some Conjectural Congruences Involving Binomial Coefficients and Apéry-like Numbers

Abstract

In this paper, we mainly establish some congruences involving binomial coefficients and Apéry-like numbers, for example, we prove the following result which was conjectured by Z.-H. Sun: Let p > 3 be a prime. Then

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\sum_{k=0}^{p-1}\left(\begin{array}{c}2k\\ k\end{array}\right)\frac{W_k}{(-12)^k}\equiv\begin{cases}L^2-2p\;({\rm{mod}}\;p^2) & {\rm{if}}\;p\equiv1\;({\rm{mod}}\;3)\;\& \;4p=L^2+27M^2,\\0\;({\rm{mod}}\;p^2) & {\rm{if}}\;p\equiv2\;({\rm{mod}}\;3),\end{cases}$$\end{document}

where L, M are integers and

W_n=\sum \begin{array}{c}\lfloor \frac{n}{3}\rfloor \\ k=0\end{array}\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)(-3{)}^{n-3k}

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$W_n=\sum\begin{array}{c}\lfloor\frac{n}{3}\rfloor\\k=0\end{array}\left(\begin{array}{c}2k\\ k\end{array}\right)\left(\begin{array}{c}3k\\k\end{array}\right)\left(\begin{array}{c}n\\ 3k\end{array}\right)(-3)^{n-3k}$$\end{document}

are the second kind Apéry-like numbers.

Keywords

Congruences / binomial coefficients / harmonic numbers / binary quadratic forms / Apéry-like numbers / 05A10 / 11A07 / 11E25

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