Proof of some conjectural congruences involving Apéry and Apéry-like numbers

Pub Date : 2024-03-07 DOI:10.1017/s0013091524000075

Guo-shuai Mao, Lilong Wang

引用次数: 0

Abstract

In this paper, we mainly prove the following conjectures of Sun [16]: Let p > 3 be a prime. Then Abstract Image \begin{align*}&A_{2p}\equiv A_2-\frac{1648}3p^3B_{p-3}\ ({\rm{mod}}\ p^4),\\&A_{2p-1}\equiv A_1+\frac{16p^3}3B_{p-3}\ ({\rm{mod}}\ p^4),\\&A_{3p}\equiv A_3-36738p^3B_{p-3}\ ({\rm{mod}}\ p^4),\end{align*}

where Abstract Image $A_n=\sum_{k=0}^n\binom{n}k^2\binom{n+k}{k}^2$ is the nth Apéry number, and Bn is the nth Bernoulli number.

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涉及阿佩里数和类阿佩里数的一些猜想全等的证明

本文主要证明 Sun [16] 的下列猜想：设 p > 3 是素数。Then\begin{align*}&A_{2p}\equiv A_2-\frac{1648}3p^3B_{p-3}\ ({\rm{mod}}\ p^4),\&A_{2p-1}\equiv A_1+\frac{16p^3}3B_{p-3}\ ({\rm{mod}}\ p^4),\&；A_{3p}equiv A_3-36738p^3B_{p-3}\ ({\rm{mod}\ p^4),end{align*}where $A_n=\sum_{k=0}^n\binom{n}k^2\binom{n+k}{k}^2$ is the nth Apéry number, and Bn is the nth Bernoulli number.

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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