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

Author:

Mao Guo-Shuai,Wang Lilong

Abstract

AbstractIn this paper, we mainly prove the following conjectures of Sun [16]: Let p > 3 be a prime. 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.

Publisher

Cambridge University Press (CUP)

Reference21 articles.

1. On the residues of the sums of products of the first p − 1 numbers, and their powers, to modulus p2 or p3;Glaisher;Quart. J. Math.,1900

2. On some congruences involving Domb numbers and harmonic numbers;Mao;Int. J. Number Theory,2019

3. Quasi-symmetric functions and mod p multiple harmonic sums;Hoffman;Kyushu J. Math.,2015

4. Super congruences and Euler numbers;Sun;Sci. China Math.,2011

同舟云学术

1.学者识别学者识别

2.学术分析学术分析

3.人才评估人才评估

"同舟云学术"是以全球学者为主线,采集、加工和组织学术论文而形成的新型学术文献查询和分析系统,可以对全球学者进行文献检索和人才价值评估。用户可以通过关注某些学科领域的顶尖人物而持续追踪该领域的学科进展和研究前沿。经过近期的数据扩容,当前同舟云学术共收录了国内外主流学术期刊6万余种,收集的期刊论文及会议论文总量共计约1.5亿篇,并以每天添加12000余篇中外论文的速度递增。我们也可以为用户提供个性化、定制化的学者数据。欢迎来电咨询!咨询电话:010-8811{复制后删除}0370

www.globalauthorid.com

TOP

Copyright © 2019-2024 北京同舟云网络信息技术有限公司
京公网安备11010802033243号  京ICP备18003416号-3