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