Three supercongruences for Apery numbers or Franel numbers

TL;DR

This paper proves three supercongruences involving Apéry and Franel numbers, confirming conjectures by Z.-W. Sun, and extends understanding of their divisibility properties in number theory.

Contribution

The paper establishes three new supercongruences for Apéry and Franel numbers, advancing the theoretical understanding of their arithmetic properties and confirming conjectured congruences.

Findings

Proved a supercongruence involving Apéry numbers modulo p^{4+3 u_p(n)}.

Established a supercongruence for Apéry numbers involving cubic powers modulo p^{6+3 u_p(n)}.

Confirmed a supercongruence for Franel numbers involving alternating signs modulo p^{3}.

Abstract

The Ap\'ery numbers $A_n$ and the Franel numbers $f_n$ are defined by $$A_n=\sum_{k=0}^{n}{\binom{n+k}{2k}}^2{\binom{2k}{k}}^2\ \ \ \ \ {\rm and }\ \ \ \ \ \ f_n=\sum_{k=0}^{n}{\binom{n}{k}}^3(n=0, 1, \cdots,).$$ In this paper, we prove three supercongruences for Ap\'ery numbers or Franel numbers conjectured by Z.-W. Sun. Let $p\geq 5$ be a prime and let $n\in \mathbb{Z}^{+}$. We show that \begin{align} \notag \frac{1}{n}\bigg(\sum_{k=0}^{pn-1}(2k+1)A_k-p\sum_{k=0}^{n-1}(2k+1)A_k\bigg)\equiv0\pmod{p^{4+3\nu_p(n)}} \end{align} and \begin{align}\notag \frac{1}{n^3}\bigg(\sum_{k=0}^{pn-1}(2k+1)^3A_k-p^3\sum_{k=0}^{n-1}(2k+1)^3A_k\bigg)\equiv0\pmod{p^{6+3\nu_p(n)}}, \end{align} where $\nu_p(n)$ denotes the $p$-adic order of $n$. Also, for any prime $p$ we have \begin{align} \notag \frac{1}{n^3}\bigg(\sum_{k=0}^{pn-1}(3k+2)(-1)^kf_k-p^2\sum_{k=0}^{n-1}(3k+2)(-1)^kf_k\bigg)\equiv0\pmod{p^{3}}. \end{align}