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A Generalized Supercongruence of Z.-W. Sun

Wei-Wei Qi

Abstract

In this paper, we employ the Wilf-Zeilberger (WZ) method to prove a supercongruence conjecture posed by Z.-W. Sun: for any prime $p$, \begin{align*} \sum_{k=0}^{\frac{p-3}{2}}\frac{92k^2+61k+9}{(2k+1)64^k}{2k \choose k}{3k \choose k}{4k \choose 2k}\equiv 6p+16p^2\left(\frac{-1}{p}\right) \pmod{p^3}, \end{align*} where $\left(\frac{\cdot}{p}\right)$ denotes the Legendre symbol. Our proof relies on combinatorial identities and symbolic summation techniques.

A Generalized Supercongruence of Z.-W. Sun

Abstract

In this paper, we employ the Wilf-Zeilberger (WZ) method to prove a supercongruence conjecture posed by Z.-W. Sun: for any prime , \begin{align*} \sum_{k=0}^{\frac{p-3}{2}}\frac{92k^2+61k+9}{(2k+1)64^k}{2k \choose k}{3k \choose k}{4k \choose 2k}\equiv 6p+16p^2\left(\frac{-1}{p}\right) \pmod{p^3}, \end{align*} where denotes the Legendre symbol. Our proof relies on combinatorial identities and symbolic summation techniques.
Paper Structure (5 sections, 6 theorems, 85 equations)

This paper contains 5 sections, 6 theorems, 85 equations.

Table of Contents

  1. Introduction
  2. Some Lemmas
  3. Proof of the Theorem 1.1
  4. Proof of the Theorem 1.2
  5. Proof of the Corollary 1.3

Key Result

Theorem 1.1

For any prime integer $p>2$. Then

Theorems & Definitions (6)

  • Theorem 1.1
  • Theorem 1.2
  • Corollary 1.3
  • Lemma 2.1
  • Lemma 2.2
  • Lemma 2.3