带p-adic Gamma函数的组合同余式是目前组合数论的研究热点之一，它们在椭圆曲线和模形式的研究中有着重要的意义。本项目将从Apéry数之和与交错和、超几何级数部分和两个方面来研究组合同余式，揭示它们与p-adic Gamma函数之间的超同余式关系，为椭圆曲线和模形式研究提供更多理论依据。主要研究内容包括：研究孙智伟教授有关Apéry数之和与交错和超同余式猜想，进而揭示它们与p-adic Gamma函数之间的超同余式关系；建立van Hamme的(A.2)和(H.2)超同余式的推广形式，解决或推动郭军伟教授有关(B.2)和(C.2)超同余式推广的猜想；研究Rodriguez-Villegas有关3F2(1)部分和超同余式的推广形式；研究孙智宏教授有关3F2(m)部分和超同余式的推广形式，并揭示它们与p-adic Gamma函数之间的超同余式关系。

Combinatorial congruences involving p-adic Gamma functions are one of the current research focuses in combinatorial number theory and play an important role in the study of elliptic curves and modular forms. This project focuses on combinatorial congruences on sums and alternating sums of Apéry numbers and partial sums of hypergeometric series, establishes supercongruences between these sums and p-adic Gamma functions, and provides more theoretical basis for the study of elliptic curves and modular forms. What we plan to do in this project are as follows: study Sun's supercongruence conjectures on sums and alternating sums of Apéry numbers, and establish supercongruences between these sums and the p-adic Gamma functions; set up generalizations of van Hamme's (A.2) and (H.2) supercongruences, solve or make progress on Guo's conjectural generalizations of (B.2) and (C.2) supercongruences; study refinements of Rodriguez-Villegas supercongruences on partial sums of 3F2(1); study some generalizations of Sun's supercongruences on partial sums of 3F2(m), and establish supercongruences between these sums and the p-adic Gamma functions.