若干组合同余式的研究

Discoveries and proofs of combinatorial congruencs form a core research topic in combinatorics and number theory. Various methods has been applied to this subject, such as analytic methods, algebraic methods, number theoretic methods and combinatorial methods. It is well known that computer proofs have been very successful in the fields such as proving and discovering combinatorial identities, mechanized proofs of geometric theorems and so on. Inspired by these facts, we will use computer proof methods to conduct further research on congruences of partition functions and combinatorial congruences whose summands contain hypergeometric terms, special functions and special combinatorial sequences. We aim to solve several conjectures on combinatorial congruences, which were proposed by some scholars such as Z.-W.Sun and Guillera and so on. We also aim to give computer proofs of some congruences of partition functions. At the same time, in order to solve more extensive and more complicated combinatorial congurences, we will continue to improve and expand current algorithms of computerproofs, and we will also explore new approaches and perspectives in the field of computer proof ways. Furthermore, we will explore combinatorial proofs of some combinatorial congruences which are simple in form and whose combinatorial meanings are explicit, by combining the method of “Combinatorial Telescoping” proposed by Chen, Hou and Sun and graph theory, group action theory.

组合同余式的发现与证明是组合数学和数论界的一个核心研究课题，其研究方法包括分析方法、代数方法、数论方法、组合方法和机器证明方法等。众所周知，机器证明方法在几何定理的机械化证明以及组合恒等式的证明和发现等领域获得了巨大成功。受此启发，本项目将采用机器证明的思想重点对求和式中含有超几何项、特殊函数和特殊组合序列的组合同余式猜想以及分拆同余式展开深入地研究，藉此解决孙智伟、Guillera等人提出的若干组合同余式猜想，以及对若干分拆同余式给出机器证明。同时，在利用机器证明方法研究这些同余式的过程中继续研究改进、拓展已有的机器证明算法，探索机器证明新的方法和观点，以便解决更广泛、更复杂的组合同余式。在此基础上，本项目将结合陈永川、侯庆虎和孙慧提出的“Combinatorial Telescoping”方法以及图论、群作用理论，针对若干形式简单、组合意义明确的组合同余式的组合证明进行探索。

组合同余式的发现与证明是组合数学和数论界的一个核心研究课题，其研究方法包括分析方法、代数方法、数论方法、组合方法和机器证明方法等。众所周知，机器证明方法在几何定理的机械化证明以及组合恒等式的证明和发现等领域获得了巨大成功。受此启发，本项目利用用机器证明和组合证明的思想重点对求和式中含有超几何项、特殊函数和特殊组合序列的组合同余式猜想进行了深入地研究。利用WZ方法给出了发现和证明组合同余式的新思路，并由此解决了孙智伟、Chow,Wiemann和Cooper等人提出的若干组合同余式猜想以及若干组合恒等式的猜想。同时，在利用机器证明方法研究这些同余式的过程中继续研究改进、拓展已有的机器证明算法，探索机器证明新的方法和观点，并由此解决了组合学中若干组合序列对数凸凹性问题。本项目所研究出来的新结果和新发现为促进组合学与数论的发展做了一些积极的贡献，同时项目研究过程中提出的一些新思想和新方法也为其他学者解决类似问题提供借鉴和参考。