零和序列与组合同余式

Due to the promotion of the work of the famous mathematician Erdos, Gowers, Alon, Green and Tao, combinatorial number theory has become a very active branch of mathematics. Zero-sum problems and combinatorial congruences are important topics in combinatorial number theory. The main research topics of this project are as follows:zero-sum problems such as Bialostocki's conjecture and the extension of the weighted EGZ theorem over a finite group, congruences involving central trinomial coefficients, Motzkin numbers and Apéry-like numbers, the p-adic order of the combinatorial sum of Stirling numbers. The tools to be used include the weighted EGZ theorem, combinatorial transformations, Legendre polynomial, Lucas congruence and p-adic analysis. The applicant has published 7 papers in international journals indexed in SCI such as Adv. in Appl. Math..

在Erdos，Gowers，Alon，Green，Tao等著名数学家工作的推动下，组合数论已成为非常活跃的数学分支。零和问题与组合同余式都是组合数论的重要课题。本项目致力于研究Bialostocki猜想与加权EGZ定理在有限群上的推广等零和问题，涉及中心三项式系数、Motzkin数及Apéry型数的组合同余式，Stirling数缺项和的p-adic阶数等。拟使用的工具包括加权EGZ定理、组合变换、Legendre多项式、Lucas同余式、p-adic分析等。申请人已在《Adv. in Appl. Math.》等SCI源刊物上发表了7 篇论文。

项目研究期间，我们共发表了 6 篇论文，其中 5 篇被 SCI 收录。中心三项式系数T_n表示(1+x+x^{-1})^n展开式中的常数项。对大于3的素数p，我们决定了T_{p-1}模p^2和Σ_{0≤k≤p-1}(-1)^kT_k模p^2，证明了几个孙智伟猜想的与Lucas序列有关的组合同余式，还得到了几个有关二项式系数和Lucas序列的恒等式。此外，我们还对Z/pZ上受限制子集和问题及整数集上的子集和问题进行了研究，并取得了一些新的结果。