关于素变量的丢番图问题

The Diophantine problem is one of the core topics in the field of number theory. This project intends to study the Diophantine problem with prime variables by using several methods such as the circle method, sieve method, etc. In the implementation of this project, we mainly study the following two aspects. The first aspect is Waring-Goldbach problem with mixed powers. We mainly focus on the cases of the representation of the cubic and quadratic powers with prime variables and almost-prime variables. Besides, we also consider the exceptional sets of these kinds of problems with all variables primes. The second aspect is Piatetski-Shapiro type Diophantine inequalities and Diophantine equations and their generalizations. The investigation of Waring-Goldbach problem with mixed powers involves both the method to estimate exponential sums over primes and the method to construct admissible exponents, while the study of Piatetski-Shapiro type Diophantine problems involves not only the ways to bound the exponential sums but also the choices of the parameters in sieve theory. During the research of this project, we shall further explore the method of constructing admissible exponents with higher powers and the method of applying Harman sieving process to deal with exponential sums estimates over primes. Especially, the former can be applied to the study of Waring-Goldbach problem with higher mixed powers, while the latter will break through the traditional patterns and greatly improve the previous results.

丢番图问题是数论领域中的核心课题之一。本项目拟利用圆法、筛法等工具研究素变数丢番图问题。在本项目实施过程中,我们主要研究以下两个方面。一是混合次幂的华林-哥德巴赫问题，重点研究含有素数变量与殆素数变量的三次和四次混合幂的全表问题,以及该问题在全素数变量下的例外集问题。二是关于Piatetski-Shapiro型的素变数丢番图不等式和素变数丢番图方程及其推广。研究混合次幂的华林-哥德巴赫问题,既涉及到处理素变数三角和的方法,又涉及到构造可允许指数组的方法。研究Piatetski-Shapiro型的丢番图问题不仅涉及到指数和的估计, 而且涉及筛法参数的选取。本项目研究过程中进一步探索构造更高次混合幂的可允许指数组的方法,以及将Harman筛法运用于处理指数和估计的方法。其中,前者可以应用于更高次混合幂的华林-哥德巴赫问题的研究;而后者将突破传统处理指数和的模式,进而可以极大地改进前人的结果。

丢番图问题是数论领域中的核心课题之一。本项目利用圆法、筛法等工具研究素变数丢番图问题。在本项目实施过程中,我们主要研究以下四个方面。一是混合次幂的华林-哥德巴赫问题，重点研究含有素数变量与殆素数变量的三次和四次混合幂的全表问题,以及该问题在全素数变量下的例外集问题。二是非齐次可允许指数组的构造及其应用。三是关于Piatetski-Shapiro型的素变数丢番图不等式和素变数丢番图方程及其推广。四是孪生素数猜想与Hardy-Littlewood猜想的相关逼近结果在Piatetski-Shapiro集合上的推广。研究混合次幂的华林-哥德巴赫问题,既涉及到处理素变数三角和的方法,又涉及到构造可允许指数组的方法。研究Piatetski-Shapiro型的丢番图问题不仅涉及到指数和的估计, 而且涉及筛法参数的选取。本项目研究过程中进一步探索构造更高次混合幂的可允许指数组的方法,以及研究Decoupling理论在处理Van der Corput型指数和估计方面的应用。其中,前者可以应用于更高次混合幂的华林-哥德巴赫问题的研究;而后者将突破传统处理指数和的模式,进而可以极大地改进前人的结果。