丢番图方程的素数解

The research on prime solutions of Diophantine equations is an interdisciplinary study, which covers two of the most important fields in number theory, Diophantine equations and the distribution of primes. The study on this topic is of great theoretical significance. We intend to study the diagonal Diophantine equations and the general Diophantine equations. Especially, unlike the diagonal Diophantine equations, the general Diophantine equations may have many cross terms. The research on prime solutions of diagonal Diophantine equations, or the so-called Waring-Goldbach problems, is badly in need of new methods and ideas. To deal with this problem, we plan to apply the new technique of Linnik's dispersion method. Moreover, there are only a few exploratory results on prime solutions of the general Diophantine equations, so that we aim to get some much stronger new results in this direction. Through this project, we plan to combine the new technology of Linnik's dispersion method, with automorphic forms and the classical analytic methods, in order to get some influential original results on the above two topics of prime solutions of Diophatine equations. It should be pointed out that there is a close relation between this project and the Sarnak conjecture which has been attracting increasing attentions recently. Therefore, this project is a popular research standing at the frontier of analytic number theory.

丢番图方程素数解的研究是丢番图方程和素数分布这两个数论重要研究领域的交叉领域，具有一定的理论意义。本项目拟研究的丢番图方程包括对角形丢番图方程和可能含有交叉项的一般丢番图方程。前者素数解的研究（即Waring-Goldbach问题）亟需新方法和新思想，本项目拟采用Linnik方差法的新技术来研究这一问题；而后者素数解的研究目前仅有一些探索性的结果，本项目力图得到新的更深刻的结果。计划通过本项目的实施，将Linnik方差法的新技术、自守形式理论与经典解析方法相结合，在上述两类丢番图方程素数解的研究领域取得有影响的原创性成果。本研究课题与已引起普遍关注的Sarnak猜想研究具有密切的联系，属于解析数论领域的热门课题和前沿课题。

丢番图方程素数解的研究是丢番图方程和素数分布这两个数论重要研究领域的交叉领域，具有较强的理论意义，属于解析数论领域的热门课题和前沿课题。本项目研究的丢番图方程包括对角形丢番图方程和可能含有交叉项的一般丢番图方程。我们深入地研究了Waring-Goldbach相关的Goldbach-Linnik型问题、Baker常数问题、丢番图逼近问题以及相关的方程组问题等，这些问题都是经典对角型丢番图方程的素数解问题的深化和推广；在这些问题中，我们利用最新的方法和最新的技术，并结合经典解析数论方法，改进了已有结果。我们正在深入研究可能含有交叉项的一般三次丢番图方程素数解的结构，目前已经有了初步结果。我们的部分研究成果在上述两类丢番图方程素数解的研究领域取得了有影响的原创性成果。