偶数的哥德巴赫猜想是至今仍未解决的数学难题之一，而Linnik问题作为哥德巴赫猜想的衍生问题在研究哥德巴赫猜想的过程中发挥着十分重要的作用。张益唐关于指数和估计的新思想和Wooley关于Vinogradov均值定理的新进展及一些数论中新研究的出现，给Linnik问题的研究提供了重要的思路。本项目主要研究如何将这些新思想和新方法应用于三次及混合型Linnik相关问题中以改进相关结果。具体研究内容包括：指数和估计、Vinogradov均值定理、三次型Linnik问题、一对整数表为三次及混合型素数和2的方幂问题、关于三次及混合型素数和2的方幂的丢番图问题等。此外，作为本项目的后续课题，我们还将研究如何将这些新思想、新方法的应用到素数分布的研究中去。

The even Goldbach conjecture is one of the mathematical conundrums remains unresolved. To research on it, its derivative --Linnik problem plays a very important role. With the emergence of Zhang Yitang’s new ideas about the exponential sum estimation and Wooley’s new advances on Vinogradov mean value theorem, a new thinking about Linnik problem is important and necessary. The main purpose of this project is to research how to apply these new ideas and new developments to cube and hybrid Linnik related problems. To be specific, we will research on exponential sum estimation, Vinogradov mean value theorem, cube Linnik problem, a pair of integers represented as cube or hybrid primes and powers of 2, Diophantine approximation with cube or hybrid primes and powers of 2, etc. In addition, our further work will be about how to apply these new developments and new methods to the study of the distribution of prime numbers, as a follow-up of this project.