Linnik型华林-哥德巴赫问题

含2的幂次的哥德巴赫问题的研究起始于著名数学家Linnik于1951-1953年的经典论文。特别，Linnik证明了充分大的偶数可以表示成两个素数与若干个2的幂次之和。该结论有时被称为几乎哥德巴赫定理。1999年，刘建亚、廖明哲和展涛得到了对应的含2的幂次的四素数平方和定理，即存在一个非负整数k，使得每个大偶数都可以表成四个素数的平方和加上k个2的幂次。这里，对2的幂次的数目的改进得到了广泛的关注。同时，在问题的处理中如何更充分的利用2的幂次这个信息是一个值得关心的问题。2的幂次是一个非常稀疏的序列，本项目侧重于研究在Linnik型华林-哥德巴赫问题中如何处理2的幂次这个稀疏序列。通过发展新的技巧和思想去利用2的幂次的性质，进而改进已有的方法和已有的结论。

The Goldbach problem involving powers of 2 was first investigaed by Linnik in his seminar papers in 1951-1953. In particular, Linnik established that every sufficiently large even integer can be represented as the sum of two primes and K powers of 2, where K is an absolutely constant. The conclusion was reviewed as an almost Goldbach theorem. In 1999, Liu, Liu and Zhan investigated the reppresentation of large even integers as the sum of four squres of primes and k powers of 2. The number of powers of 2 in the above two problems was improved in a series of papers. Furthermore, how to make use of the information on powers of 2 plays a key role in these problems. The power of 2 is a very thin sequence, and this project will focus on how to make full use of this thin sequence in the proof. By developing new techniques and new ideas, we wish to improve some known conclusions.