An improvement on Waring–Goldbach problem for unlike powers

Abstract

Let pi be prime numbers. In this paper, it is proved that for any integer k≧5, with at most \(O\big(N^{1-\frac{1}{3k\times2^{k-2}}+\varepsilon}\big)\) exceptions, all positive even integers up to N can be expressed in the form \(p_{2}^{2}+p_{3}^{3}+p_{5}^{5}+p_{k}^{k}\). This improves the result \(O\big(\frac{N}{\log^{c}N}\big)\) for some c>0 due to Lu and Shan [12], and it is a generalization for a series of results of Ren and Tsang [15], [16] and Bauer [1–4] for the problem in the form \(p_{2}^{2}+p_{3}^{3}+p_{4}^{4}+p_{5}^{5}\). This method can also be used for some other similar forms.