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 N 1− 1 3k×2 k−2 +ε 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 N log c N 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.