AN IMPROVEMENT ON WARING-GOLDBACH PROBLEM FOR UNLIKE POWERS

Let p(i) be prime numbers. In this paper, it is proved that for any integer k >= 5, with at most O (N(1-1/3kx2k-2+epsilon)) 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.