Exceptional Sets for Sums of Prime Cubes in Short Interval

Gongrui Chen,Jie Wu

doi:10.1155/2021/5255880

Abstract

Let E N , X denote the number of even integers n , with N − X ≤ n ≤ N , such that n cannot be written as n = p 1 3 + ⋯ + p 8 3 . We prove that if X > N 1 / 36 + ɛ , then E N , X = o X .

Highlights

Introduction e WaringGoldbach problem is to study the representation of positive integers as sums of powers of prime numbers
Roughout, we assume that N is a large natural number, and X < N
Zhao [2] proved that E8(N) ≪ N1/6+ε, which implies E(N, X) o(X) if X > N1/6+ε. e main result in the paper is as follows

Summary

Preliminaries

Before we prove eorem 1, we introduce the following theorem. We can get eorems 1 from 2 immediately. Let R(n) be the weighted number of solutions of n p31 + · · · + p36 + p37 + p38 with U ≤ p1, . We can find the proof in ([2], Section 9). Let S be the number of solutions of. On recalling the set ξ(N, N5/6) defined in Section 1 and by means of an argument of Wooley (see [4]), we have. Is completes the proof of eorem 2

Proof of Lemma 3

Findings

Proof of Lemma 1