Small prime solutions of a nonlinear equation

JOURNAL OF NUMBER THEORY

卷:185

Article

Liu, Zhixin¹ 

[1] Tianjin Univ, Sch Math, Tianjin 300072, Peoples R China

关键词: Small prime;    Waring-Goldbach problem;    Circle method;   

DOI  :  10.1016/j.jnt.2017.09.014

来源: Elsevier

【 摘 要 】

Let a(1), ... , a(4) be non-zero integers and n any integer. Suppose that a(1), ... , a(4) and n satisfy some related conditions. In this paper we prove that (i) if a(j) are not all of the same sign, then the equation a(1)p(1) + a(2)p(2)(2)+a(3)p(3)(2)+a(4)p(4)(2)=n has prime solutions satisfying max{p(1),p(2)(2),p(3)(2),p(4)(2)} << I vertical bar n vertical bar + max{vertical bar a(j)vertical bar}(14+epsilon); (ii) if all a(j) are positive and n >> max{vertical bar a(j)vertical bar}(15+epsilon), then the equation a(1)p(1) + a(2)p(2)(2) + a(3)p(3)(2) + a(4)p(4)(2) = n is soluble in primes p(j). (C) 2017 Elsevier Inc. All rights reserved.

【 授权许可】

Free