The values of cubic forms at prime arguments

JOURNAL OF NUMBER THEORY

卷:180

Article

Ge, Wenxu¹  Zhao, Feng¹ 

[1] North China Univ Water Resources & Elect Power, Sch Math & Informat Sci, Zhengzhou 450046, Henan, Peoples R China

关键词: Diophantine inequalities;    Primes;    Exceptional set;    Cubic forms;   

DOI  :  10.1016/j.jnt.2017.05.014

来源: Elsevier

【 摘 要 】

Let lambda(1), lambda(2), lambda(3), lambda(4), lambda(5) be non-zero real numbers, not all negative. Let nu be a well-spaced sequence, delta > 0. If lambda(1)/lambda(2) is irrational and algebraic, then we prove that E(nu, X, delta) << X17/18+2 delta+epsilon , where E(nu, X, delta) denotes the number of nu is an element of nu with nu <= X such that the inequality |lambda(1)p(1)(3) + lambda(2)p(2)(3) + lambda(3)p(3)(3) + lambda(4)p(4)(3) + lambda(5)p(5)(3) - nu| < upsilon(-delta) has no solution in primes p(1), p(2), p(3), p(4), p(5). Further, we assume that except for one, all other the ratios lambda(k)/lambda(l) (1 <= k < l <= 5) are irrational and algebraic, then 17/18 can be replaced by 11/12. These improve the earlier results. (C) 2017 Elsevier Inc. All rights reserved.

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