Small Prime Solutions to Cubic Diophantine Equations II

Canadian mathematical bulletin

Zhixin Liu¹ 

[1] Department of Mathematics, School of Science, Tianjin University, Tianjin 300072, P. R. China

关键词: small prime;    Waring-Goldbach problem;    circle method;   

DOI  :  10.4153/CMB-2015-079-6

学科分类：数学（综合）

来源: University of Toronto Press * Journals Division

【 摘 要 】

Let $a_1, cdots, a_9$ be non-zero integers and $n$ any integer.Supposethat $a_1+cdots+a_9 equiv n( extrm{mod},2)$ and $(a_i, a_j)=1$for $1 leq i lt j leq 9$.In this paper we prove that(i) if $a_j$ are not all of the same sign, then the cubicequation $a_1p_1^3+cdots +a_9p_9^3=n$ has prime solutions satisfying$p_j ll |n|^{1/3}+extrm{max}{|a_j|}^{8+varepsilon};$(ii) if all $a_j$ are positive and $n gg extrm{max}{|a_j|}^{25+varepsilon}$,then$a_1p_1^3+cdots +a_9p_9^3=n$ is soluble in primes $p_j$.This results improve our previous results (Canad. Math. Bull.,56 (2013), 785-794)with the bounds $extrm{max}{|a_j|}^{14+varepsilon}$ and$extrm{max}{|a_j|}^{43+varepsilon}$in place of $extrm{max}{|a_j|}^{8+varepsilon}$ and $extrm{max}{|a_j|}^{25+varepsilon}$above, respectively.

【 授权许可】

Unknown