Small Prime Solutions to Cubic Diophantine Equations

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-2012-025-0

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 above cubicequation has prime solutions satisfying$p_j ll |n|^{1/3}+extrm{max}{|a_j|}^{14+varepsilon};$and (ii) if all $a_j$ are positive and $n gg extrm{max}{|a_j|}^{43+varepsilon}$, then the cubicequation $a_1p_1^3+cdots +a_9p_9^3=n$ is soluble in primes $p_j$.This result is the extension of the linear and quadratic relative problems.

【 授权许可】

Unknown