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# Small Prime Solutions to Cubic Diophantine Equations

Published:2012-08-27
Printed: Dec 2013
• Zhixin Liu,
Department of Mathematics, School of Science, Tianjin University, Tianjin 300072, P. R. China
 Format: LaTeX MathJax PDF

## Abstract

Let $a_1, \cdots, a_9$ be non-zero integers and $n$ any integer. Suppose that $a_1+\cdots+a_9 \equiv n( \textrm{mod}\,2)$ 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 cubic equation has prime solutions satisfying $p_j \ll |n|^{1/3}+\textrm{max}\{|a_j|\}^{14+\varepsilon};$ and (ii) if all $a_j$ are positive and $n \gg \textrm{max}\{|a_j|\}^{43+\varepsilon}$, then the cubic equation $a_1p_1^3+\cdots +a_9p_9^3=n$ is soluble in primes $p_j$. This result is the extension of the linear and quadratic relative problems.
 Keywords: small prime, Waring-Goldbach problem, circle method
 MSC Classifications: 11P32 - Goldbach-type theorems; other additive questions involving primes 11P05 - Waring's problem and variants 11P55 - Applications of the Hardy-Littlewood method [See also 11D85]

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