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Small Prime Solutions to Cubic Diophantine Equations II

  Published:2016-04-13
 Printed: Sep 2016
  • Zhixin Liu,
    Department of Mathematics, School of Science, Tianjin University, Tianjin 300072, P. R. China
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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 cubic equation $a_1p_1^3+\cdots +a_9p_9^3=n$ has prime solutions satisfying $p_j \ll |n|^{1/3}+\textrm{max}\{|a_j|\}^{8+\varepsilon};$ (ii) if all $a_j$ are positive and $n \gg \textrm{max}\{|a_j|\}^{25+\varepsilon}$, 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 $\textrm{max}\{|a_j|\}^{14+\varepsilon}$ and $\textrm{max}\{|a_j|\}^{43+\varepsilon}$ in place of $\textrm{max}\{|a_j|\}^{8+\varepsilon}$ and $\textrm{max}\{|a_j|\}^{25+\varepsilon}$ above, respectively.
Keywords: small prime, Waring-Goldbach problem, circle method small prime, Waring-Goldbach problem, circle method
MSC Classifications: 11P32, 11P05, 11P55 show english descriptions Goldbach-type theorems; other additive questions involving primes
Waring's problem and variants
Applications of the Hardy-Littlewood method [See also 11D85]
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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