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Small Prime Solutions to Cubic Diophantine Equations II


Published:20160413
Printed: Sep 2016
Zhixin Liu,
Department of Mathematics, School of Science, Tianjin University, Tianjin 300072, P. R. China
Abstract
Let $a_1, \cdots, a_9$ be nonzero 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), 785794)
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.