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Small Prime Solutions to Cubic Diophantine Equations


Published:20120827
Printed: Dec 2013
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 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 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.