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1. CMB 2012 (vol 56 pp. 785)
Small Prime Solutions to Cubic Diophantine Equations 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 Categories:11P32, 11P05, 11P55 |