Small Prime Solutions to Cubic Diophantine Equations II
Small Prime Solutions to Cubic Diophantine Equations II
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DOI：
10.4153/cmb-2015-079-6
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发表时间：
2016-09
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影响因子：
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通讯作者：
Zhixin Liu
中科院分区：
文献类型：
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作者：
Zhixin Liu
设${a}_{1}，\，.\，，\，{{a}_{9}}$为非零整数，$n为任意整数。假设${a}_{1}}\，+\，.\，.\，+\，{{a}_{9}}\，\等价\，n$$\Left(\bmod\，2\right)$和$\Left({a}_{i}}，\，{a}_{i}}\right)\，=\，$1\，\le\，i\，<；\，j\le\，9$。本文证明了：(I)如果${a}_{j}$不都是同号的，则三次方程${a}_{1}p_{1}^{3}\，+\，.\，.\，+\，{a}_{9}}p_{9}^{3}，=，n$有素数解满足${p}_{j}}，\ll\，{{\Left|n\Right|}^{{1}/{3}\；}}\，+\，\max{{\Left\{\Left|{{a}_{j}}\Right|\Right\}^{8+\varepsilon}}$；(Ii)如果所有${a}_{j}$都为正，且$n\，\gg\，\max{{\Left\{\Left|{{a}_{j}}\Right|\Right\}^{25+\varepsilon}}$，则${a}_{1}p_{1}^{3}\，+\，.\，.\，+\，{{a}_{j}}p_{9}^{3}\，=\，N$可溶于素数$Pj$。这些结果改进了我们以前的结果，得到了下界$\max{{\Left\{\Left|{{a}_{j}}\Right|\Right\}^{14+\varepsilon}}$和$\max\，{{\Left\{\Left|{{a}_{j}}\Right|\Right\}^{43+\varepsilon}}$分别代替上面的$\max{{\Left\{\Left|{{a}_{j}}\Right|\Right\}^{8+\varepsilon}}$和$\max{{\Left\{\Left|{{a}_{j}}\Right|\Right\}^{25+\varepsilon}}$。
Abstract Let ${{a}_{1}},\,.\,.\,.\,,\,{{a}_{9}}$ be non-zero integers and $n$ any integer. Suppose that ${{a}_{1}}\,+\,.\,.\,.\,+\,{{a}_{9}}\,\equiv \,n$ $\left( \bmod \,2 \right)$ and $\left( {{a}_{i}},\,{{a}_{i}} \right)\,=\,1$ for $1\,\le \,i\,<\,j\le \,9$ . In this paper we prove that (i) if ${{a}_{j}}$ are not all of the same sign, then the cubic equation ${{a}_{1}}p_{1}^{3}\,+\,.\,.\,.\,+\,{{a}_{9}}p_{9}^{3}\,=\,n$ has prime solutions satisfying ${{p}_{j}}\,\ll \,{{\left| n \right|}^{{1}/{3}\;}}\,+\,\max {{\left\{ \left| {{a}_{j}} \right| \right\}}^{8+\varepsilon }}$ ; (ii) if all ${{a}_{j}}$ are positive and $n\,\gg \,\max {{\left\{ \left| {{a}_{j}} \right| \right\}}^{25+\varepsilon }}$ , then ${{a}_{1}}p_{1}^{3}\,+\,.\,.\,.\,+\,{{a}_{j}}p_{9}^{3}\,=\,n$ is soluble in primes $Pj$ . These results improve our previous results with the bounds $\max {{\left\{ \left| {{a}_{j}} \right| \right\}}^{14+\varepsilon }}$ and $\max \,{{\left\{ \left| {{a}_{j}} \right| \right\}}^{43+\varepsilon }}$ in place of $\max {{\left\{ \left| {{a}_{j}} \right| \right\}}^{8+\varepsilon }}$ and $\max {{\left\{ \left| {{a}_{j}} \right| \right\}}^{25+\varepsilon }}$ above, respectively.