Small Prime Solutions to Cubic Diophantine Equations II

Liu, Zhixin

Canadian mathematical bulletin, Tome 59 (2016) no. 3, pp. 599-605

Résumé

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.

Détail

DOI : 10.4153/CMB-2015-079-6

Mots-clés : 11P32, 11P05, 11P55, small prime, Waring-Goldbach problem, circle method