On a System of Two Diophantine Inequalities with Five Prime Variables

Abstract

Suppose that c, d, α, β are real numbers satisfying the inequalities $1 < d < c < {39\over 37}$ and 1 < α < β < 5^1−d/c. In this paper, it is proved that, for sufficiently large real numbers N₁ and N₂ subject to $\alpha\leq {N_{2}\over N_{1}^{d/c}}\leq \beta$, the following Diophantine inequalities system $\begin{cases}|p_{1}^{c}+p_{2}^{c}+p_{3}^{c}+p_{4}^{c}+p_{5}^{c}-N_{1}|<\varepsilon_{1}(N_{1}),\\ |p_{1}^{d}+p_{2}^{d}+p_{3}^{d}+p_{4}^{d}+p_{5}^{d}-N_{2}|<\varepsilon_{2}(N_{2})\end{cases}$ is solvable in prime variables p₁, p₂, p₃, p₄, p₅, where $\begin{cases}\varepsilon_{1}(N_{1})=N_{1}^{-(1/c)(39/37-c)}(\log \ N_{1})^{201},\\ \varepsilon_{2}(N_{2})=N_{2}^{-(1/d)(39/37-d)}(\log\ N_{2})^{201}.\end{cases}$

This result constitutes an improvement upon a series of previous results of Zhai [Acta Arith., 2000, 92(1): 31–46] and Tolev [Acta Arith., 1995, 69(4): 387–400].

Keywords

Diophantine inequality / circle method / exponential sum / prime variable / 11D75 / 11P05 / 11L07 / 11L20

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