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Keywords:
Diophantine inequalities; Davenport-Heilbronn method; prime
Summary:
Let $k\geq 5$ be an odd integer and $\eta $ be any given real number. We prove that if $\lambda _1$, $\lambda _2$, $\lambda _3$, $\lambda _4$, $\mu $ are nonzero real numbers, not all of the same sign, and $\lambda _1/\lambda _2$ is irrational, then for any real number $\sigma $ with $0<\sigma <1/(8\vartheta (k))$, the inequality $$ |\lambda _1p_1^2+\lambda _2p_2^2+\lambda _3p_3^2+\lambda _4p_4^2+\mu p_5^k+ \eta |<\Bigl (\max _{1\leq j\leq 5} p_j\Bigr )^{-\sigma } $$ has infinitely many solutions in prime variables $p_1, p_2, \cdots , p_5$, where $\vartheta (k)=3\times 2^{(k-5)/2}$ for $k=5,7,9$ and $\vartheta (k)=[(k^2+2k+5)/8]$ for odd integer $k$ with $k\geq 11$. This improves a recent result in W. Ge, T. Wang (2018).
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