Czechoslovak Mathematical Journal, Vol. 69, No. 2, pp. 353-363, 2019


A Diophantine inequality with four squares and one $k$th power of primes

Quanwu Mu, Minhui Zhu, Ping Li

Received July 3, 2017.   Published online November 19, 2018.

Abstract:  Let $k\geq5$ 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 $|\lambda_1p_1^2+\lambda_2p_2^2+\lambda_3p_3^2+\lambda_4p_4^2+\mu p_5^k+ \eta|<(\max_{1\leq j\leq5} p_j)^{-\sigma}$ has infinitely many solutions in prime variables $p_1, p_2, \cdots, p_5$, where $\vartheta(k)=3\times2^{(k-5)/2}$ for $k=5,7,9$ and $\vartheta(k)=[(k^2+2k+5)/8]$ for odd integer $k$ with $k\geq11$. This improves a recent result in W. Ge, T. Wang (2018).
Keywords:  Diophantine inequalities; Davenport-Heilbronn method; prime
Classification MSC:  11D75, 11P55
DOI:  10.21136/CMJ.2018.0316-17


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Affiliations:   Quanwu Mu (corresponding author), Minhui Zhu, Ping Li, School of Science, Xi'an Polytechnic University, No. 19 Jinhua South Road, Xi'an 710048, Shaanxi, P. R. China, e-mail: muquanwu@163.com, xiao-yu123@126.com, liping7487@163.com


 
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