Czechoslovak Mathematical Journal

Czechoslovak Mathematical Journal, first online, pp. 1-14

A Diophantine equation involving one Linnik prime

Yuhui Liu

Received September 4, 2024. Published online December 13, 2024.

Abstract: Let $[\theta]$ denote the integral part of the real number $\theta.$ We prove that for $1<c<\frac{25 908}{18 905}$, the Diophantine equation $[p_1^c]+[p_2^c]+[p_3^c]+[p_4^c]+[p_5^c]=N$ is solvable in prime variables $p_1$, $p_2$, $p_3$, $p_4$, $p_5$ such that $p_1=x^2+y^2+1$ with integers $x$ and $y$ for sufficiently large integer $N$, and we also establish the corresponding asymptotic formula. This result constitutes a refinement upon that of S. Dimitrov (2023).

Keywords: Diophantine equation; exponential sum; prime

Classification MSC: 11L07, 11L20, 11P32

DOI: 10.21136/CMJ.2024.0372-24

PDF available at: Springer Institute of Mathematics CAS

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Affiliations: Yuhui Liu, School of Science, Jiangnan University, No. 1800, Lihu Avenue, Wuxi 214122, Jiangsu, P. R. China, e-mail: 8202106018@jiangnan.edu.cn

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