Czechoslovak Mathematical Journal, first online, pp. 1-27


On the Waring-Goldbach problem for one square and five cubes in short intervals

Fei Xue, Min Zhang, Jinjiang Li

Received January 6, 2020.   Published online December 11, 2020.

Abstract:  Let $N$ be a sufficiently large integer. We prove that almost all sufficiently large even integers $n\in[N-6U,N+6U]$ can be represented as $\cases n=p_1^2+p_2^3+p_3^3+p_4^3+p_5^3+p_6^3, \\ \Bigl| p_1^2-\frac{N}6\Bigr| \leq U, \Bigl| p_i^3-\frac{N}6\Bigr|\leq U, i=2,3,4,5,6 \endcases $, where $U=N^{1-\delta+\varepsilon}$ with $\delta\leq8/225$.
Keywords:  Waring-Goldbach problem; Hardy-Littlewood method; exponential sum; short interval
Classification MSC:  11P05, 11P32, 11P55
DOI:  10.21136/CMJ.2020.0013-20

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Affiliations:   Fei Xue, Department of Mathematics, China University of Mining and Technology, Beijing 100083, P. R. China, e-mail: fei.xue.math@gmail.com; Min Zhang (corresponding author), School of Applied Science, Beijing Information Science and Technology University, Beijing 100192, P. R. China, e-mail: min.zhang.math@gmail.com; Jinjiang Li, Department of Mathematics, China University of Mining and Technology, Beijing 100083, P. R. China, e-mail: jinjiang.li.math@gmail.com


 
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