Czechoslovak Mathematical Journal, Vol. 71, No. 2, pp. 563-589, 2021
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
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