Czechoslovak Mathematical Journal, Vol. 72, No. 4, pp. 1167-1174, 2022
On an additive problem of unlike powers in short intervals
Qingqing Zhang
Received November 2, 2021. Published online June 6, 2022.
Abstract: We prove that almost all positive even integers $n$ can be represented as $p_2^2+p_3^3+p_4^4+p_5^5$ with $|p_k^k-\tfrac14 N|\leq N^{1-1/54+\varepsilon}$ for $2\leq k\leq5$. As a consequence, we show that each sufficiently large odd integer $N$ can be written as $p_1+p_2^2+p_3^3+p_4^4+p_5^5$ with $|p_k^k- \tfrac15 N|\leq N^{1-1/54+\varepsilon}$ for $1\leq k\leq5$.
Keywords: Waring-Goldbach problem; exponential sum over prime in short interval; circle method
Affiliations: Qingqing Zhang, School of Mathematics, Shandong University, 27 Shanda Nanlu, Jinan, Shandong 250100 , P. R. China, e-mail: qingqingzhang@mail.sdu.edu.cn
