On an additive problem of unlike powers in short intervals

Zhang, Qingqing. "On an additive problem of unlike powers in short intervals." Czechoslovak Mathematical Journal 72.4 (2022): 1167-1174. <http://eudml.org/doc/298884>.

@article{Zhang2022,
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 $|p_\{k\}^\{k\}-\tfrac\{1\}\{4\} N|\le N^\{1-1/54+\varepsilon \}$ for $2\le k\le 5$. 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 $|p_\{k\}^\{k\}- \tfrac\{1\}\{5\} N|\le N^\{1-1/54+\varepsilon \}$ for $1\le k\le 5$.},
author = {Zhang, Qingqing},
journal = {Czechoslovak Mathematical Journal},
keywords = {Waring-Goldbach problem; exponential sum over prime in short interval; circle method},
language = {eng},
number = {4},
pages = {1167-1174},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {On an additive problem of unlike powers in short intervals},
url = {http://eudml.org/doc/298884},
volume = {72},
year = {2022},
}

TY - JOUR
AU - Zhang, Qingqing
TI - On an additive problem of unlike powers in short intervals
JO - Czechoslovak Mathematical Journal
PY - 2022
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 72
IS - 4
SP - 1167
EP - 1174
AB - 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 $|p_{k}^{k}-\tfrac{1}{4} N|\le N^{1-1/54+\varepsilon }$ for $2\le k\le 5$. 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 $|p_{k}^{k}- \tfrac{1}{5} N|\le N^{1-1/54+\varepsilon }$ for $1\le k\le 5$.
LA - eng
KW - Waring-Goldbach problem; exponential sum over prime in short interval; circle method
UR - http://eudml.org/doc/298884
ER -