Slim exceptional sets of Waring-Goldbach problems involving squares and cubes of primes
Sbornik. Mathematics, Tome 214 (2023) no. 5, pp. 744-756
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Let $p_{1},p_{2},\dots,p_{6}$ be prime numbers. First we show that, with at most $O(N^{1/12+\varepsilon})$ exceptions, all even positive integers not exceeding $N$ can be represented in the form $p_{1}^{2}+p_{2}^{2}+p_{3}^{3}+p_{4}^{3}+p_{5}^{3}+p_{6}^{3}$, which improves the previous result $O(N^{1/4+\varepsilon})$ obtained by Y. H. Liu. Moreover, we also prove that, with at most $O(N^{5/12+\varepsilon})$ exceptions, all even positive integers not exceeding $N$ can be represented in the form $p_{1}^{2}+p_{2}^{3}+p_{3}^{3}+p_{4}^{3}+p_{5}^{3}+p_{6}^{3}$.
Bibliography: 21 titles.
Keywords:
Waring-Goldbach problem, exceptional set, Hardy-Littlewood method.
@article{SM_2023_214_5_a5,
author = {X. Han and H. Liu},
title = {Slim exceptional sets of {Waring-Goldbach} problems involving squares and cubes of primes},
journal = {Sbornik. Mathematics},
pages = {744--756},
publisher = {mathdoc},
volume = {214},
number = {5},
year = {2023},
language = {en},
url = {http://geodesic.mathdoc.fr/item/SM_2023_214_5_a5/}
}
X. Han; H. Liu. Slim exceptional sets of Waring-Goldbach problems involving squares and cubes of primes. Sbornik. Mathematics, Tome 214 (2023) no. 5, pp. 744-756. http://geodesic.mathdoc.fr/item/SM_2023_214_5_a5/