Affiliation:
1. School of Applied Science, Beijing Information Science and Technology University , Beijing 100192 , P. R. China
2. College of Data Science and Software Engineering, Baoding University , Baoding 071000 , Hebei , P. R. China
3. Department of Mathematics, China University of Mining and Technology , Beijing 100083 , P. R. China
Abstract
Abstract
Let
N
N
be a sufficiently large integer. In this article, it is proved that, with at most
O
(
N
1
12
+
ε
)
O\left({N}^{\tfrac{1}{12}+\varepsilon })
exceptions, all even positive integers up to
N
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
{p}_{1}^{2}+{p}_{2}^{2}+{p}_{3}^{3}+{p}_{4}^{3}+{p}_{5}^{3}+{p}_{6}^{3}
, where
p
1
,
p
2
,
p
3
,
p
4
,
p
5
{p}_{1},{p}_{2},{p}_{3},{p}_{4},{p}_{5}
, and
p
6
{p}_{6}
are prime variables. This result constitutes a large improvement upon the previous result of Liu [On a Waring-Goldbach problem involving squares and cubes, Math. Slovaca. 69 (2019), no. 6, 1249–1262].
Reference15 articles.
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2. C. Hooley, On a new approach to various problems of Waring’s type, in: Recent Progress in Analytic Number Theory, Academic Press, London, New York, 1981, pp. 127–191.
3. Y. C. Cai, Waring-Goldbach problem: two squares and higher powers, J. Théor. Nombres Bordeaux 28 (2016), no. 3, 791–810.
4. M. Zhang and J. Li, On the Waring-Goldbach problem for squares, cubes and higher powers, Ramanujan J. 56 (2021), no. 3, 1123–1150.
5. Y. Liu, On a Waring-Goldbach problem involving squares and cubes, Math. Slovaca 69 (2019), no. 6, 1249–1262.