On the Waring-Goldbach problem for two squares and four cubes
Let NN be a sufficiently large integer. In this article, it is proved that, with at most O(N112+ε)O\left({N}^{\tfrac{1}{12}+\varepsilon }) exceptions, all even positive integers up to NN can be represented in the form p12+p22+p33+p43+p53+p63{p}_{1}^{2}+{p}_{2}^{2}+{p}_{3}^{3}+{p}_{4}^{3}+{p}_{5}^{3}...
| Main Authors: | , , |
|---|---|
| Format: | Article |
| Language: | English |
| Published: |
De Gruyter
2023-08-01
|
| Series: | Open Mathematics |
| Subjects: | |
| Online Access: | https://doi.org/10.1515/math-2022-0608 |
| _version_ | 1797752473407979520 |
|---|---|
| author | Zhang Min Bai Hongxin Li Jinjiang |
| author_facet | Zhang Min Bai Hongxin Li Jinjiang |
| author_sort | Zhang Min |
| collection | DOAJ |
| description | Let NN be a sufficiently large integer. In this article, it is proved that, with at most O(N112+ε)O\left({N}^{\tfrac{1}{12}+\varepsilon }) exceptions, all even positive integers up to NN can be represented in the form p12+p22+p33+p43+p53+p63{p}_{1}^{2}+{p}_{2}^{2}+{p}_{3}^{3}+{p}_{4}^{3}+{p}_{5}^{3}+{p}_{6}^{3}, where p1,p2,p3,p4,p5{p}_{1},{p}_{2},{p}_{3},{p}_{4},{p}_{5}, and p6{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]. |
| first_indexed | 2024-03-12T17:03:53Z |
| format | Article |
| id | doaj.art-55409c4672f34cb2bbc155680720edfe |
| institution | Directory Open Access Journal |
| issn | 2391-5455 |
| language | English |
| last_indexed | 2024-03-12T17:03:53Z |
| publishDate | 2023-08-01 |
| publisher | De Gruyter |
| record_format | Article |
| series | Open Mathematics |
| spelling | doaj.art-55409c4672f34cb2bbc155680720edfe2023-08-07T06:56:44ZengDe GruyterOpen Mathematics2391-54552023-08-01211688010.1515/math-2022-0608On the Waring-Goldbach problem for two squares and four cubesZhang Min0Bai Hongxin1Li Jinjiang2School of Applied Science, Beijing Information Science and Technology University, Beijing 100192, P. R. ChinaCollege of Data Science and Software Engineering, Baoding University, Baoding 071000, Hebei, P. R. ChinaDepartment of Mathematics, China University of Mining and Technology, Beijing 100083, P. R. ChinaLet NN be a sufficiently large integer. In this article, it is proved that, with at most O(N112+ε)O\left({N}^{\tfrac{1}{12}+\varepsilon }) exceptions, all even positive integers up to NN can be represented in the form p12+p22+p33+p43+p53+p63{p}_{1}^{2}+{p}_{2}^{2}+{p}_{3}^{3}+{p}_{4}^{3}+{p}_{5}^{3}+{p}_{6}^{3}, where p1,p2,p3,p4,p5{p}_{1},{p}_{2},{p}_{3},{p}_{4},{p}_{5}, and p6{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].https://doi.org/10.1515/math-2022-0608waring-goldbach problemhardy-littlewood methodexceptional set11p0511p3211p55 |
| spellingShingle | Zhang Min Bai Hongxin Li Jinjiang On the Waring-Goldbach problem for two squares and four cubes Open Mathematics waring-goldbach problem hardy-littlewood method exceptional set 11p05 11p32 11p55 |
| title | On the Waring-Goldbach problem for two squares and four cubes |
| title_full | On the Waring-Goldbach problem for two squares and four cubes |
| title_fullStr | On the Waring-Goldbach problem for two squares and four cubes |
| title_full_unstemmed | On the Waring-Goldbach problem for two squares and four cubes |
| title_short | On the Waring-Goldbach problem for two squares and four cubes |
| title_sort | on the waring goldbach problem for two squares and four cubes |
| topic | waring-goldbach problem hardy-littlewood method exceptional set 11p05 11p32 11p55 |
| url | https://doi.org/10.1515/math-2022-0608 |
| work_keys_str_mv | AT zhangmin onthewaringgoldbachproblemfortwosquaresandfourcubes AT baihongxin onthewaringgoldbachproblemfortwosquaresandfourcubes AT lijinjiang onthewaringgoldbachproblemfortwosquaresandfourcubes |