On the number of unit solutions of cubic congruence modulo n
For any positive integer $ n $, let $ \mathbb Z_n: = \mathbb Z/n\mathbb Z = \{0, \ldots, n-1\} $ be the ring of residue classes module $ n $, and let $ \mathbb{Z}_n^{\times}: = \{x\in \mathbb Z_n|\gcd(x, n) = 1\} $. In 1926, for any fixed $ c\in\mathbb Z_n $, A. Brauer studied the linear congruence...
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| Format: | Article |
| Language: | English |
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AIMS Press
2021-09-01
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| Series: | AIMS Mathematics |
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| Online Access: | https://www.aimspress.com/article/doi/10.3934/math.2021784?viewType=HTML |
| _version_ | 1819147722321559552 |
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| author | Junyong Zhao |
| author_facet | Junyong Zhao |
| author_sort | Junyong Zhao |
| collection | DOAJ |
| description | For any positive integer $ n $, let $ \mathbb Z_n: = \mathbb Z/n\mathbb Z = \{0, \ldots, n-1\} $ be the ring of residue classes module $ n $, and let $ \mathbb{Z}_n^{\times}: = \{x\in \mathbb Z_n|\gcd(x, n) = 1\} $. In 1926, for any fixed $ c\in\mathbb Z_n $, A. Brauer studied the linear congruence $ x_1+\cdots+x_m\equiv c\pmod n $ with $ x_1, \ldots, x_m\in\mathbb{Z}_n^{\times} $ and gave a formula of its number of incongruent solutions. Recently, Taki Eldin extended A. Brauer's result to the quadratic case. In this paper, for any positive integer $ n $, we give an explicit formula for the number of incongruent solutions of the following cubic congruence
$ x_1^3+\cdots +x_m^3\equiv 0\pmod n\ \ \ {\rm with} \ x_1, \ldots, x_m \in \mathbb{Z}_n^{\times}. $ |
| first_indexed | 2024-12-22T13:34:20Z |
| format | Article |
| id | doaj.art-a8b30d439cc343d1bb3d762e3c172e66 |
| institution | Directory Open Access Journal |
| issn | 2473-6988 |
| language | English |
| last_indexed | 2024-12-22T13:34:20Z |
| publishDate | 2021-09-01 |
| publisher | AIMS Press |
| record_format | Article |
| series | AIMS Mathematics |
| spelling | doaj.art-a8b30d439cc343d1bb3d762e3c172e662022-12-21T18:24:06ZengAIMS PressAIMS Mathematics2473-69882021-09-01612135151352410.3934/math.2021784On the number of unit solutions of cubic congruence modulo nJunyong Zhao01. Mathematical College, Sichuan University, Chengdu 610064, China 2. School of Mathematics and Physics, Nanyang Institute of Technology, Nanyang 473004, ChinaFor any positive integer $ n $, let $ \mathbb Z_n: = \mathbb Z/n\mathbb Z = \{0, \ldots, n-1\} $ be the ring of residue classes module $ n $, and let $ \mathbb{Z}_n^{\times}: = \{x\in \mathbb Z_n|\gcd(x, n) = 1\} $. In 1926, for any fixed $ c\in\mathbb Z_n $, A. Brauer studied the linear congruence $ x_1+\cdots+x_m\equiv c\pmod n $ with $ x_1, \ldots, x_m\in\mathbb{Z}_n^{\times} $ and gave a formula of its number of incongruent solutions. Recently, Taki Eldin extended A. Brauer's result to the quadratic case. In this paper, for any positive integer $ n $, we give an explicit formula for the number of incongruent solutions of the following cubic congruence $ x_1^3+\cdots +x_m^3\equiv 0\pmod n\ \ \ {\rm with} \ x_1, \ldots, x_m \in \mathbb{Z}_n^{\times}. $https://www.aimspress.com/article/doi/10.3934/math.2021784?viewType=HTMLcubic congruenceexponential sumsunit solutions |
| spellingShingle | Junyong Zhao On the number of unit solutions of cubic congruence modulo n AIMS Mathematics cubic congruence exponential sums unit solutions |
| title | On the number of unit solutions of cubic congruence modulo n |
| title_full | On the number of unit solutions of cubic congruence modulo n |
| title_fullStr | On the number of unit solutions of cubic congruence modulo n |
| title_full_unstemmed | On the number of unit solutions of cubic congruence modulo n |
| title_short | On the number of unit solutions of cubic congruence modulo n |
| title_sort | on the number of unit solutions of cubic congruence modulo n |
| topic | cubic congruence exponential sums unit solutions |
| url | https://www.aimspress.com/article/doi/10.3934/math.2021784?viewType=HTML |
| work_keys_str_mv | AT junyongzhao onthenumberofunitsolutionsofcubiccongruencemodulon |