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}. $
Citation: Junyong Zhao. On the number of unit solutions of cubic congruence modulo $ n $[J]. AIMS Mathematics, 2021, 6(12): 13515-13524. doi: 10.3934/math.2021784
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}. $
| [1] | T. M. Apostol, Introduction to analytic number theory, New York: Springer, 1976. |
| [2] | A. Brauer, Lösung der Aufgabe 30, Jahresber. Dtsch. Math.-Ver., 35 (1926), 92–94. |
| [3] |
S. Chowla, J. Cowles, M. Cowles, On the number of zeros of diagonal cubic forms, J. Number Theory, 9 (1977), 502–506. doi: 10.1016/0022-314X(77)90010-5
|
| [4] |
S. F. Hong, C. X. Zhu, On the number of zeros of diagonal cubic forms over finite fields, Forum Math., 33 (2021), 697–708. doi: 10.1515/forum-2020-0354
|
| [5] | K. Ireland, M. Rosen, A classical introduction to modern number theory, 2 Eds., Graduate Texts in Mathematics, New York: Springer-Verlag, 1990. |
| [6] |
S. Li, Y. Ouyang, Counting the solutions of $ {\lambda _1}x_{11}^k + \ldots + {\lambda _t}x_{tt}^k \equiv c $ mod $n$, J. Number Theroy, 187 (2018), 41–65. doi: 10.1016/j.jnt.2017.10.017
|
| [7] | H. Rademacher, Aufgabe 30, Jahresber. Dtsch. Math.-Ver., 34 (1925), 158. |
| [8] |
R. F. Taki Eldin, On the number of incongruent solutions to a quadratic congruence over algebraic integers, Int. J. Number Theory, 15 (2019), 105–130. doi: 10.1142/S1793042118501762
|
| [9] |
C. Sun, Q. Yang, On the sumset of atoms in cyclic groups, Int. J. Number Theory, 10 (2014), 1355–1363. doi: 10.1142/S1793042114500328
|
| [10] |
Q. Yang, M. Tang, On the addition of squares of units and nonunits modulo $n$, J. Number Theory, 155 (2015), 1–12. doi: 10.1016/j.jnt.2015.02.019
|
Junyong Zhao. On the number of unit solutions of cubic congruence modulo $ n $[J]. AIMS Mathematics, 2021, 6(12): 13515-13524. doi: 10.3934/math.2021784
DownLoad: