Let $ R $ be a commutative ring with the identity $ 1_{R} $, and let $ R^* $ be the multiplicative group of units in $ R $. An element $ a\in R^* $ is called an exceptional unit if there exists a $ b\in R^* $ such that $ a+b = 1_{R} $. We set $ R^{**} $ to be the set of all exceptional units in $ R $. In this paper, we consider the residue-class ring $ \mathbb{Z}_n $. For any positive integers $ n, s $, and $ c\in\mathbb{Z}_n $, let $ {\mathcal N}_{s}(n, c): = \sharp\big\{(x_1, ..., x_s)\in (\mathbb{Z}_n^{**})^s : x_1+...+x_s\equiv c \pmod n\big\} $. In 2016, Sander (J.Number Theory 159 (2016)) got a formula for $ {\mathcal N}_{2}(n, c) $. Later on, Yang and Zhao (Monatsh. Math. 182 (2017)) extended Sander's theorem to finite terms by using exponential sum theory. In this paper, using matrix theory, we present an explicit formula for $ {\mathcal N}_{s}(n, c) $. This extends and improves earlier results.
Citation: Junyong Zhao. Counting sums of exceptional units in $ \mathbb{Z}_n $[J]. AIMS Mathematics, 2024, 9(9): 24546-24554. doi: 10.3934/math.20241195
Let $ R $ be a commutative ring with the identity $ 1_{R} $, and let $ R^* $ be the multiplicative group of units in $ R $. An element $ a\in R^* $ is called an exceptional unit if there exists a $ b\in R^* $ such that $ a+b = 1_{R} $. We set $ R^{**} $ to be the set of all exceptional units in $ R $. In this paper, we consider the residue-class ring $ \mathbb{Z}_n $. For any positive integers $ n, s $, and $ c\in\mathbb{Z}_n $, let $ {\mathcal N}_{s}(n, c): = \sharp\big\{(x_1, ..., x_s)\in (\mathbb{Z}_n^{**})^s : x_1+...+x_s\equiv c \pmod n\big\} $. In 2016, Sander (J.Number Theory 159 (2016)) got a formula for $ {\mathcal N}_{2}(n, c) $. Later on, Yang and Zhao (Monatsh. Math. 182 (2017)) extended Sander's theorem to finite terms by using exponential sum theory. In this paper, using matrix theory, we present an explicit formula for $ {\mathcal N}_{s}(n, c) $. This extends and improves earlier results.
[1] | T. M. Apostol, Introduction to analytic number theory, Springer-Verlag, New York, 1976. |
[2] |
M. Deaconescu, Adding units mod $n$, Elem. Math., 55 (2000), 123–127. https://doi.org/10.1007/s000170050078 doi: 10.1007/s000170050078
![]() |
[3] | J. Harrington, L. Jones, On the iteration of a function related to Euler's $\varphi$-function, Integers, 10 (2010), 497–515. |
[4] |
J. Houriet, Exceptional units and Euclidean number fields, Arch. Math., 88 (2007), 425–433. https://doi.org/10.1007/s00013-006-1019-0 doi: 10.1007/s00013-006-1019-0
![]() |
[5] |
H. W. Lenstra, Euclidean number fields of large degree, Invent. Math., 38, (1976/1977), 237–254. https://doi.org/10.1007/BF01403131 doi: 10.1007/BF01403131
![]() |
[6] | A. Leutbecher, G. Niklasch, On cliques of exceptional units and Lenstra's construction of Euclidean fields, In: H.P. Schlickewei, E. Wirsing (eds.), Number Theory, Springer, 1989,150–178. https://doi.org/10.1007/BFb0086541 |
[7] |
T. Nagell, Sur un type particulier d'unites algebriques, Ark. Mat., 8 (1969), 163–184. https://doi.org/10.1007/BF02589556 doi: 10.1007/BF02589556
![]() |
[8] |
J. W. Sander, On the addition of units and nonunits mod $m$, J. Number Theory, 129 (2009), 2260–2266. https://doi.org/10.1016/j.jnt.2009.04.010 doi: 10.1016/j.jnt.2009.04.010
![]() |
[9] |
J. W. Sander, Sums of exceptional units in residue class rings, J. Number Theory, 159 (2016), 1–6. https://doi.org/10.1016/j.jnt.2015.07.018 doi: 10.1016/j.jnt.2015.07.018
![]() |
[10] |
J. H. Silverman, Exceptional units and numbers of small Mahler measure, Exp. Math., 4 (1995), 69–83. https://doi.org/10.1080/10586458.1995.10504309 doi: 10.1080/10586458.1995.10504309
![]() |
[11] | J. H. Silverman, Small Salem numbers, exceptional units, and Lehmer's conjecture, Rocky Mt. J. Math., 26 (1996), 1099–1114. |
[12] |
N. P. Smart, Solving discriminant form equations via unit equations, J. Symbolic Comput., 21 (1996), 367–374. https://doi.org/10.1006/jsco.1996.0018 doi: 10.1006/jsco.1996.0018
![]() |
[13] |
C. L. Stewart, Exceptional units and cyclic resultants, Acta Arith., 155 (2012), 407–418. https://doi.org/10.4064/aa155-4-5 doi: 10.4064/aa155-4-5
![]() |
[14] | C. L. Stewart, Exceptional units and cyclic resultants, Contemp. Math., 587 (2013), 191–200. |
[15] |
N. Tzanakis, B. M. M. deWeger, On the practical solution of the Thue equation, J. Number Theory, 31 (1989), 99–132. https://doi.org/10.1016/0022-314X(89)90014-0 doi: 10.1016/0022-314X(89)90014-0
![]() |
[16] | N. Tzanakis, B. M. M. deWeger, How to explicitly solve a Thue-Mahler equation, Compos. Math., 84 (1992), 223–288. |
[17] |
Q. H. Yang, Q. Q. Zhao, On the sumsets of exceptional units in $\mathbb{Z}_n$, Monatsh. Math., 182 (2017), 489–493. https://doi.org/10.1007/s00605-015-0872-y doi: 10.1007/s00605-015-0872-y
![]() |