## Taiwanese Journal of Mathematics

### ENUMERATION PROBLEMS FOR A LINEAR CONGRUENCE EQUATION

#### Abstract

Let $m \geq 2$ and $r \geq 1$ be integers and let $c \in Z_m = \{0, 1,\dots, m - 1\}$. In this paper, we give an upper bound and a lower bound for the number of unordered solutions $x_1, \dots, x_n \in Z_m$ of the congruence $x_1 + x_2 + \cdots + x_r \equiv c \mod m$. Exact formulae are also given when $m$ or $r$ is prime. This solution number involves the Catalan number or generalized Catalan number in some special cases. Moreover, the enumeration problem has relationship with the restricted integer partition.

#### Article information

Source
Taiwanese J. Math., Volume 18, Number 1 (2014), 265-275.

Dates
First available in Project Euclid: 10 July 2017

Permanent link to this document
https://projecteuclid.org/euclid.twjm/1499706342

Digital Object Identifier
doi:10.11650/tjm.18.2014.2295

Mathematical Reviews number (MathSciNet)
MR3162124

Zentralblatt MATH identifier
1357.05006

#### Citation

Chou, Wun-Seng; He, Tian-Xiao; Shiue, Peter. ENUMERATION PROBLEMS FOR A LINEAR CONGRUENCE EQUATION. Taiwanese J. Math. 18 (2014), no. 1, 265--275. doi:10.11650/tjm.18.2014.2295. https://projecteuclid.org/euclid.twjm/1499706342

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