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.