Rainbow Numbers of ${\mathbb{Z}}_n$ for $a_1{x}_1+a_2{x}_2+a_3{x}_3=b$

Abstract

An exact $r$-coloring of a set $S$ is a surjective function $c:S\to [r]$. The rainbow number of a set $S$ for equation $eq$ is the smallest integer $r$ such that every exact $r$-coloring of $S$ contains a rainbow solution to $eq$. In this paper, the rainbow number of ${\mathbb{Z}}_p$, for $p$ prime and the equation $a_1{x}_1+a_2{x}_2+a_3{x}_3=b$ is determined. The rainbow number of ${\mathbb{Z}}_n$, for a natural number $n$, is determined under certain conditions.