# Rainbow Numbers of $\mathbb{Z}_n$ for $a_1x_1+a_2x_2+a_3x_3 =b$

## TL;DR

This paper determines the rainbow numbers for the equation $a_1x_1 + a_2x_2 + a_3x_3 = b$ over the cyclic groups $ ext{Z}_n$, providing exact values for prime and certain composite cases.

## Contribution

It extends the concept of rainbow numbers to $ ext{Z}_n$ for a specific linear equation, including explicit results for prime and some composite moduli.

## Key findings

- Rainbow number of $ ext{Z}_p$ for prime $p$ is determined.
- Rainbow number of $ ext{Z}_n$ for composite $n$ is established under certain conditions.
- Provides exact rainbow numbers for specific equations over cyclic groups.

## 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 $\Z_p$, for $p$ prime and the equation $a_1x_1 + a_2x_2 + a_3x_3 = b$ is determined. The rainbow number of $\Z_{n}$, for a natural number $n$, is determined under certain conditions.