Rainbow numbers for $x_1+x_2=k{x}_3$ in ${\mathbb{Z}}_n$

Abstract

In this work, we investigate the fewest number of colors needed to guarantee a rainbow solution to the equation $x_1+x_2=k{x}_3$ in ${\mathbb{Z}}_n$. This value is called the Rainbow number and is denoted by $rb({\mathbb{Z}}_n,k)$ for positive integer values of $n$ and $k$. We find that $rb({\mathbb{Z}}_p,1)=4$ for all primes greater than $3$ and that $rb({\mathbb{Z}}_n,1)$ can be deterimined from the prime factorization of $n$. Furthermore, when $k$ is prime, $rb({\mathbb{Z}}_n,k)$ can be determined from the prime factorization of $n$.