Rainbow numbers for $x_1+x_2=kx_3$ in $\mathbb{Z}_n$

TL;DR

This paper determines the minimum number of colors needed to ensure a rainbow solution to the equation x_1 + x_2 = k x_3 in cyclic groups, revealing explicit formulas based on prime factorizations.

Contribution

It provides explicit formulas for rainbow numbers in cyclic groups for the equations with k=1 and prime k, based on prime factorization of n.

Findings

rb(Z_p, 1) = 4 for primes p > 3

rb(Z_n, 1) determined by prime factorization of n

rb(Z_n, k) for prime k determined by prime factorization of 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 $n$.