Abstract
An exact $r$-coloring of a set $S$ is a surjective function $c:S\to \{1,2,\cdots, r\}$. Given an equation $eq$, a solution in $S$ is a rainbow solution if each element is colored distinctly by the coloring $c$. 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 $eq$. The rainbow numbers of $\mathbb{Z}_p$, for prime $p$, for the equation $x_1 + x_2 = 4x_3$ are known to be either $3$ or $4$. This paper investigates which primes yield rainbow number $3$ or $4$. Additionally, the rainbow numbers of $\mathbb{Z}_n$ for this equation are discussed.
Funding Statement
The first author's contribution is supported by a University Research Scholar fellowship from his institution.
Acknowledgments
This work was inspired by the first author's attendance to the 2018 Research Experiences for Undergraduate Faculty Workshop (REUF) hosted at the American Institute of Mathematics (AIM) in San Jose, CA. The authors are very thankful for Dr. Ryan Tully-Doyle for his help with a SAGE code and Dr. Salam Turki for her help with Theorem 2.4.
Citation
Houssein El Turkey. Nathan Waskiewicz. "On the Rainbow Numbers of $\mathbb{Z}_n$ for $x_1 + x_2 = 4x_3$." Missouri J. Math. Sci. 35 (1) 117 - 122, May 2023. https://doi.org/10.35834/2023/3501117
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