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arXiv · 1809.04576

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

Erin Bevilacqua·Samuel King·Jürgen Kritschgau·Michael Tait·Suzannah Tebon·Michael Young
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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 $n$.

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Erin Bevilacqua, Samuel King, Jürgen Kritschgau, Michael Tait, Suzannah Tebon, Michael Young. 2018-09-12. Rainbow numbers for $x_1+x_2=kx_3$ in $\mathbb{Z}_n$. https://arxiv.org/abs/1809.04576

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