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arXiv · 2010.04127

Rainbow Solutions to the Sidon Equation in Cyclic Groups

Zhanar Berikkyzy·Jürgen Kritschgau
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Abstract

Given a coloring of group elements, a rainbow solution to an equation is a solution whose every element is assigned a different color. The rainbow number of $\mathbb{Z}_n$ for an equation $eq$, denoted $rb(\mathbb{Z}_n,eq)$, is the smallest number of colors $r$ such that every exact $r$-coloring of $\mathbb{Z}_n$ admits a rainbow solution to the equation $eq$. We prove that for every exact $4$-coloring of $\mathbb{Z}_p$, where $p\geq 3$ is prime, there exists a rainbow solution to the Sidon equation $x_1+x_2=x_3+x_4$. Furthermore, we determine the rainbow number of $\mathbb{Z}_n$ for the Sidon equation.

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Zhanar Berikkyzy, Jürgen Kritschgau. 2020-10-08. Rainbow Solutions to the Sidon Equation in Cyclic Groups. https://arxiv.org/abs/2010.04127

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