Rainbow Solutions to the Sidon Equation in Cyclic Groups

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 $p\ge 3$ 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.