Rainbow solutions to the Sidon equation in cyclic groups and the interval

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 X∈{Zn,[n]} for an equation eq, denoted rb(X,eq), is the smallest number of colors r such that every exact r-coloring of X admits a rainbow solution to the equation eq. We prove that for every exact 4-coloring of Zp, where p≥3 is prime, there exists a rainbow solution to the Sidon equation x1+x2=x3+x4. Furthermore, we determine the rainbow number of Zn and [n] for the Sidon equation.