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 is an element of {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 Znand [n] for the Sidon equation. (c) 2024 Elsevier B.V. All rights are reserved, including those for text and data mining, AI training, and similar technologies.