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Given a coloring of group elements, a rainbow solution to an equation is asolution whose every element is assigned a different color. The rainbow numberof $\mathbb{Z}_n$ for an equation $eq$, denoted $rb(\mathbb{Z}_n,eq)$, is thesmallest number of colors $r$ such that every exact $r$-coloring of$\mathbb{Z}_n$ admits a rainbow solution to the equation $eq$. We prove thatfor every exact $4$-coloring of $\mathbb{Z}_p$, where $p\geq 3$ is prime, thereexists 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.

Rainbow Solutions to the Sidon Equation in Cyclic Groups