Rainbow-free colorings for x+y=czx+y=cz in ZpZp

Let pp be a prime number and ZpZp be the cyclic group of order pp. A 3-coloring of ZpZp is rainbow-free for some equation if it contains no rainbow solution of the equation. In  [3] Jungić et al. (2003) proved that every 3-coloring of ZpZp, with the cardinality of the smallest color class greater than four, has a rainbow solution of “almost” all linear equations in three variables in ZpZp. In this work we handle the “small” cases and give a structural description of rainbow-free colorings for the particular case of x+y=czx+y=cz, which includes the Schur equation.