  Let p be a prime number and Zp be the cyclic group of order p. A coloring of
Zp is called rainbow-free with respect to a certain equation, if it contains no
rainbow solution of the same, that is, a solution whose elements have pairwise
distinct colors. In this paper we describe the structure of rainbow-free
3-colorings of Zp with respect to all linear equations on three variables.
Consequently, we determine those linear equations on three variables for which
every 3-coloring (with nonempty color classes) of Zp contains a rainbow
solution of it.
