This question is answered (affirmatively and somewhat more generally) in the following paper: Chat Yin Ho, [Projective planes with a regular collineation group and a question about powers of a prime][1], J. Algebra 154 (1993), no. 1, 141–151. The proof there uses the ring of Eisenstein integers. (The issue is available online via Open Access).


  [1]: http://www.sciencedirect.com/science/article/pii/S0021869383710094