On Residue Difference Sets

Abstract

In recent years the subject of difference sets has attracted a considerable amount of attention in connection with problems in finite geometries [4]. Difference sets arising from higher power residues were first discussed by Chowla [1], who proved that biquadratic residues modulo p form a difference set if (p — l )/4 is an odd square. In this paper we shall prove a similar result for octic residues and develop some necessary conditions which will eliminate all odd power residue difference sets and many others. We also prove that a perfect residue difference set (that is, one in which every difference appears exactly once) contains all the powers of 2 modulo p.