Some Theorems on Difference Sets

Abstract

A set a₁ …, a_k of different residues mod v is called a difference set (v, k, λ) (v>k > λ) if the congruence a_i — a_j ≡ d (mod v) has exactly λ solutions for d ≢ 0 (mod v). Singer [4] has demonstrated the existence of a difference set (v, k, 1) if k — 1 is a prime power, and difference sets for λ > 1 have been constructed by various authors; but necessary and sufficient conditions for the existence of a (v, k, λ) are not known. It has not been possible so far to find a difference set with λ = 1 if k — 1 is not a prime power and it has therefore been conjectured that no such difference set exists.