On the Non-Existence of Difference Sets with Perfect Square Order

Abstract

The existence of Paley-Hadamard (4t − 1, 2t − 1, t − 1) difference sets provides a platform for solving the equation δ¯δ = n in the cyclotomic ring Z[ζ4t−1], where ζ4t−1 is root of unity, n > 1 and t > 1 are integers. We look at cases where ⟨n⟩ = ⟨δ⟩⟨¯δ⟩ in Z[ζ4t−1] but δ¯δ = n has trivial solutions. This criterion is combined with other results to conclude non-existence of some difference set parameters.