Extraneous multipliers of cyclic difference sets

AbstractMann (Canad. J. Math. (1952), 222–226) has proved that 2 is a multiplier for a cyclic difference set only if 2 | n = k − λ, and 3 is a multiplier for a simple cyclic difference set only if 3 | n = k − λ = k − 1, provided that these cyclic difference sets are not trivial. In this paper, we prove that 3 is a multiplier for a nontrivial cyclic difference set only if 3 | n = k − λ for arbitrary λ, and 5 is a multiplier for a non-trivial simple cyclic difference set only if 5 | n = k − λ = k − 1, as well as a necessary and sufficient condition for a prime to be an extraneous multiplier—multipliers not dividing n = k − λ