The Multiplier Conjecture

The First Multiplier Theorem says that for a (v,k,λ)-difference set D in an abelian group G, any prime p > λ dividing n=k-λ and not v is a multipler of D. The Multiplier Conjecture (MC) is that the condition p > λ is not necessary. Bernhard Schmidt and I have written a new paper surveying known results towards the MC. For this paper we looked at possible difference set parameters with v < 106, seeing how often existing results prove that primes covered by the MC must be multipliers. The paper gives an overview of the computations. The full list of results is given here, as a python list. Each line contains a list:

The primes are not identified; the status for each, from smallest to largest, are given in order for each set of parameters.

Of the 413586 (v,k,λ,G,p) possibilities, 268592 must be multipliers, leaving 35% of the cases open. Most of these are open cases, most of which presumably don't have any difference sets. 99% of Paley parameter primes are covered by known theorems.

As further calculations are done the numbers will be revised. This page was last updated October 4, 2015, with the numbers from the paper.