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
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:
- an abelian group G, given by its invariant factors
- the difference set's status (Yes, No, Open)
- for each p dividing n, whether any known theorems imply that
must be a multiplier:
- 'FMT': First Multiplier Theorem
- 'SMT': Second Multiplier Theorem
- 'McF': McFarland
- 'LMS': Leung, Ma and Schmidt
- '4.3': Corollary 4.3 from the paper
- '5.1': Result 5.1 from the paper
- 'Trivial': p ≡ 1 modulo exp(G)
- 'Exhaust': a search of unions of orbits found all such difference
sets, and p was a multiplier for all of them
- 'OPEN': none of the above results apply
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