Note on a conjecture of Wegner

Abstract

The optimal packings of n unit discs in the plane are known for those ${n \in \mathbb N}$, which satisfy certain number theoretic conditions. Their geometric realizations are the extremal Groemer packings (or Wegner packings). But an extremal Groemer packing of n unit discs does not exist for all ${n\in\mathbb N}$ and in this case, the number n is called exceptional. We are interested in number theoretic characterizations of the exceptional numbers. A counterexample is given to a conjecture of Wegner concerning such a characterization. We further give a characterization of the exceptional numbers, whose shape is closely related to that of Wegner’s conjecture.