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Note on a conjecture of Wegner


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.

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Correspondence to Dominik Kenn.

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Kenn, D. Note on a conjecture of Wegner. Beitr Algebra Geom 52, 45–50 (2011).

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Mathematics Subject Classification (2000)