Instytut Matematyczny Polskiej Akademii Nauk / Institute of Mathematics / Publishing house / Journals and Serials / Acta Arithmetica / All issues

Abstract

Let $\mathcal O$ be an order in an algebraic number field and suppose that the set of distances $\varDelta (\mathcal O)$ of $\mathcal O$ is nonempty (equivalently, $\mathcal O$ is not half-factorial). If $\mathcal O$ is seminormal (in particular, if $\mathcal O$ is a principal order), then $\min \varDelta (\mathcal O)=1$. So far, only a few examples of orders were found with $\min \varDelta (\mathcal O) \gt 1$. We say that $\varDelta (\mathcal O)$ is unusual if $\min \varDelta (\mathcal O) \gt 1$. In the present paper, we establish algebraic characterizations of orders $\mathcal O$ in real quadratic number fields with $\min \varDelta (\mathcal O) \gt 1$. We also provide a classification of the real quadratic number fields that possess an order whose set of distances is unusual. As a consequence thereof, we revisit certain squarefree integers (cf. OEIS A135735) that were studied by A. J. Stephens and H. C. Williams.