Let $\mathcal{ O}$ be an order in an algebraic number field and suppose that the set of distances $\Delta(\mathcal{ O})$ 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 $\min\Delta(\mathcal{ O})=1$. So far, only a few examples of orders were found with $\min\Delta(\mathcal{ O})>1$. We say that $\Delta(\mathcal{ O})$ is unusual if $\min\Delta(\mathcal{ O})>1$. In the present paper, we establish algebraic characterizations of orders $\mathcal{ O}$ in real quadratic number fields with $\min\Delta(\mathcal{ O})>1$. 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.
Categories: math.NT math.AC