On orders in quadratic number fields with unusual sets of distances

2023/05/16 by Andreas Reinhart, Reinhart, Andreas
Mathematics · #Algebraic Geometry and Number Theory #Analytic Number Theory Research #Advanced Differential Equations and Dynamical Systems

paper · pdf · doi:10.48550/arxiv.2305.09267

Abstract

Let O be an order in an algebraic number field and suppose that the set of distances Δ(O) of O is nonempty (equivalently, O is not half-factorial). If O is seminormal (in particular, if O is a principal order), then minΔ(O)=1. So far, only a few examples of orders were found with minΔ(O)>1. We say that Δ(O) is unusual if minΔ(O)>1. In the present paper, we establish algebraic characterizations of orders O in real quadratic number fields with minΔ(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.

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