On the distribition of residue classes of quadratic forms and integer-detecting sequences in number fields

Abstract

The extensive study of residue classes to given moduli which are represented by a quadratic form leads to an application to sequences in algebraic number fields. So-called integer-detecting sequences were introduced over the rationals in earlier work. We discuss the respective concept for sequences in number fields. 1. Introduction and definitions In this paper, we study quadratic forms x 2 + Dy 2 mod s for given moduli s and try to decide which residue classes are represented by them. Quadratic forms of this type may be considered as norms in quadratic number fields. These norms and, more generally, norms in arbitrary number fields occur naturally when studying certain sequences, so-called integer-detecting sequences. We first shall explain this notion for rational numbers. Integer-detecting sequences were introduced in [12] (see also [3]), and a bunch of necessary or sufficient criteria was derived. For the convenience of the reader let us review the definition: A sequence (mn) n 0 of positive integers with mn 0) if any rational number r ,s atisfy� � .