Constructing minimal periods of quadratic irrationalities in Zagier's reduction theory

JOURNAL OF NUMBER THEORY

卷:187

Article

Smith, Barry R.¹ 

[1] Lebanon Valley Coll, Dept Math Sci, 101 N Coll Ave, Annville, PA 17003 USA

关键词: Binary quadratic form;    Continued fraction;   

DOI  :  10.1016/j.jnt.2017.09.002

来源: Elsevier

【 摘 要 】

Dirichlet's version of Gauss's reduction theory for indefinite binary quadratic forms includes a map from Gauss-reduced forms to strings of natural numbers. It attaches to a form the minimal period of the continued fraction of a quadratic irrationality associated with the form. When Zagier developed his own reduction theory, parallel to Dirichlet's, he omitted an analogue of this map. We define a new map on Zagier-reduced forms that serves as this analogue. We also define a map from the set of Gauss-reduced forms into the set of Zagier-reduced forms that gives a near-embedding of the structure of Gauss's reduction theory into that of Zagier's. From this perspective, Zagier-reduction becomes a refinement of Gauss-reduction. (C) 2017 Elsevier Inc. All rights reserved.

【 授权许可】

Free