Bulletin of the
Korean Mathematical Society
BKMS

ISSN(Print) 1015-8634 ISSN(Online) 2234-3016

Article

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Bull. Korean Math. Soc. 2021; 58(5): 1149-1161

Online first article August 24, 2021      Printed September 30, 2021

https://doi.org/10.4134/BKMS.b200744

Copyright © The Korean Mathematical Society.

New bounds for fundamental units and class numbers of real quadratic fields

Sevcan I\c{s}ikay, Ayten Pek{$\dot{\textsc i}$}n

Gebze Technical University; Istanbul University

Abstract

In this paper, we present new bounds on the fundamental units of real quadratic fields $\mathbb{Q}(\sqrt{d})$ using the continued fraction expansion of the integral basis element of the field. Furthermore, we apply these bounds to Dirichlet's class number formula. Consequently, we provide computational advantages to estimate the class numbers of such fields. We also give some numerical examples.

Keywords: Continued fraction, class number, fundamental unit, real quadratic field

MSC numbers: 11A55, 11R27, 11R29, 11R11