Tokyo Journal of Mathematics

Continued Fractions and Gauss' Class Number Problem for Real Quadratic Fields

Fuminori KAWAMOTO and Koshi TOMITA

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Abstract

The main purpose of this article is to present a numerical data which shows relations between real quadratic fields of class number 1 and a mysterious behavior of the period of simple continued fraction expansion of certain quadratic irrationals. For that purpose, we define a class number, a fundamental unit,a discriminant and a Yokoi invariant for a non-square positive integer, and then see that a generalization of theorems of Siegel and of Yokoi holds. These and a theorem of Friesen and Halter-Koch imply several interesting conjectures for solving Gauss' class number problem for real quadratic fields.

Article information

Source
Tokyo J. of Math., Volume 35, Number 1 (2012), 213-239.

Dates
First available in Project Euclid: 19 July 2012

Permanent link to this document
https://projecteuclid.org/euclid.tjm/1342701351

Digital Object Identifier
doi:10.3836/tjm/1342701351

Mathematical Reviews number (MathSciNet)
MR2977452

Zentralblatt MATH identifier
1276.11177

Subjects
Primary: 11R29: Class numbers, class groups, discriminants
Secondary: 11A55: Continued fractions {For approximation results, see 11J70} [See also 11K50, 30B70, 40A15] 11R11: Quadratic extensions 11R27: Units and factorization

Citation

KAWAMOTO, Fuminori; TOMITA, Koshi. Continued Fractions and Gauss' Class Number Problem for Real Quadratic Fields. Tokyo J. of Math. 35 (2012), no. 1, 213--239. doi:10.3836/tjm/1342701351. https://projecteuclid.org/euclid.tjm/1342701351


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