REAL QUADRATIC FIELDS, CONTINUED FRACTIONS, AND A CONSTRUCTION OF PRIMARY SYMMETRIC PARTS OF ELE TYPE

• KAWAMOTO Fuminori

  Faculty of Science Gakushuin University

• KISHI Yasuhiro

  Faculty of Education Aichi University of Education

• SUZUKI Hiroshi

  Graduate School of Mathematics Nagoya University

• TOMITA Koshi

  Faculty of Science and Technology Meijo University

抄録

For a non-square positive integer d with 4 ∤ d, put ω(d) :=(1 + √d)/2 if d is congruent to 1 modulo 4 and ω(d) := √d otherwise. Let a₁, a₂, . . . , a_ℓ-1 be the symmetric part of the simple continued fraction expansion of ω(d). We say that the sequence a₁, a₂, . . . , a_[ℓ/2] is the primary symmetric part of the simple continued fraction expansion of ω(d). A notion of ‘ELE type' for a finite sequence was introduced in Kawamoto et al (Comment. Math. Univ. St. Pauli 64(2) (2015), 131-155). The aims of this paper are to introduce a notion of ‘pre-ELE type' for a finite sequence and to give a way of constructing primary symmetric parts of ELE type. As a byproduct, we show that there exist infinitely many real quadratic fields with period ℓ of minimal type for each even ℓ ≥ 6.

収録刊行物

• 九州数学雑誌

  九州数学雑誌 73 (1), 165-187, 2019

  九州大学大学院数理学研究院

キーワード

• continued fractions
• real quadratic fields
• class numbers
• fundamental units

詳細情報

• CRID: 1390564227322242048
• NII論文ID: 130007728836
• DOI: 10.2206/kyushujm.73.165
• ISSN: 18832032
• ISSN: 13406116
• Web Site: https://www.jstage.jst.go.jp/article/kyushujm/73/1/73_165/_pdf
• 本文言語コード: en
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