实数二次域，连分式，以及e型的初级对称部分的构造

IF 0.6 4区数学 Q3 MATHEMATICS

Kyushu Journal of Mathematics Pub Date : 2019-01-01 DOI:10.2206/kyushujm.73.165

Fuminori Kawamoto, Y. Kishi, H. Suzuki, Koshi Tomita

{"title":"实数二次域，连分式，以及e型的初级对称部分的构造","authors":"Fuminori Kawamoto, Y. Kishi, H. Suzuki, Koshi Tomita","doi":"10.2206/kyushujm.73.165","DOIUrl":null,"url":null,"abstract":"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`−1 be the symmetric part of the simple continued fraction expansion of ω(d). We say that the sequence a1, a2, . . . , a[`/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 ` of minimal type for each even `≥ 6.","PeriodicalId":49929,"journal":{"name":"Kyushu Journal of Mathematics","volume":"1 1","pages":""},"PeriodicalIF":0.6000,"publicationDate":"2019-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.2206/kyushujm.73.165","citationCount":"4","resultStr":"{\"title\":\"REAL QUADRATIC FIELDS, CONTINUED FRACTIONS, AND A CONSTRUCTION OF PRIMARY SYMMETRIC PARTS OF ELE TYPE\",\"authors\":\"Fuminori Kawamoto, Y. Kishi, H. Suzuki, Koshi Tomita\",\"doi\":\"10.2206/kyushujm.73.165\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"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`−1 be the symmetric part of the simple continued fraction expansion of ω(d). We say that the sequence a1, a2, . . . , a[`/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 ` of minimal type for each even `≥ 6.\",\"PeriodicalId\":49929,\"journal\":{\"name\":\"Kyushu Journal of Mathematics\",\"volume\":\"1 1\",\"pages\":\"\"},\"PeriodicalIF\":0.6000,\"publicationDate\":\"2019-01-01\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"https://sci-hub-pdf.com/10.2206/kyushujm.73.165\",\"citationCount\":\"4\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Kyushu Journal of Mathematics\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://doi.org/10.2206/kyushujm.73.165\",\"RegionNum\":4,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q3\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Kyushu Journal of Mathematics","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.2206/kyushujm.73.165","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"MATHEMATICS","Score":null,"Total":0}

引用次数: 4

摘要

对于具有4d的非平方正整数d，如果d等于1模4，则设ω(d):=(1+√d)/2，否则设ω(d):=√d。设a1 a2。， a '−1是ω(d)的简单连分式展开式的对称部分。我们说序列a1, a2，…， a[' /2]是ω(d)的简单连分式展开式的初等对称部分。Kawamoto等人(评论)引入了有限序列的“ELE型”概念。数学。圣保利大学学报，64(2)(2015)，131-155。本文的目的是为有限序列引入“前ELE型”的概念，并给出构造ELE型的初级对称部分的一种方法。作为副产物，我们证明了存在无穷多个周期为最小型的实二次域，对于每一个偶数≥6。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

查看原文本刊更多论文

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

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`−1 be the symmetric part of the simple continued fraction expansion of ω(d). We say that the sequence a1, a2, . . . , a[`/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 ` of minimal type for each even `≥ 6.

求助全文

通过发布文献求助，成功后即可免费获取论文全文。去求助

来源期刊

Kyushu Journal of Mathematics 数学-数学

CiteScore

0.80

自引率

0.00%

发文量

审稿时长

>12 weeks

期刊介绍： The Kyushu Journal of Mathematics is an academic journal in mathematics, published by the Faculty of Mathematics at Kyushu University since 1941. It publishes selected research papers in pure and applied mathematics. One volume, published each year, consists of two issues, approximately 20 articles and 400 pages in total. More than 500 copies of the journal are distributed through exchange contracts between mathematical journals, and available at many universities, institutes and libraries around the world. The on-line version of the journal is published at "Jstage" (an aggregator for e-journals), where all the articles published by the journal since 1995 are accessible freely through the Internet.