On some properties of partial quotients of the continued fraction expansion of d with even period
Let d be a non-square positive integer such that the period of the continued fraction expansion of d is even. We give some relations between some properties of partial quotients of the continued fraction expansion of d , which emerge from numerical experiments.
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| Published in | Archiv der Mathematik Vol. 114; no. 6; pp. 649 - 660 |
|---|---|
| Main Authors | , , |
| Format | Journal Article |
| Language | English |
| Published |
Cham
Springer International Publishing
01.06.2020
Springer Nature B.V |
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| Abstract | Let
d
be a non-square positive integer such that the period of the continued fraction expansion of
d
is even. We give some relations between some properties of partial quotients of the continued fraction expansion of
d
, which emerge from numerical experiments. |
|---|---|
| AbstractList | Let d be a non-square positive integer such that the period of the continued fraction expansion of d is even. We give some relations between some properties of partial quotients of the continued fraction expansion of d, which emerge from numerical experiments. Let d be a non-square positive integer such that the period of the continued fraction expansion of d is even. We give some relations between some properties of partial quotients of the continued fraction expansion of d , which emerge from numerical experiments. |
| Author | Tomita, Koshi Kawamoto, Fuminori Kishi, Yasuhiro |
| Author_xml | – sequence: 1 givenname: Fuminori surname: Kawamoto fullname: Kawamoto, Fuminori organization: Gakushuin University – sequence: 2 givenname: Yasuhiro surname: Kishi fullname: Kishi, Yasuhiro email: ykishi@auecc.aichi-edu.ac.jp organization: Aichi University of Education – sequence: 3 givenname: Koshi surname: Tomita fullname: Tomita, Koshi organization: Meijo University |
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| DOI | 10.1007/s00013-020-01434-7 |
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| References | Kubo, K.: Relations between the primary symmetric parts and positive integers of minimal type in continued fraction expansions. Tokyo University of Science, Master thesis (2019) (in Japanese) LouboutinSContinued fractions and real quadratic fieldsJ. Number Theory19883016717696191410.1016/0022-314X(88)90015-7 Perron, O.: Die Lehre von den Kettenbrüchen, Band I: Elementare Kettenbrüche, 3te Aufl. B.G. Teubner Verlagsgesellschaft, Stuttgart (1954) KawamotoFTomitaKContinued fractions and certain real quadratic fields of minimal typeJ. Math. Soc. Japan2008603865903244041610.2969/jmsj/06030865 Golubeva, E.P.: Quadratic irrationalities with a fixed length of the period of continued fraction expansion, Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 196 (1991), Modul. Funktsii Kvadrat. Formy. 2, 5–30, 172; translation in J. Math. Sci. 70(6), 2059–2076 (1994) LouboutinSChakrabortyKHoqueAPandeyPOn the continued fraction expansions of p\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sqrt{p}$$\end{document} and 2p\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sqrt{2p}$$\end{document} for primes p≡3(mod4)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p\equiv 3\ (\text{mod}\ 4)$$\end{document}Class Groups of Number Fields and Related Topics2020SingaporeSpringer17517810.1007/978-981-15-1514-9_16 Halter-KochFQuadratic Irrationals: An Introduction to Classical Number Theory2013Boca Raton, FLCRC Press10.1201/b14968 KawamotoFKishiYTomitaKContinued fraction expansions with even period and priary symmetric parts with extremely large endComm. Math. Univ. Sancti Pauli20156421311551415.11161 KawamotoFTomitaKContinued fractions with even period and an infinite family of real quadratic fields of minimal typeOsaka J. Math.200946494999326049171247.11140 |
| References_xml | – reference: LouboutinSChakrabortyKHoqueAPandeyPOn the continued fraction expansions of p\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sqrt{p}$$\end{document} and 2p\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sqrt{2p}$$\end{document} for primes p≡3(mod4)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p\equiv 3\ (\text{mod}\ 4)$$\end{document}Class Groups of Number Fields and Related Topics2020SingaporeSpringer17517810.1007/978-981-15-1514-9_16 – reference: Golubeva, E.P.: Quadratic irrationalities with a fixed length of the period of continued fraction expansion, Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 196 (1991), Modul. Funktsii Kvadrat. Formy. 2, 5–30, 172; translation in J. Math. Sci. 70(6), 2059–2076 (1994) – reference: KawamotoFTomitaKContinued fractions with even period and an infinite family of real quadratic fields of minimal typeOsaka J. Math.200946494999326049171247.11140 – reference: LouboutinSContinued fractions and real quadratic fieldsJ. Number Theory19883016717696191410.1016/0022-314X(88)90015-7 – reference: Halter-KochFQuadratic Irrationals: An Introduction to Classical Number Theory2013Boca Raton, FLCRC Press10.1201/b14968 – reference: Kubo, K.: Relations between the primary symmetric parts and positive integers of minimal type in continued fraction expansions. Tokyo University of Science, Master thesis (2019) (in Japanese) – reference: KawamotoFKishiYTomitaKContinued fraction expansions with even period and priary symmetric parts with extremely large endComm. Math. Univ. Sancti Pauli20156421311551415.11161 – reference: KawamotoFTomitaKContinued fractions and certain real quadratic fields of minimal typeJ. Math. Soc. Japan2008603865903244041610.2969/jmsj/06030865 – reference: Perron, O.: Die Lehre von den Kettenbrüchen, Band I: Elementare Kettenbrüche, 3te Aufl. B.G. Teubner Verlagsgesellschaft, Stuttgart (1954) |
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| Snippet | Let
d
be a non-square positive integer such that the period of the continued fraction expansion of
d
is even. We give some relations between some properties of... Let d be a non-square positive integer such that the period of the continued fraction expansion of d is even. We give some relations between some properties of... |
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| SubjectTerms | Mathematics Mathematics and Statistics Quotients |
| Title | On some properties of partial quotients of the continued fraction expansion of d with even period |
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