A+ CATEGORY SCIENTIFIC UNIT

PDF files of articles are only available for institutions which have paid for the online version upon signing an Institutional User License.

Some congruences connecting quadratic class numbers with continued fractions

Volume 191 / 2019

Weidong Cheng, Xuejun Guo Acta Arithmetica 191 (2019), 309-340 MSC: Primary 11R29; Secondary 11A55, 11F20. DOI: 10.4064/aa8640-4-2019 Published online: 7 October 2019

Abstract

Let $p$ be a prime number, and $h(-p)$ and $h(p)$ be the ideal class numbers of the quadratic fields $\mathbb {Q}(\sqrt {-p})$ and $\mathbb {Q}(\sqrt {p})$ respectively. We prove that if $p\equiv 1 \pmod 8$ then $h(-p)\equiv h(p)m(4p) \pmod 8$, and if $p\equiv 5 \pmod 8$ then $h(-p)\equiv h(p)m(4p) \pmod 4$ under some further restrictions on the fundamental unit of $\mathbb {Q}(\sqrt {p})$, where $m(4p)$ is an integer depending on the minimal period of the negative continued fraction expansion of $\sqrt {4p}$.

Authors

  • Weidong ChengDepartment of Mathematics
    Nanjing University
    Nanjing, 210093 Jiangsu, China
    E-mail: chengwd@smail.nju.edu.cn
    Current address:
    School of Science
    Chongqing University of Posts and Telecommunications
    Chongqing, 400065 Chongqing, China
    e-mail
  • Xuejun GuoDepartment of Mathematics
    Nanjing University
    Nanjing, 210093 Jiangsu, China
    e-mail

Search for IMPAN publications

Query phrase too short. Type at least 4 characters.

Rewrite code from the image

Reload image

Reload image