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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

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