Some congruences connecting quadratic class numbers with continued fractions
Volume 191 / 2019
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 $\sqrt {4p}$.
