ON RELATIVE CLASS NUMBER AND CONTINUED FRACTIONS

Title & Authors
ON RELATIVE CLASS NUMBER AND CONTINUED FRACTIONS
CHAKRABORTY, DEBOPAM; SAIKIA, ANUPAM;

Abstract
The relative class number $\small{H_d(f)}$ of a real quadratic field $\small{K=\mathbb{Q}(\sqrt{m})}$ of discriminant d is the ratio of class numbers of $\small{O_f}$ and $\small{O_K}$, where $\small{O_K}$ denotes the ring of integers of K and $\small{O_f}$ is the order of conductor f given by $\small{\mathbb{Z}+fO_K}$. In a recent paper of A. Furness and E. A. Parker the relative class number of $\small{\mathbb{Q}(\sqrt{m})}$ has been investigated using continued fraction in the special case when $\small{(\sqrt{m})}$ has a diagonal form. Here, we extend their result and show that there exists a conductor f of relative class number 1 when the continued fraction of $\small{(\sqrt{m})}$ is non-diagonal of period 4 or 5. We also show that there exist infinitely many real quadratic fields with any power of 2 as relative class number if there are infinitely many Mersenne primes.
Keywords
relative class number;continued fraction;
Language
English
Cited by
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