Debopam Chakraborty and Anupam Saikia Indian Institute of Technology, Indian Institute of Technology

Abstract : The relative class number $H_{d}(f)$ of a real quadratic field $K=\mathbb{Q}(\sqrt{m})$ of discriminant $d$ is the ratio of class numbers of $\mathcal{O}_{f}$ and $\mathcal{O}_{K}$, where $\mathcal{O}_{K}$ denotes the ring of integers of $K$ and $\mathcal{O}_{f}$ is the order of conductor $f$ given by $\mathbb{Z}+f\mathcal{O}_{K}$. In a recent paper of A. Furness and E. A. Parker the relative class number of $\mathbb{Q}(\sqrt{m})$ has been investigated using continued fraction in the special case when $\sqrt{m}$ 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 $\sqrt{m}$ 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