Distribution of class numbers in continued fraction families of real quadratic fields

TL;DR

This paper models the distribution of class numbers in specific families of real quadratic fields derived from continued fraction expansions, extending known results to new generalized families.

Contribution

It introduces a new random model for analyzing class number distribution in generalized continued fraction families of real quadratic fields.

Findings

Develops a probabilistic framework for class number distribution

Extends results to families generalizing Chowla's quadratic fields

Provides insights into the behavior of class numbers in these families

Abstract

We construct a random model to study the distribution of class numbers in special families of real quadratic fields $\mathbb Q(\sqrt d)$ arising from continued fractions. These families are obtained by considering periodic continued fraction expansions of the form $\sqrt {D(n)}=[f(n), [u_1, u_2, \dots, u_{s-1}, 2f(n)]]$ with fixed coefficients $u_1, \dots, u_{s-1}$ and generalize well-known families such as Chowla's $4n^2+1$, for which analogous results were recently proved by Dahl and Lamzouri.