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

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.

20 pages

References in corpus (4)

• Number fields without n-ary universal quadratic forms
• Universal quadratic forms and elements of small norm in real quadratic fields
• Polynomial Solutions to Pell's Equation and Fundamental Units in Real Quadratic Fields
• Norms of indecomposable integers in real quadratic fields

Cited by in corpus (5)

• Universal quadratic forms, small norms and traces in families of number fields
• Consecutive real quadratic fields with large class numbers
• Universal quadratic forms and indecomposables in number fields: A survey
• Small class number fields in the family $\mathbb{Q}(\sqrt{9m^2+4m})$
• On continued fraction partial quotients of square roots of primes