Small class number fields in the family $\mathbb{Q}(\sqrt{9m^2+4m})$

Abstract

We study the class number one problem for real quadratic fields $\mathbb{Q}(\sqrt{9m^2+4m})$, where $m$ is an odd integer. We show that for $m\equiv 1(\mathrm{mod}3)$ there is only one such field with class number one and only one such field with class number two.

20 pages. Comments are welcome