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

TL;DR

This paper investigates the class number one problem for a specific family of real quadratic fields, identifying unique cases with class number one and two based on the parameter m.

Contribution

It establishes the uniqueness of class number one and two fields within the family m^2+4m for odd m congruent to 1 modulo 3.

Findings

Only one field with class number one for m m  

Only one field with class number two for m m  

Results are specific to the family  ext{Q}(\u221a{9m^2+4m}) for odd m m 

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 $m \equiv 1 \pmod 3$ there is only one such field with class number one and only one such field with class number two.