Class number one problem for the real quadratic fields $\mathbb{Q}({\sqrt{m^2+2r}})$

TL;DR

This paper studies the class number one problem for a specific family of real quadratic fields defined by parameters m and r, aiming to identify when these fields have class number one.

Contribution

It provides new insights into the class number one problem for quadratic fields of the form Q(√(m^2+4r)), expanding understanding of their class number properties.

Findings

Identifies conditions under which the fields have class number one.

Provides classifications for certain parameter ranges.

Advances the understanding of class number distribution in quadratic fields.

Abstract

We investigate the class number one problem for a parametric family of real quadratic fields of the form $\mathbb{Q}( \sqrt{m^2+4r})$ for certain positive integers $m$ and $r$.