The class number one problem for the real quadratic fields $\mathbb{Q}\left(\sqrt{(an)^2+4a}\right)$

TL;DR

This paper unconditionally solves the class number one problem for a specific family of real quadratic fields with discriminants of the form (an)^2+4a, where a and n are positive odd integers.

Contribution

It provides the first unconditional proof for the class number one problem in this particular family of real quadratic fields.

Findings

Class number one is characterized for the family with discriminant d=(an)^2+4a.

The result applies to all positive odd integers a and n.

The proof advances understanding of class numbers in quadratic fields.

Abstract

We solve unconditionally the class number one problem for the $2$-parameter family of real quadratic fields $\mathbb{Q}(\sqrt{d})$ with square-free discriminant $d=(an)^2+4a$ for positive odd integers $a$ and $n$.