Using continued fractions with prescribed period for universal quadratic forms

TL;DR

This paper investigates the properties of continued fractions of square roots of integers with specific periods and applies these findings to improve understanding of universal quadratic forms over real quadratic fields, especially regarding their discriminants.

Contribution

It introduces a method to analyze continued fractions with prescribed periods and applies this to refine results on the ranks of universal quadratic forms with certain discriminance conditions.

Findings

Characterization of continued fractions with prescribed periods for square roots.

Refined bounds on the ranks of universal quadratic forms over real quadratic fields.

Imposition of congruence conditions on discriminants to improve existing results.

Abstract

We study the congruence classes attained by positive integers $D$ with a prescribed period of the continued fraction of $\sqrt D$. As an application, we refine the available results on large ranks of universal quadratic forms over real quadratic fields by also imposing congruence conditions on their discriminants.