Lower bounds for the principal genus of definite binary quadratic forms

TL;DR

This paper uses Tatuzawa's version of Siegel's theorem to establish new lower bounds on the size of the principal genus of positive definite binary quadratic forms, advancing understanding of their algebraic structure.

Contribution

It provides the first explicit lower bounds for the principal genus size using Tatuzawa's refinement of Siegel's theorem.

Findings

Derived two explicit lower bounds for the principal genus size.

Applied Tatuzawa's version of Siegel's theorem to quadratic forms.

Enhanced theoretical understanding of algebraic properties of quadratic forms.

Abstract

We apply Tatuzawa's version of Siegel's theorem to derive two lower bounds on the size of the principal genus of positive definite binary quadratic forms.