Bounds on the largest prime factor of a negative discriminant with one class per genus

TL;DR

This paper proves that negative discriminants with many genera cannot have very large prime factors, using methods inspired by Baker's approach to Gauss' class number one problem.

Contribution

It introduces bounds on the largest prime factor of negative discriminants with one class per genus, extending understanding of their prime factorization properties.

Findings

Negative discriminants with many genera have bounded prime factors.

The proof adapts Baker's techniques from class number one problem.

Results limit the size of prime factors in specific discriminants.

Abstract

It was conjectured by Gauss that any negative discriminant with at least 32 genera has at least two classes of binary quadratic forms in each genus. We prove that such a discriminant cannot have a particularly large prime factor, by an argument similar to that of Baker's solution to Gauss' class number one problem.