On the divisibility of class numbers and discriminants of imaginary quadratic fields

TL;DR

This paper proves the existence of infinitely many imaginary quadratic fields with discriminants divisible by a given odd squarefree integer and containing class groups with elements of that order, linking to sums of three squares.

Contribution

It establishes the infinite occurrence of such quadratic fields with specific divisibility and class group properties, extending understanding of class number divisibility.

Findings

Existence of infinitely many quadratic fields with discriminant divisible by n

Presence of elements of order n in class groups of these fields

Connection between class group structure and sums of three squares

Abstract

Let $n$ be a squarefree positive odd integer. We will show that there exist infinitely many imaginary quadratic number fields with discriminant divisible by $n$ and-at the same time-having an element of order $n$ in the class group. We then apply our result to prove that for a given squarefree positive odd integer $n$ there exist infinitely many $N$ such that $n$ divides both $N$ and $r(N)$, where $r(N)$ is the representation number of $N$ as sums of three squares.