Divisibility of class numbers of quadratic fields and a conjecture of Iizuka

TL;DR

This paper proves that under certain conditions, the class number of specific imaginary quadratic fields is divisible by a fixed odd integer, and constructs infinite families of such fields with all class numbers divisible by that integer.

Contribution

It establishes a divisibility result for class numbers of quadratic fields and constructs infinite families with this property, addressing a conjecture related to Iizuka.

Findings

Class number divisibility by n under specified conditions

Construction of infinite families of quadratic fields with class numbers divisible by n

Verification of the divisibility for infinitely many successive fields

Abstract

Assume $x,\ y,\ n$ are positive integers and $n$ is odd. In this note, we show that the class number of the imaginary quadratic field $\mathbb{Q}(\sqrt{x^{2}-y^{n}})$ is divisible by $n$ for fixed $x, n$ if $\gcd(2x,y)=1$ and $y>C$ where $C$ is a constant depending only on $x$ and $n$. Based on this result, for any odd integer $n$ and any positive integer $m$, we construct an infinite family of $m+1$ successive imaginary quadratic fields $\mathbb{Q}(\sqrt{d})$, $\mathbb{Q}(\sqrt{d+1^{2}})$, $\cdots$, $\mathbb{Q}(\sqrt{d+m^{2}})$ $(d\in \mathbb{Z})$ whose class numbers are all divisible by $n$.