On an indivisibility version of Iizuka's conjecture

TL;DR

This paper investigates the distribution of class numbers in quadratic and biquadratic fields, focusing on the divisibility properties by powers of 3, and explores an indivisibility variant of Iizuka's conjecture.

Contribution

It introduces a new perspective on Iizuka's conjecture by examining the proportion of fields with class numbers not divisible by certain powers of 3, extending the understanding of class number divisibility.

Findings

Quantifies the proportion of quadratic fields with class numbers not divisible by 3^k.

Analyzes the distribution of class numbers in imaginary biquadratic fields.

Provides evidence supporting the indivisibility variant of Iizuka's conjecture.

Abstract

Iizuka's conjecture predicts that, given $m \in \mathbb{N}$ and a prime $p$, there exists infinitely many integers $n$ such that the class numbers of \textit{all} of the following quadratic number fields, \[ \mathbb{Q}(\sqrt{n}),\ \mathbb{Q}(\sqrt{n+1}),\ \ldots,\ \mathbb{Q}(\sqrt{n+m}), \] are divisible by $p$. In this article, given $k$ and $m$, we study the proportion of $n$ such that the class numbers of \textit{none} of the successive fields \[ \mathbb{Q}(\sqrt{n}),\ \mathbb{Q}(\sqrt{n+1}),\ \ldots,\ \mathbb{Q}(\sqrt{n+m}), \] are divisible by \( 3^k \). Moreover, we study the proportion of imaginary biquadratic fields whose class numbers are not divisible by $3$.