On a conjecture of Iizuka

TL;DR

This paper constructs infinite families of imaginary quadratic fields with class numbers divisible by a given odd integer, proving Iizuka's conjecture for certain cases and extending its scope.

Contribution

It provides the first explicit infinite families of quadratic fields with class numbers divisible by a fixed odd integer, confirming and generalizing Iizuka's conjecture for specific cases.

Findings

Constructed infinite families of quadratic fields with class numbers divisible by n.

Proved Iizuka's conjecture for the case m=1.

Confirmed a weaker version of the conjecture for m≥4.

Abstract

For a given odd positive integer $n$ and an odd prime $p$, we construct an infinite family of quadruples of imaginary quadratic fields $\mathbb{Q}(\sqrt{d})$, $\mathbb{Q}(\sqrt{d+1})$, $\mathbb{Q}(\sqrt{d+4})$ and $\mathbb{Q}(\sqrt{d+4p^2})$ with $d\in \mathbb{Z}$ such that the class number of each of them is divisible by $n$. Subsequently, we show that there is an infinite family of quintuples of imaginary quadratic fields $\mathbb{Q}(\sqrt{d})$, $\mathbb{Q}(\sqrt{d+1})$, $\mathbb{Q}(\sqrt{d+4})$, $\mathbb{Q}(\sqrt{d+36})$ and $\mathbb{Q}(\sqrt{d+100})$ with $d\in \mathbb{Z}$ whose class numbers are all divisible by $n$. Our results provide a complete proof of Iizuka's conjecture (in fact a generalization of it) for the case $m=1$. Our results also affirmatively answer a weaker version of (a generalization of) Iizuka's conjecture for $m\geq 4$.