# On the simultaneous $3$-divisibility of class numbers of triples of imaginary quadratic fields

## TL;DR

This paper proves the existence of infinitely many triples of imaginary quadratic fields with class numbers divisible by 3, addressing a weaker form of Iizuka's conjecture through number-theoretic constructions.

## Contribution

It establishes the infinite existence of such triples for specific parameters, advancing understanding of class number divisibility in quadratic fields.

## Key findings

- Existence of infinitely many triples with class number divisible by 3
- Construction of such triples for specific cube-free integers
- Partial resolution of Iizuka's conjecture

## Abstract

Let $k \geq 1$ be a cube-free integer with $k \equiv 1 \pmod {9}$ and $\gcd(k, 7\cdot 571)=1$. In this paper, we prove the existence of infinitely many triples of imaginary quadratic fields $\mathbb{Q}(\sqrt{d})$, $\mathbb{Q}(\sqrt{d+1})$ and $\mathbb{Q}(\sqrt{d+k^2})$ with $d \in \mathbb{Z}$ such that the class number of each of them is divisible by $3$. This affirmatively answers a weaker version of a conjecture of Iizuka \cite{iizuka-jnt}.