The divisibility of the class number of the imaginary quadratic fields $\mathbb{Q}(\sqrt{1-2m^k})$

TL;DR

This paper proves divisibility properties of class numbers of certain imaginary quadratic fields, establishing new results on their divisibility by odd integers and primes, and constructing infinite families related to Iizuka's conjecture.

Contribution

It demonstrates that for all odd k, class numbers of specific quadratic fields are divisible by k, and constructs infinite families of such fields, advancing understanding of class number divisibility.

Findings

For odd k, class numbers are divisible by k beyond a certain m.

For odd m ≥ 3, divisibility by k or p holds under certain conditions.

Constructs infinite families of quadratic fields with class numbers divisible by k.

Abstract

Let $h_{(m,k)}$ be the class number of $\mathbb{Q}(\sqrt{1-2m^k}).$ We prove that for any odd natural number $k,$ there exists $m_0$ such that $k \mid h_{(m,k)}$ for all odd $m > m_0.$ We also prove that for any odd $m \geq 3,$ $k \mid h_{(m,k)}$ (when $k$ and $1-2m^k$ square-free numbers) and $p \mid h_{(m,p)}$ (except finitely many primes $p$). We deduce that for any pair of twin primes $p_1,p_2=p_1+2$, $p_1 \mid h_{(m,p_1)}$ or $p_2 \mid h_{(m,p_2)}.$ For any odd natural number $k$, we construct an infinite family of pairs of imaginary quadratic fields $\mathbb{Q}(\sqrt{d}), \mathbb{Q}(\sqrt{d+1})$ whose class numbers are divisible by $k$, which settles a generalized version of Iizuka's conjecture (cf : Conjecture 2.2) for the case $n=1$.