On an indivisibility version 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 $3$.