Lehmer sequence approach to the divisibility of class numbers of imaginary quadratic fields

TL;DR

This paper investigates the divisibility properties of class numbers in certain imaginary quadratic fields using Lehmer sequences, establishing new divisibility results and constructing infinite families of fields with class numbers divisible by a given integer.

Contribution

It introduces a novel approach using Lehmer sequences to analyze class number divisibility and constructs infinite families of quadratic fields with prescribed divisibility properties.

Findings

Class numbers are divisible by n or its divisors in specified quadratic fields.

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

Uses deep results on primitive divisors of Lehmer sequences.

Abstract

Let $k\geq 3$ and $n\geq 3$ be odd integers, and let $m\geq 0$ be any integer. For a prime number $\ell$, we prove that the class number of the imaginary quadratic field $\mathbb{Q}(\sqrt{\ell^{2m}-2k^n})$ is either divisible by $n$ or by a specific divisor of $n$. Applying this result, we construct an infinite family of certain tuples of imaginary quadratic fields of the form $$\left(\mathbb{Q}(\sqrt{d}), \mathbb{Q}(\sqrt{d+1}), \mathbb{Q}(\sqrt{4d+1}), \mathbb{Q}(\sqrt{2d+4}), \mathbb{Q}(\sqrt{2d+16}), \cdots, \mathbb{Q}(\sqrt{2d+4^t}) \right)$$ with $d\in \mathbb{Z}$ and $1\leq 4^t\leq 2|d|$ whose class numbers are all divisible by $n$. Our proofs use some deep results about primitive divisors of Lehmer sequences.