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

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-...

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Bibliographic Details
Published in:arXiv.org 2024-03
Main Authors: Krishnamoorthy, Srilakshmi, Muneeswaran, R
Format: Article
Language:English
Subjects:
Fields (mathematics)
Integers
Number theory
Online Access:Get full text
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Summary: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 \(n=1\).
ISSN:2331-8422