Abstract
Let \(h_{(m,k)}\) be the class number of \(Q(\sqrt{1-2m^k})\). 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 \(m \ge 3,\) \(k \mid h_{(m,k)}\) (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 \(Q(\sqrt{d}), {\mathbb {Q}}(\sqrt{d+1})\) whose class numbers are divisible by k, which settles a generalized version of Iizuka’s conjecture (cf: Conjecture [2.2]) for the case \(n=1\).
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The data that support the findings of this study are available from the corresponding author upon reasonable request.
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Acknowledgements
We thank IISER, TVM for providing the excellent working conditions. We would like to thank Azizul Hoque and Sunil Kumar Pasupulati for helpful discussions and suggestions. We also would to thank Jaitra Chattopadhyay, Subham Bhakta and Jayanta Manoharmayum for helpful comments. We used Sage Math for calculations, hence we thank Sage Math for this.
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S. Krishnamoorthy—Framed the problem, wrote the main manuscript text, contribution as first author for the results appear in the paper. R. Muneeswaran—Collaborative discussions with S. Krishnamoorthy. All authors reviewed the manuscript.
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The first author’s research was supported by SERB Grant CRG/2023/009035. The second author wishes to thank CSIR for financial support.
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Krishnamoorthy, S., Muneeswaran, R. The divisibility of the class number of the imaginary quadratic fields \({\mathbb {Q}}(\sqrt{1-2m^k})\). Ramanujan J (2024). https://doi.org/10.1007/s11139-024-00860-3
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DOI: https://doi.org/10.1007/s11139-024-00860-3