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On the simultaneous 3-divisibility of class numbers of triples of imaginary quadratic fields

Volume 197 / 2021

Jaitra Chattopadhyay, Subramani Muthukrishnan Acta Arithmetica 197 (2021), 105-110 MSC: Primary 11R11; Secondary 11R29. DOI: 10.4064/aa200221-16-6 Published online: 31 August 2020

Abstract

Let $k \geq 1$ be a cube-free integer with $k \equiv 1 \pmod {9}$ and $\gcd (k, 7\cdot 571)= 1$. We prove the existence of infinitely many triples of imaginary quadratic fields $\mathbb {Q}(\sqrt {d})$, $\mathbb {Q}(\sqrt {d+1})$ and $\mathbb {Q}(\sqrt {d+k^2})$ with $d \in \mathbb {Z}$ such that the class number of each of them is divisible by $3$. This affirmatively answers a weaker version of a conjecture of Iizuka (2018).

Authors

  • Jaitra ChattopadhyayHarish-Chandra Research Institute
    HBNI, Chhatnag Road, Jhunsi
    Prayagraj (Allahabad) 211019
    Uttar Pradesh, India
    e-mail
  • Subramani MuthukrishnanIndian Institute of Information Technology,
    Design and Manufacturing, Kancheepuram
    Chennai 600127, India
    e-mail

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