On the simultaneous 3-divisibility of class numbers of triples of imaginary quadratic fields
Volume 197 / 2021
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).
