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

Abstract

Let $k\ge 1$ be a cube-free integer with $k\equiv 1(\mathrm{mod}9)$ and $\gcd (k,7\cdot 571)=1$. In this paper, 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 \cite{iizuka-jnt}.

To appear in Acta Arithmetica