On the $3$-divisibility of class numbers of pairs of quadratic fields with splitting conditions

Abstract

Let $m_1$ and $m_2$ be distinct square-free integers. We show that there exist infinitely many pairs of quadratic fields $\mathbb{Q}(\sqrt{m_1D})$ and $\mathbb{Q}(\sqrt{m_2D})$ whose class numbers are both divisible by $3$ under the splitting conditions of prime numbers. This improves results of T. Komatsu and the author.