Simultaneous indivisibility of class numbers of pairs of real quadratic fields

For a square-free integer $t$, Byeon \cite{byeon} proved the existence of infinitely many pairs of quadratic fields $\mathbb{Q}(\sqrt{D})$ and $\mathbb{Q}(\sqrt{tD})$ with $D>0$ such that the class numbers of all of them are indivisible by $3$. In the same spirit, we prove that for a given integer $t\ge 1$ with $t\equiv 0(\mathrm{m}\mathrm{o}\mathrm{d}4)$, a positive proportion of fundamental discriminants $D>0$ exist for which the class numbers of both the real quadratic fields $\mathbb{Q}(\sqrt{D})$ and $\mathbb{Q}(\sqrt{D+t})$ are indivisible by $3$. This also addresses the complement of a weak form of a conjecture of Iizuka in \cite{iizuka}. As an application of our main result, we obtain that for any integer $t\ge 1$ with $t\equiv 0(\mathrm{m}\mathrm{o}\mathrm{d}12)$, there are infinitely many pairs of real quadratic fields $\mathbb{Q}(\sqrt{D})$ and $\mathbb{Q}(\sqrt{D+t})$ such that the Iwasawa $\lambda$-invariants associated with the basic ${\mathbb{Z}}_3$-extensions of both $\mathbb{Q}(\sqrt{D})$ and $\mathbb{Q}(\sqrt{D+t})$ are $0$. For $p=3$, this supports Greenberg’s conjecture which asserts that ${\lambda }_p(K)=0$ for any prime number $p$ and any totally real number field $K$.