A note on the simultaneous 3-divisibility of class numbers of tuples of real quadratic fields

Let r be a positive integer, and let k1,k2,⋯,kr\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$k_1, k_2, \cdots ,k_r$$\end{document} be integers such that ki≡±1(mod9)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$k_i \equiv \pm 1 \pmod 9$$\end{document} for all 1≤i≤r\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$1 \le i \le r$$\end{document}. In this article, we prove the existence of infinitely many positive integers D such that the class numbers of the real quadratic fields Q(3D),Q(3(D-1)),Q(3(D-k12)),…,Q(3(D-kr2))\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ {\mathbb {Q}}(\sqrt{3D}), ~ {\mathbb {Q}}(\sqrt{3(D -1)}), ~ {\mathbb {Q}}(\sqrt{3(D -k_1^2)}), \ldots , ~ {\mathbb {Q}}(\sqrt{3(D-k_r^2)})$$\end{document} are simultaneously divisible by 3. This result gives an affirmative answer to a weaker version of a conjecture of Iizuka (J Number Theory 184:122–127, 2018).