Let r be a positive integer, and let \(k_1, k_2, \cdots ,k_r\) be integers such that \(k_i \equiv \pm 1 \pmod 9\) for all \(1 \le i \le r\) . In this article, we prove the existence of infinitely many positive integers D such that the class numbers of the real quadratic fields \( {\mathbb {Q}}(\sqrt{3D}), ~ {\mathbb {Q}}(\sqrt{3(D -1)}), ~ {\mathbb {Q}}(\sqrt{3(D -k_1^2)}), \ldots , ~ {\mathbb {Q}}(\sqrt{3(D-k_r^2)})\) 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).
