For a square-free integer t, Byeon (Proc. Am. Math. Soc. 132:3137–3140, 2004) 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 \(t \equiv 0 \pmod {4}\), 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 (J. Numb. Theory, 184:122–127, 2018). As an application of our main result, we obtain that for any integer \(t \ge 1\) with \(t \equiv 0 \pmod {12}\), 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.

对于无平方整数t，Byeon (Proc. Am. Math. Soc. 132:3137–3140, 2004) 证明了存在无限多对二次域\(\mathbb {Q}(\sqrt{D}) \)和\(\mathbb {Q}(\sqrt{tD})\)和\(D > 0\)使得它们的类数都不能被 3 整除。本着同样的精神，我们证明对于一个给定的整数\(t \ge 1\)与\(t \equiv 0 \pmod {4}\)，基本判别式\(D > 0\) 的正比例存在，对于这两个实二次方的类数字段\(\mathbb {Q}(\sqrt{D})\)和\(\mathbb {Q}(\sqrt{D + t})\)不能被 3 整除。 这也解决了 Iizuka 猜想的弱形式的补充（J. Numb. Theory, 184:122–127, 2018）。作为我们主要结果的应用，我们得到对于任何具有\(t \equiv 0 \pmod {12}\) 的整数\(t \ge 1 \)，有无穷多对实二次场\(\mathbb {Q}(\sqrt{D})\)和\(\mathbb {Q}(\sqrt{D + t})\)使得 Iwasawa \(\lambda \) -与基本\(\ mathbb {Z}_{3}\) - \(\mathbb {Q}(\sqrt{D})\)和\(\mathbb {Q}(\sqrt{D + t})\) 的扩展是0. 对于\(p = 3\)，这支持格林伯格猜想，该猜想断言\(\lambda _{p}(K) = 0\)对于任何素数p和任何全实数域K。