Recall that D is a half-factorial domain (HFD) when D is atomic and every equation \(\pi _1\cdots \pi _k = \rho _1 \cdots \rho _\ell \), with all \(\pi _i\) and \(\rho _j\) irreducible in D, implies \(k=\ell \). We explain how techniques introduced to attack Artin’s primitive root conjecture can be applied to understand half-factoriality of orders in real quadratic number fields. In particular, we prove that (a) there are infinitely many real quadratic orders that are half-factorial domains, and (b) under the generalized Riemann hypothesis, \({\mathbb {Q}}(\sqrt{2})\) contains infinitely many HFD orders.

回想一下，当D是原子且每个方程\(\pi _1\cdots \pi _k = \rho _1 \cdots \ rho _\ell \)且所有\(\pi _i \)和\(\rho _j\)在D中不可约，意味着\(k=\ell \)。我们解释了如何应用用于攻击 Artin 原根猜想的技术来理解实二次数域中阶的半阶乘性。特别是，我们证明（a）存在无限多个半阶乘域的实二次阶，以及（b）在广义黎曼假设下，\({\mathbb {Q}}(\sqrt{2})\ )包含无限多个 HFD 订单。