Half-factorial real quadratic orders

Abstract

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.