A refinement of B\'ezout's Lemma, and order 3 elements in some quaternion algebras over $\mathbb{Q}$

Given coprime positive integers $d',d''$, B\'ezout's Lemma tells us that there are integers $u,v$ so that $d'u-d''v=1$. We show that, interchanging $d'$ and $d''$ if necessary, we may choose $u$ and $v$ to be Loeschian numbers, i.e., of the form $|\alpha|^2$, where $\alpha\in\mathbb{Z}[j]$, the ring of integers of the number field $\mathbb{Q}(j)$, where $j^2+j+1=0$. We do this by using Atkin-Lehner elements in some quaternion algebras $\mathcal{H}$. We use this fact to count the number of conjugacy classes of elements of order 3 in an order $\mathcal{O}\subset\mathcal{H}$.