I was reading that the proof of the fact that $R =\mathbb Z [(1+\sqrt{-19})/2]$ is a principal ideal domain from here
It actually shows that $R$ is a Dedekind-Hasse domain, that is let $ \alpha , \beta \in R $ then there exists $ \gamma , \delta \in R$ such that $N(\alpha/\beta*\gamma-\delta) <1$, where $N$ is the D-H norm.
To prove that he comes up many cases. I am not able to understand how does he come up with these cases.