Nice proof that $\mathbb{Z}[\sqrt{6}]$ is a Euclidean domain wrt absolute norm map I know that $\mathbb{Z}\left[\sqrt{6}\,\right]$ is a Euclidean domain with respect to the absolute valued norm map $x+y\sqrt{6} \mapsto |x^2-6y^2|$. I think I proved this result with some common techniques, but the proof is a bit sloppy and it requires a lot of cases. (Basically, I checked that for every $z \in \mathbb{Q}(\sqrt{6})$ there is $\gamma \in \mathbb{Z}[\sqrt{6}]$ such that $|N(z-\gamma)|<1$.)
Does there exist a short proof for this result, with less cases or a more, say, enlightening method?
Many thanks.
 A: this question has been answered here: https://math.stackexchange.com/a/124573/4997

this is a picture of one of the "starfish" - a region bounded by two hyperbolas. 

Reading this review by Franz Lemmermeyer, The Euclidean Algorithm in Algebraic Number Fields 

Theorem The rings $\mathcal{O}_{\mathbb{Q}[\sqrt{d}]}$ are norm-Euclidean if and only if
  $m \in \{ 2, 3, 5, 6, 7, 11, 13, 17, 19, 21, 29, 33, 37, 41, 57, 73\}$.

Fortunately, in the internet age.  Oppenheim's 1934 paper Quadratic Fields with and Without Euclid's Algorithm is online.  The preview page is all you need.

Forget about the case $m \equiv 1 \mod 4$ (since $m = 6$), we want to find $m$ such that for any rational point $(a,b)$ we can find $(x,y) \in \mathbb{Z}^2$ such that $|(x-a)^2 - m (y-b)^2 | < 1 $  
Oppenheim proceeds by process of eliination: if we can find $(a,b) \in \mathbb{Q}^2$ such that for all integers $(x,y) \in \mathbb{Z}^2$ either


*

*$(x-a)^2 - m (y-b)^2 \geq 1 $  

*$m (y-b)^2 - (x-a)^2 \geq 1 $  


Then we have ruled out the possibility of $m$ as a Euclidean domain (with this particular norm).  This then becomes a quadratically constrained quadratic programming problem.  Let's assume $0 < a,b < \tfrac{1}{2}$.
Examining the constraints around the points $(0,0), (1,0), (-1,0)$ one can show $mb^2 \geq 1 + (1 + a)^2  \geq 2$ and $m \geq 8$.  
$m = 2,3,\mathbf{6},7$ work.  
To get $m = 5,13,17,21,29$, a modified Euclidean algorithm gives the same constraints with norm $|(x + \tfrac{1}{2}y)^2 - \tfrac{1}{4}my^2   | $

I read this proof and would like to understand better how these constraints dictate problems with factorization.
