## Count the possible numbers of quotients in the Gaussian integers

Recall that for all Gaussian integers $\alpha$ and $\beta$ with $\beta \neq 0$, there exist $\gamma$, $\delta$ such that $\alpha = \gamma\beta + \delta$ and $N(\delta) < N(\beta)$. In general these $(\gamma,\delta)$ are not unique. Determine the possible number of such pairs.

First, note that once $\gamma$ is chosen, $\delta$ is uniquely determined and vice versa.

Suppose $\alpha = \gamma\beta + \delta$. In $\mathbb{Q}[i]$, we have $\frac{\alpha}{\beta} = \gamma + \frac{\delta}{\beta}$. Note that $N(\delta/\beta) < 1$, and moreover that $N(z)$ is merely the square of the Euclidean norm on $\mathbb{Q}[i]$. That is, if $\alpha = \gamma\beta + \delta$, then $\gamma$ is interior to a circle of radius 1 centered at $\alpha/\beta$. Moreover, with a Gaussian integer $\gamma$ chosen such that $\alpha/\beta = \gamma + \delta^\prime$ for some $\delta^\prime \in \mathbb{Q}[i]$ such that $N(\delta^\prime) < 1$, Certainly $\delta^\prime = \delta/\beta$ for some $\delta \in \mathbb{Z}[i]$. Thus the number of solutions $(\gamma,\delta)$ of the equation $\alpha = \gamma\beta + \delta$ is precisely the number of Gaussian integers interior to the circle of (Euclidean) radius 1 centered at $\alpha/\beta$ in the complex plane.

Note that any set of five Gaussian integers must contain a pair whose Euclidean distance from one another is at least 2. So the number of Gaussian integers interior to a circle of radius 1 is at most 4. Indeed, 4 such points are possible, as is the case for the circle centered at $\frac{1}{2} + \frac{1}{2}i$.