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.

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