Recall that for all Gaussian integers and with , there exist , such that and . In general these are not unique. Determine the possible number of such pairs.
First, note that once is chosen, is uniquely determined and vice versa.
Suppose . In , we have . Note that , and moreover that is merely the square of the Euclidean norm on . That is, if , then is interior to a circle of radius 1 centered at . Moreover, with a Gaussian integer chosen such that for some such that , Certainly for some . Thus the number of solutions of the equation is precisely the number of Gaussian integers interior to the circle of (Euclidean) radius 1 centered at 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 .