prime factors, this doubling is also true for them. Hence, and are squares. Thus, the first factor can be written The real and imaginary parts of this equation give the two formulas: For any primitive Pythagorean triple, there must be integers and such that these two equations are satisfied. Hence, every Pythagorean triple can be generated from some choice of these integers. As perfect square Gaussian integers If we consider the square of a Gaussian integer we get the following direct interpretation of Euclid's formula as representing the perfect square of a Gaussian integer. Using the facts that the Gaussian integers are a Euclidean domain and that for a Gaussian integer p is always a square it is possible to show that a Pythagorean triple corresponds to the square of a prime Gaussian integer if the hypotenuse is prime. If the Gaussian integer is not prime then it is the product of two Gaussian integers p and q with and integers. Since magnitudes multiply in the Gaussian integers, the product must be , which when squared to find a Pythagorean triple must be composite. The contrapositive completes the proof. Distribution of triples There are a number of results on the distribution of Pythagorean triples. In the scatter plot, a number of obvious patterns are already apparent. Whenever the legs of a primitive triple appear in the plot, all integer multiples of must also appear in the plot, and this property produces the appearance of lines radiating from the origin in the diagram. Within the scatter, there are sets of parabolic patterns with a high density of points and all their foci at the origin, opening up in all four directions. Different parabolas intersect at the axes and appear to reflect off the axis with an incidence angle of 45 degrees, with a third parabola entering in a perpendicular fashion. Within this quadrant, each arc centered on the origin shows that section of the parabola that lies between its tip and its intersection with its semi-latus rectum. These patterns can be explained as follows. If is an integer, then (, , ) is a Pythagorean triple. (In fact every Pythagorean triple can be written in this way with integer , possibly after exchanging and , since and and cannot both be odd.) The Pythagorean triples thus lie on curves given by , that is, parabolas reflected at the -axis, and the