Structure of Primitive Pythagorean Triples in Generating Trees

A Pythagorean triple is a triple of positive integers (a, a, b, c ) such that a2 2 + b2 2 = c 2 . If a, b are coprime, then it is called a primitive Pythagorean triple. It is known that every primitive Pythagorean triple can be generated from the triple (3, , 4, , 5) using multiplication by unique number and order of three specific 3 x 3 matrices, which yields a ternary tree of triplets. Two such trees were described by Berggren and Price, respectively. A different approach is to view the primitive Pythagorean triples as points in the three-dimensional Euclidean space. In this paper, we prove that the triple of descendants of any primitive Pythagorean triple in Berggren's or Price's tree forms a triangle (and therefore defines a plane), and we present our results related to these triangles (and these planes).