Trees of integral triangles with given rectangular defect (Q1759810)

The rectangular defect of a triangle with side lengths \(a,\) \(b,\) and \(c\) where \(a,b \leq c\) is given by \(a^2 + b^2 - c^2,\) a measure of how far a triangle is from being a right triangle (defect zero). Let \(\text{PIT}(d)\) denote the set of all primitive (\(\gcd(a,b,c)=1\)) integral (side lengths are integers) triangles with defect \(d\) (also an integer). It is known that there are three transformation that produce new primitive integral triangles from a given triangle while preserving the defect. These transformations induce a partial ordering on \(\text{PIT}(d)\), where a given triangle is less than one that is obtained from it via one of the three transformations. The poset \(\text{PIT}(0)\) consisting of Pythagorean triples has been well-studied; it can be described as a regular rooted tree of valence three whose root corresponds to the right triangle with sides \(3,4\), and \(5\) (i.e., every primitive integral right triangle can be obtained via a unique series of transformations starting with \(\triangle(3,4,5)\). The structure of \(\text{PIT}(1)\) and \(\text{PIT}(-1)\) have also been completely determined. In this paper, the author gives a complete description of the structure of the poset \(\text{PIT}(d)\) for an arbitrary integer \(d\). It is proved that \(\text{PIT}(d)\) is isomorphic to a disjoint union of finitely many rooted trees, where each tree is either the rooted tree of valence three or a specific subtree of it. As a result, the roots of these trees form minimal elements from which every triangle of defect \(d\) can be obtained via a unique sequence of transformations. The proofs of these results yield methods to construct explicitly the minimal elements.