Trees of integral triangles with given rectangular defect

The rectangular defect of a triangle with side lengths aa, bb and cc is a2+b2−c2a2+b2−c2 where a,b≤ca,b≤c. For a given integer dd we examine the set PIT(d)PIT(d) consisting of all primitive integral triangles with rectangular defect equal to dd. There are simple transformations τ1τ1, τ2τ2 and τ3τ3 which produce new elements of PIT(d)PIT(d) from any triangle with defect dd. They determine a partial ordering on PIT(d)PIT(d) in which applying any τiτi moves upward. We will show that the poset PIT(d)PIT(d) has finitely many components and that each of these components is isomorphic to one of two rooted trees TT or T˜ (where TT is the regular rooted tree of valence three and T˜ is a subtree of it). It follows that the minimal elements of PIT(d)PIT(d) form a finite set from which any triangle in PIT(d)PIT(d) can be uniquely obtained by applying a finite sequence of the τiτi’s.In order to prove these statements we will analyze a larger poset Σ(d)Σ(d) which contains copies of both PIT(d)PIT(d) and its inverse −PIT(d)−PIT(d) as subposets. The elements of Σ(d)Σ(d) are equivalence classes of solutions to the equation x12+x22+x32−2x2x1−2x2x3=d. The key result will assert that the complement of ±PIT(d)±PIT(d) in Σ(d)Σ(d) is a finite poset, denoted by Core(d). The proof of this key result is very different according to whether dd is nonpositive (the obtuse case) or dd is positive (the acute case), and the two cases must be analyzed separately. In the obtuse case we will see that the components of Core(d) are singletons while in the acute case they are poset segments or poset circuits (these are the finite connected posets in which each element has at most two neighbors). For all values of dd the analysis of Σ(d)Σ(d) will produce algorithms for constructing both Core(d) and the minimal elements of PIT(d)PIT(d).