Formulas For Generating Pythagorean Triples - Height-excess Enumeration Theorem

Height-excess Enumeration Theorem

Wade and Wade first introduced the categorization of Pythagorean triples by their height, defined as c - b, linking 3,4,5 to 5,12,13 and 7,24,25 and so on.

McCullough and Wade extended this approach, which produces all Pythagorean triples when Write a positive integer h as pq2 with p square-free and q positive. Set d = 2pq if p is odd, or d= pq if p is even. For all pairs (h,k) of positive integers, the triples are given by

The primitive triples occur when gcd(k, h) = 1 and either h=q2 with q odd or h=2q2.