Pythagorean numbers, more typically referred to as Pythagorean Triples, are those sets of three integers that define all integral primitive and non-primitive Pythagorean right triangles. General formulas for deriving all integer sided right-angled Pythagorean triangles, have been known since the days of Diophantus and the early Greeks. For the right triangle with sides x, y, and z, z being the hypotenuse, the lengths of the three sides of the triangle can be derived as follows: x = k(m^2 - n^2), y = k(2mn), and z = k(m^2 + n^2) where k = 1 for primitive triangles (x, y, and z having no common factor), m and n are arbitrarily selected integers, one odd, one even, usually called generating numbers, with m greater than n. Another set that was attributed to Pythagoras took the form of x = 2n + 1, y = 2n^2 + 2n, and z = 2n^2 + 2n + 1 where n is any integer. (It was ultimately discovered that these formulas created only triangles where the hypotenuse exceeded the larger leg by one.) Another set of expressions that produce triples is x = n^2, y = (n^2 - 1)^2/2, and z = (n^2 + 1)^2/2. For any positive integer m, 2m, m^2 - 1, and m^2 + 1 are Pythagorean Triples.