Recursive Enumeration of Pythagorean Triples

Abstract

In [9], P. W. Wade and W. R. Wade (no relation to the second author) gave a recursion formula that produces Pythagorean triples. In fact, it produces all Pythagorean triples (a, b, c) having a given value of the height, which is defined to be h = c − b. For the cases when h is a square or twice a square, they gave a complete proof that the recursion generates all Pythagorean triples. In this note, we give a quick proof of this for all values of h, using a formula that gives all Pythagorean triples. We call the formula the height-excess enumeration because its parameters are the height and certain multiples of the excess e = a + b − c. This enumeration method appears several times in the literature, but does not seem to be widely known. We will discuss these origins after giving the formula. A more extensive treatment of the height-excess enumeration and other applications of it appear in [7]. To set terminology, a Pythagorean triple (PT) is an ordered triple (a, b, c) of positive integers such that a2 + b2 = c2. A PT is primitive when it is not a multiple of a smaller triple. A PT with a < b is called a Pythagorean triangle. A number is called square-free if it is not divisible by the square of any prime number.