In order to describe all Pythagorean triples, therefore, it is enough to do thejob for all triples(a, b, c)where the hcf of the three numbers is 1, as all others are merely scaled-up versions of these.The recipe is as follows. Take any pair of coprime positive integers mand n, with one of them even, and let mdenote the larger. Form the triple given by a=2mn, b=m2-n2 and c=m2+n2. The three numbers a, b, and c then give you a Pythagorean triple(the algebra is easily checked)and the three numbers have no common factor(also not difficult to verify). The three examples above arise bytaking m=2 and n=1 in the first case, m=3 and n=2 in the second, while for the last triangle we have m=9, n=4. It takes more work to verify the converse:any such Pythagorean triple arises in this fashion for suitably chosen values of m and n, and what is more, the representation is unique so that two different pairs(m, n)cannot yield the same triple(a, b, c).