Pythagorean Triples

Find all integral solutions to

x²+y²=z².

A triple (x,y,z) of integers satisfying the above equation is called a Pythagorean triple, due to the fact that by Pythagoras' theorem these integers may be taken to be the sides of a right triangle. Some sample solutions are (3,4,5), (5,12,13), (8,15,17), (9,40,41). Later, we shall arrive at a formula for all such triples.