A Pythagorean triple is a triplet , where are integers such that .

It is called a Pythagorean triple because it can represent 3 sides of a right triangle. You probably already know two Pythagorean triples: and . Are there other triples? How do we find them?

Let’s consider the equation

where are integers. If have a common factor , say , , and . Then we have

which is true if and only if

So let’s assume that have no common factors. Such triples will be called *primitive*. Since odd perfect squares are congruent to 1 mod 4 and even squares are congruent to 0 mod 4, then must be odd and exactly one of or is even. Let’s suppose that is even: for some integer . Then

So both and must be even. Say and . Then and we have and . Since and have no common factors, we must have that and have no common factors. Since is a square, then both and are squares. Say and . So if *(a,b,c)* is a primitive triple.

Hence, any Pythagorean triple, with even, is either of the form or it is a multiple of that form. You can plug in different values for and to get different right triangles with integer sides. Fun!