Pythagorean Triples

Are integer length sides of right triangles. Here's a trick for finding them.

What might one want to do with a list of pythagorean triples? Here are a few ideas:

• Impress your friends who have only just learned about Pythagorean triples.
• It gets boring working with (3,4,5) all the time.
• If designing a contaner that is to be decorated by gluing lots of some small object to the sides in a diagonal x-shape, try laying out a Pythagorean triangle in the objects -- choosing a Pythagorean triangle that has proportions that fit your materials or goals.
• Chain two together to make a rectangular box with an integer length diagonal. For example, take 3,4,5 and 12,5,13... make two edges of a box 3 and 4 units long. Then the diagonal of a face they share will be 5. Now make the third edge length of the box 12 units long. The long diagonal of the box is the 13-unit-long hypotenuse of the triangle formed by one of the 12-unit-long edges and the 5-unit-long diagonal of a face.
  Does an example exist of such a 4-dimensional box?

How it works

The formulas for x, y, and z are as follows: x = 2*s*t, y = t^2 -s^2, and z = t^2 +s^2, you pick s and t from the positive integers. For example, if s=1 and t=2 you get (3,4,5).

If you want to ensure that the triple can't be reduced by dividing by a common factor, follow this rule: make s and t coprime (having no common divisors other than 1) and not both odd. A non-reducable Pythagorean triple is called a primitive Pythagorean triple.

It can easily be verified that x^2 +y^2 = z^2, with x, y, and z in terms of s and t, the tough part is in deriving these equations from scratch.

I've even gone so far as to generate Pythagorean triples from one number.