Pythagorean triangles are right triangles with integer numbers for lengths of sides. An example would be a triangle with sides 3, 4 and 5. This triangle is a right triangle since 3^2 + 4^2 = 5^2 or 9 + 16 = 25. You can tell we are using the "Pythagorean" idea that for sides a, b and c of a right triangle a^2 + b^2 = c^2. We will call a collection of three natural numbers a Pythagorean Triple if they represent lengths of sides of a Pythagorean triangle.

If {3, 4, 5} is a Pythagorean Triple, PT, then so is {6, 8, 10} and we will let you prove that. You may want to demonstrate that for any natural number, n, {na, nb, nc} is a Pythagorean Triple when {a, b, c} is a Pythagorean Triple. We will call {a, b, c} a Unique Pythagorean Triple, UPT, if a, b and c do not have a common factor n. This also means that Unique Pythagorean Triples have unique angles since we are eliminating similar triangles.

We can make Prime Pythagorean Triples using two numbers x and y. To accomplish this let a = 2xy, b = x^2 - y^2 and c = x^2 + y^2 where y is less than x. Now show that a^2 + b^2 = c^2. What about reversing the direction where you are given any Pythagorean Triple a, b and c then find an x and y that yields a = 2xy , b = x^2 - y^2 and c = x^2 + y^2 where y is less than x?

How can we pick x and y, with y less than x, in order that we get a Unique Pythagorean Triple? Try using natural numbers that have a greatest commom factor of one and opposite parity. In other words, x and y should be relatively prime and if one is odd the other is even. Of course part of the parity issue is taken care of since x even and y even implies a common factor of 2.

Write a short program for your computer that gives a list of Unique Pythagorean Triples. One outside loop can give natural numbers x and for each x an inside loop could give natural numbers y from one to the number x. For each x and y in this nested loop you can check to see if the greatest common factor is 1 and check to see if x and y have opposite parity. For opposite parity: Mod(x,2) not equal to Mod(y,2) would check the natural number remainder of x divided by 2 against the remainder of y divided by 2. Try creating a few UPTs by hand and with a program on your calculator or computer.

For any Pythagorean Triple {a, b, c} we can show that a/c, b/c and 1 satisfy the Pythagorean identity m^2 + n^2 = 1 and therefore (a/c, b/c) is a rational point on a unit circle.