Finding Pythagorean triples can be thought of as a special case of determining when an integer is the sum of two squares; a Pythagorean triple consists of three positive integers x, y, and z such that
x2 + y2 = z2. You may recognize this equation as being the relation satisfied by the lengths of the sides of a right triangle. Historically, interest in Pythagorean triples started from this geometric connection.
We can search for Pythagorean triples with Java. For the following applet, you provide a bound B and the applet will find all Pythagorean triples with x and y less than or equal to B. Let's try it out.
As you start to look at these triples, you may notice that some are related to others. For example, we have the first triple, (3, 4, 5), and also (6, 8, 10) which can be gotten from (3, 4, 5) by multiplying each number by 2. We also have (9, 12, 15), which can be gotten by multiplying each term of (3, 4, 5) by 3.
More generally, if (x, y, z) satisfies x2 + y2 = z2 and k is an integer, then
(kx)2 + (ky)2 = k2(x2 + y2) = k2z2 = (kz)2. So, (kx, ky, kz) satisfies the same equation. By reversing this argument, we see that if (x, y, z) is a Pythagorean triple, then we can cancel a factor of gcd(x, y, z) to get a triple with gcd(x, y, z) = 1. A Pythagorean triple with gcd(x, y, z) = 1 is called a primitive Pythagorean triple. Every Pythagorean triple is a multiple of a primitive triple.
The next applet works just like the one above, but it only outputs the primitive triples. Here we will try the new applet with the same initial bound as earlier.
As you can see, there are considerably fewer primitive triples in this range. We are now ready for our next Research Question.
Research Question 4
Discover as much as you can about primitive Pythagorean triples.
Research Question 4 is more open ended than any previous Research Question in this course. Do as much as you can with it. We will give you two hints:
- The strongest known conjecture here is connected to the earlier parts of the lab! You may want to start by looking at the numbers which come out as hypotenuses.
- You may want to consider sums and/or differences between the legs and hypotenuse of a Pythagorean triangle.
Section 12.1 | Section 12.2 | Section 12.3 | Section 12.4 | Section 12.5
Copyright © 2001 by W. H. Freeman and Company