# Pythagorean Triples

We can make Prime Pythagorean Triples using two natural or counting numbers m and n. To accomplish this let m be greater than n then:

a = 2mn,

b = m2 - n2 and

c = m2 + n2 where n is less than m.

Now show that:

a2 + b2 = c2.

What about reversing the direction where you are given any Pythagorean Triple a, b and c then find an m and n that yields:

a = 2mn,

b = m2 - n2 and

c = m2 + n2 where n is less than m?

How can we pick m and n, with n less than m, 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, m and n 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 m even and n even implies a common factor of 2.

The following picture is an example MS Excel table for your computer that gives a list of Unique Pythagorean Triples.

Student Comments

Student _________
There is way more out there [In Web Space] about Pythagorean Triples than I thought there would be, or could be. I thought the Pythagorean theorem was just a simple way to find sides of a right triangle, but now it seems to me like you'd have to be a genius to figure out all those formulas. It's overwhelming when you start looking at some of the twists and turns and secrets some mathematicians try to find from the basic Pythagorean theorem. Another thing that I thought was cool on the web was a site, from the School of Mathematics and Statistics and University of St. Andrews, Scotland, that talked about Pythagorean Triples being in ancient Babylonian/Mesopotamian math as early as 1900-1600 B. C. There must be something special about the Pythagorean theorem if it has fascinated so many people throughout history.

Student _________
What I learned about Pythagorean Triples: I learned that the triples are only three positive integers that satisfy the equation c squared = a squared + b squared. I was surprised to learn that there are so many numbers that can make this work and how long it took me to find three on my own outside of class. It was also surprising to learn that the numbers 1,2, & 7 do not work. The numbers that I found on my own were 40, 30, and50, /15, 20, and 25 / 12, 16, and 20.

Student _________
I chose to prove the Pythagorean triple formula using one of my sets of numbers. The numbers are: 18, 80, & 82. I found my numbers using trial and error. I found a different Pythagorean triple formula on a Web site called cut-the-knot.com. This site had a different formula to provide Pythagorean Triples:

a = n^2 - m^2     b = 2 n m     c = n^2 + m^2       for n > m

So a =(80)^2 - (18)^2     b =2(80*18)     c=(80)^2+(18)^2

a=6076     b=2880     c=6724

(6076)^2 + (2880)^2 = (6724)^2   or   36917776 + 8294400 = 45212176

This n and m formula worked. It works (as demonstrated in class) for all coprime triples. It's a lot easier than trial and error.

Student _________
The information [ ... ] I found on the Internet was helpful after I wrote it on the board and had it explained. I think the method that we did in class is easier and helpful in finding the triple sets. It's interesting that this formula is fool proof.

Student _________
Pythagorean Triples are cool because once you work on three or four you start to see patterns and formulas develop just by working out the problems. For example by working out Pythagorean Triples you start to see the pattern that the square root of numbers are just consecutive odd numbers. Also I think they are fun to solve, because by just being given 2 points you are able to solve the entire problem. I think once people learn how Pythagorean Triples work they have a good foundation to further their understanding of more complicated math problems, but with a simple approach to understand and learn.

Student _________
While researching Pythagorean Triples, I came across an interesting website. This link featured pallindromic Pythagorean Triples. It defines pallindromic as numbers that read left to right (forward), and right to left (backward). Here is the example given:

4004, 630036, 1559551

Student _________
In our study of Pythagorean triples, I became fascinated enough to apply the theory to our past escalator project. If you go to the escalator fun at Plaza Frontanac we had a right triangle with 363 feet for a and 208 feet for side b.

I applied a^2 + b^2 = c^2 much to my surprise this did not work. c = 418.37 feet, not an integer!

In the end I have learned that all Pythagorean triple are right triangles, but not all right triangles are Pythagorean triples. While many may be astounded by how many numbers work, I am left wondering about the more vast collection that do not work?

Student _________

A brief history on early use of Pythagorean triples gives:

"Pythagorean triples are shown by the Pythagorean theorem. The sum of the squares of the sides of a right triangle is equal to the square of the hypotenuse. This relationship has been known since the days of ancient Babylonians and Egyptians. A portion of a 4000 year old Babylonian tablet lists columns of numbers showing what is now known as Pythagorean triples. These sets of numbers satisfy the equation A^2+B^2=C^2."

Student _________
Finding Pythagorean triples is equivalent to locating rational points on the unit circle. http://www.cut-the-knot.com/pythagoras/pythTriple.html Pythagorean triples are fun. It's interesting that the square root of adding two counting number squares sometimes equals another counting number. Also, the equation is related to a circle.

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For more information and ideas about working with Pythagorean Numbers you may return to the last Web page

Danielle, Darlene, Kellie, Jill, Nanyal, Kevin, Rachel, Christina, Jennifer, Candice, Vance, Esther and Melinda demonstrate calculations using distance formulas and equations of circles. Both ideas are develpoed from a2 + b2 = c2.     A "Service Learning" Project to provide web pages for our community's better understanding of mathematics.