Pythagorean Triples: Introduction

Sometimes I get a bug in my head about something and it won't go away until I write it down and work it out. This time I recorded it in a journal, so I decided to upload it here on a non-comic days. Comments are welcome, especially from my students.

A few years back, I got tired of every right triangle problem involving a 3-4-5, 6-8-10, 30-40-50 or 5-12-13 triangle. I started creating a list of other triples that I could use when avoiding irrational numbers.

First discovery: Triples are either E² + O² = O² or E² + E² = E²
(where E is an even number, and O is an odd number), and the latter form could always be reduced to the first form. For need of a term, let's say that the triples that are reduced are in the simplest form, which means that the numbers are relatively prime.

Second discovery: Many (a, b, c) triples could be reduced to (a, b, b + 1), when a was odd, or (a, b, b + 2), when a was even. (There are some that don't fit either, but those will have to wait.)

Squares can be made by summing consecutive odd numbers. When that addend is a perfect square, the result is a Pythagorean Triple.
9 = 3² ==> 3² + 4² = 5²
25 = 5² ==> 5² + 12² = 13²

Was there a quicker way to find the triples for 7, 9, 11, etc?
I'll get to that.

Also notice that 31 + 33 = 64, which is 8², so 8² + 15² = 17².
And it had been under my nose all the time that 4² + 3² = 5².
The simplest Pythagorean Triple of them all, fits both these models.

(to be continued)

Questions for my students

Extra credit for answering or for participating in the discussion.

1. Can you explain in your own words what I meant by "relatively prime"? What do you know about prime numbers that might give you a hint?
   
   
2. Can you find Pythagorean Triples where the smallest side is 9, 11, and 13?
   Remember: 9-12-15 doesn't count. It can be reduced to 3-4-5.
   
   
3. Why can't there be any Triples of the form: Even² + Even² = Odd²?