## Tuesday, April 07, 2009

### 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?

#### 1 comment:

Kartik Kwatra said...

1.Relatively prime numbers are the ones which don't have any factor in common , i.e their HCF is 1. :)

2. 9^2 + b^2 = (b+1)^2 ==> 2b+1 = 81
==> b = 40 , so triple is 9,40,41

3. E^2 = E, and E + E is always Even

MY QUESTION: so 9,40,41 will be the only triple with 9 as the shortest side ?
I'm preparing for MBA entrance exams like GMAT and CAT, so studying from that perspective (solve big problems in minimum time).
If I wanted to find all the triples with 9 as the shortest side, how to do it?

:) I really loved your post on Pythagorean triples (this post) and i found out the triple so easily. I'm glad.
Thanks a lot :)
Will be eagerly waiting for your response.