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.