Sunday, April 11, 2010

Pythagorean triples

Do you ever need to come up with an unusual Pythagorean triple (a set of 3 natural numbers such that a² + b² = c²) quickly?  If you're not a math teacher, maybe you don't.

Here's what I use:

3,4,5 and multiples (6,8,10; 9,16,20; ...)

5,12,13 and multiples

7, 24, 25 and multiples

8,15,17 and multiples

But then my memory gives out, so I have to revert to a formula. For any natural number p, you get

2p, p² - 1, p² + 1.

So for p = 8  you get 16,63,65.

I wondered what other algebraic tricks I could find, and came upon

4p, p² - 4, p² + 4.

So for p = 5 I got 20, 21, 29. Since 29 is prime I know the triple isn't just a multiple of an earlier one.  Odd p values work well with this one:

p = 7: 28, 45, 53

p = 9: 36, 77, 85

Where can we go from here?

No comments:

Post a Comment