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?
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?