*a*² +

*b*² =

*c*²) 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

2

*p*,

*p*² - 1,

*p*² + 1.

So for

*p*= 8 you get 16,63,65.

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

4

*p*,

*p*² - 4,

*p*² + 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?