**Ref: Professor Stewart’s Cabinet of Mathematical Curiosities: Ian Stewart**

**Pythagorean Triples:**

I can’t really get away without telling you Diophantus’s method for finding all Pythagorean triples, can I?

OK, here it is. Take any two whole numbers, and form:

- twice their product
- the difference between their squares
- the sum of their squares

Then, the resulting three numbers are the sides of a Pythagorean triangle.

For instance, take the numbers 2 and 1. Then,

- twice their product =
- the difference between their squares =
- the sum of their squares =

and, we obtain the famous 3-4-5 triangle. If instead we take numbers 3 and 2, then

- twice their product =
- the difference between their squares =
- the sum of their squares =

and, we get to the next-most-famous 5-12-13 triangle. Taking numbers 42 and 23, on the other hand, leads to

- twice their product =
- the difference between their squares =
- the sum of their squares =

and no one has ever heard of the 1235-1932-2293 triangle. But these numbers do work:

.

There’s a final twist to Diophantus’s rule. Having worked out the three numbers, we can choose any other number we like and multiply them all by that. So, the 3-4-5 triangle can be converted to a 9-12-15 triangle by multiplying all three numbers by 3, or to an 18-24-30 triangle by multiplying all three numbers by 6. We can’t get these two triples from the above prescription using whole numbers, Diophantus knew that.

More later,

Nalin Pithwa