# Pythagoras, triples, 3,4,5, a calculator.

How to generate Pythagorean triples (example: 3,4,5), well one way at least.

Starting with (x + y)2 = x2 + y2 + 2xy and (x – y)2 = x2 + y2 – 2xy

we can write the difference of two squares

(x + y)2  –  (x – y)2 = 4xy

and if we write  x = A2 and y = B2 the right hand side is a square as well.

Thus:

(A2  +  B2) 2 – (A2 – B2) 2 = 4A2 B2 = (2AB) 2

which can be written as

(A2  +  B2) 2 = (A2 – B2) 2 + (2AB) 2

the Pythagoras form.

Now just put in some integers for A and B

2 and 1 gives 3,4,5

Conjecture1: This process generates ALL the Pythagorean triples.

Conjecture2: Every odd number belongs to some  Pythagorean triple.

Have fun…….

My next post will be about finding the radius of the inscribed circle in a right angled triangle…..

Filed under algebra, geometry, teaching

### 3 responses to “Pythagoras, triples, 3,4,5, a calculator.”

1. Elaine Watson

On the first line, did you mean y^2 rather than t^2? I can’t figure out where the t^2 came from.

2. howardat58

The ‘t’ is a mysterious key right next to the ‘y’.
I have fixed it !
Gracias !

• Elaine Watson

Whew! I thought that I was missing some important mathematical concept that I must have skipped in school! I’m glad it’s just bad typing and not a hole in my mathematics education!