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

Starting with (x + y)^{2} = x^{2} + y^{2} + 2xy and (x – y)^{2} = x^{2} + y^{2} – 2xy

we can write the difference of two squares

(x + y)^{2} – (x – y)^{2} = 4xy

and if we write x = A^{2} and y = B^{2} the right hand side is a square as well.

Thus:

(A^{2} + B^{2})^{ 2} – (A^{2} – B^{2})^{ 2} = 4A^{2} B^{2} = (2AB)^{ 2}

which can be written as

(A^{2} + B^{2})^{ 2} = (A^{2} – B^{2})^{ 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…..