Reply to Comment on "Pythagoras' Theorem"

I was scrolling through my comments, and
I found one on one of my older posts, titled,
"Pythagoras' Theorem." I have shown it below.

Therefore, I have decided to prove my
statement, which is also in the picture above.

I start off with saying that 'a', the first part of
my Pythagorean triplet, has to be an odd number.
B, then has to be (a² - 1)/2. C is then (a² + 1)/2.

Now, input these into Pythagoras' Theorem, and
see if it works out.

a² + b² = c²


a² + [ (a² - 1)/2 ]² = [ (a² + 1)/2 ]²


a² + (a² - 1)² / 4 = (a² + 1)² / 4

a² = (a² + 1)² / 4  -  (a² - 1)² / 4

a² = [(a² + 1)²   -  (a² - 1)² ]/ 4

a² = [(a⁴ + 2a² + 1) - (a⁴ - 2a² + 1)]/ 4

a² = [a⁴ + 2a² + 1 - a⁴ + 2a² - 1]/ 4

a² = [4a²]/ 4

a² = a²

Thus, we can see that the sum works out.
It's a little messy trying to write mathematical
equations in a blog post, so I'll try and put it in
LaTeX.

I'll also be looking to expand the initial post,
to include even numbers. I think, in the post,
I mentioned "two short algorithms", meaning
that I had the intention of including both even
and odd, but might have forgotten.

Edit (On 28-Feb-2016):

I received a comment asking me how I derived it.
I realized that I had forgotten to include that in the
initial post and this one as well.

My initial idea came when I noticed that the
difference between two consecutive squares was
the odd numbers. This emerged from the fact that
a² - b² = (a+b)(a-b)
And when a and b are consecutive, a-b becomes 1.
We can apply this to Pythagoras' Theorem, as below:

a² + b² = c²
c² - b² = (b+c)

As long as b+c is a square number, we can use this
to derive the Pythgorean triplet from an initial odd
value.

Let the odd number be given as a. This is part of the
Pythagorean triplet, but its square will be used in the
calculation.

a² = (b+c)

Now, because b and c are consecutive, c = b+1.
This makes the equation.

a² = 2b + 1

Isolating b, we get:

b = (a² - 1)/2
c = (a² - 1)/2 + 1 = (a² + 1)/2

In the following post, I hope to include a similar
conclusion for even Pythagorean values.