## JSH: Pythagorean Triplets and Pell's Equation

Now that I have a general theory for all 2 variable quadratic Diophantine equations it's worth coming back to note again the weird connection I found between certain Pythagorean Triplets and Pell's Equation in the form

x^2 - Dy^2 = 1

when D-1 is a perfect square. For instance for D=2, I have that for every solution of Pell's Equation you have a Pythagorean Triplet!

But the triplets are special in that with u^2 + v^2 = w^2, v = u+1. The connection is that w is x+y from Pell's Equation.

The more general result is that u = sqrt(D-1)j, and v = j+1, while w still equals x+y.

Intriguingly that means that proof that there are an infinite number of solutions for certain Pell's Equations is proof that there are an infinity of Pythagorean Triplets of a certain form!

An easy example with D=2, is x=17, y=12, where notice you are paired with the triplet 20, 21, 29.

That is just some low-hanging fruit that I thought I'd mention. Kind of been a whirlwind of results flowing from playing with my Diophantine Quadratic Theorem.
New argument now I'm starting to see is that I've found nothing new, though I will add that for me the Pell's Equation result is just a fun tidbit which is nothing compared to the main result of generally solving the 2 variable Diophantine equation.

A succinct example of the tidbit result claimed to not be new is the easy to show case that for EVERY solution to

x^2 - 2y^2 = 1

you have a solution to the negative Pell's Equation:

z^2 - 2(x+y)^2 = -1.

For instance, x=17, y = 12 is a solution to the first as

17^2 - 2(12)^2 = 1

and with x+y=29, you get z=41 for the second, as

41^2 - 2(29)^2 = -1.

To me that it's easy to explain so I have to wonder why no one it seems has said it in that way in human history before…

2000 years of mathematical history with Pell's Equation.

The will to lie about a subject that old is a powerful and demonic one, and for those of you who have wondered how I could be right, if so many people are arguing with me, here you can see.

They argue with me because these battles are supposed to be hard.

If it were easy then there wouldn't be a choice, now would there?

I'm set. It's you who has a fate in the balance.

It's your life that is being decided now. Not mine.