## JSH: Awesome simplification

Sorry to add a talking about post to my other post giving a route to a general solution to 2 variable Diophantine equations but it's just so freaking cool. I achieved the dream of every decent physics student: stunning simplification.

Now whenever you have some complicated Diophantine in 2 variables, like

c_1*x^2 + c_2*xy + c_3*y^2 = c_4 + c_5*x + c_6*y

you can turn it into

A(x+y)^2 - B(x+y) + C = S^2

where I gave how you calculate A, B and C in my earlier post.

That is HUGE. As in awesomely huge, and it's just so cool to me that I was the one to discover the mathematics.

And some of the other results that have followed quickly have been just kind of cool, as in, who knew?

Who knew that for ever Pell's Equation solution, solutions to x^2 - Dy^2 = 1, you had an integer solution to a circle or an ellipse?

And you can just SEE it in action with x^2 - 2y^2 = 1, where scan through any list of Pythagorean Triplets—integer solutions to u^2 + v^2 = w^2—and the cases where x and y can be integers just jump out at you, for instance Wikipedia has such a list at

http://en.wikipedia.org/wiki/Pythagorean_triple

conveniently at the top of the page, and I can just look through it and see 3,4,5 and 20,21,29 as the ONLY cases where solutions for x^2 - 2y^2 = 1 can exist with x+y less than 100, as w=x+y, and v=u+1, for every solution to x^2 - 2y^2 = 1.

That is for EVERY solution of x^2 - 2y^2 = 1, there is a Pythagorean triplet where w=x+y, and v=u+1, where

u^2 + v^2 = w^2.

And did you know that for EVERY solution to x^2 - 2y^2 = 1, there is a corresponding solution to

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

There are these DEEP mathematical connections that have physics implications and I'm glad to be able to share them with you.

Hope you go to my post: Solving Quadratic Diophantine equations in 2 variables

The mathematics is a stunning simplification over a previously complex area. The implications are huge. And the Gold Rush is on!!!

What can you now prove in physics that you couldn't even touch before?

The prizes of knowledge and that other stuff await those who figure it out first.