### Saturday, September 13, 2008

## Complete theory for 2 variable Diophantine equations, paper now available

I have completed the basic theory for 2 variable Diophantine equations. That is, the mathematical theory covering equations of the form

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

That theory gives the equation for determining existence of integer solutions, as well as a method using what I call Diophantine chains to actually find solutions when they exist.

The paper does include some basic Pell's Equation results as well, like the result that for every solution to Pell's Equation of the form

x^2 - 2y^2 = 1

you have a solution to the negative Pell's equation z^2 - 2(x+y)^2 = -1.

I am publishing through Google Docs:

http://docs.google.com/Doc?id=dfs4mntm_35gc7sc4dv

I did want to put the new mathematical tools forward but also I was curious to find out things like, can people actually access the document? And more importantly, will anyone reply with issues with the details of the theory itself?

After all I found a proof of Fermat's Last Theorem over 6 years ago. I no longer believe in the academic system. I read news reports of supposed findings, or listen to babbling about "String theory" or supposedly how old the universe is, and now kind of yawn, if I bother at all.

It is weird though. I discretized conic sections using my tautological spaces. And added to my math yet another remarkable result, which took me 8 days to do, with mathematical tools so powerful they can do that with little to no effort.

2000 years of mathematical research in that area. 8 days for me to cover with a more powerful and succinct mathematical theory.

So I had this one idea I call tautological spaces and it can do so much, but it doesn't seem like that big of a deal. But I remember being a kid playing with parabolas, graphing them over and over again, and being excited about their properties. Reading other people's math. Now I read my math.

And to my math that I read I now have a full discrete theory for conics in general, figured out within a few days, using a mathematical technique, I invented.

And the world calls me a crackpot.

I no longer believe in people, with good reasons. Over six years of good reasons with the knowledge of my accomplishments and the world's reaction to them. But at least I still like people, mostly. I just kind of see most as, primitive.

I believe in mathematics.

Long after all of you are dead. Long after the sun has cooled and the Milky Way has drifted apart, what I've discovered will still be true.

And I defined mathematical proof. So I know of what I speak.

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

That theory gives the equation for determining existence of integer solutions, as well as a method using what I call Diophantine chains to actually find solutions when they exist.

The paper does include some basic Pell's Equation results as well, like the result that for every solution to Pell's Equation of the form

x^2 - 2y^2 = 1

you have a solution to the negative Pell's equation z^2 - 2(x+y)^2 = -1.

I am publishing through Google Docs:

http://docs.google.com/Doc?id=dfs4mntm_35gc7sc4dv

I did want to put the new mathematical tools forward but also I was curious to find out things like, can people actually access the document? And more importantly, will anyone reply with issues with the details of the theory itself?

After all I found a proof of Fermat's Last Theorem over 6 years ago. I no longer believe in the academic system. I read news reports of supposed findings, or listen to babbling about "String theory" or supposedly how old the universe is, and now kind of yawn, if I bother at all.

It is weird though. I discretized conic sections using my tautological spaces. And added to my math yet another remarkable result, which took me 8 days to do, with mathematical tools so powerful they can do that with little to no effort.

2000 years of mathematical research in that area. 8 days for me to cover with a more powerful and succinct mathematical theory.

So I had this one idea I call tautological spaces and it can do so much, but it doesn't seem like that big of a deal. But I remember being a kid playing with parabolas, graphing them over and over again, and being excited about their properties. Reading other people's math. Now I read my math.

And to my math that I read I now have a full discrete theory for conics in general, figured out within a few days, using a mathematical technique, I invented.

And the world calls me a crackpot.

I no longer believe in people, with good reasons. Over six years of good reasons with the knowledge of my accomplishments and the world's reaction to them. But at least I still like people, mostly. I just kind of see most as, primitive.

I believe in mathematics.

Long after all of you are dead. Long after the sun has cooled and the Milky Way has drifted apart, what I've discovered will still be true.

And I defined mathematical proof. So I know of what I speak.