Monday, February 09, 2009


JSH: Pell's Equation, factoring, and two way method

I will admit I've been a bit puzzled by recent false claims about my latest research linking Pell's Equation to a discrete ellipse and in so doing to integer factorization. I've heard of academic jealousy but what's going on here is almost beyond belief.

To understand why, here are the equations again in rationals:

x^2 - Dy^2 = 1

you have solutions for an ellipse or Pythagorean triples with

(D-1)j^2 + (j+/-1)^2 = (x+y)^2

where j = ((x+Dy) -/+1)/D, and j can be a fraction.

I use rationals so that it is all more compact.

The result above links solving the Pell's Equation to solving a discrete ellipse—a much easier problem!!!

One approach I've given is to let j = uv, introducing yet another

degree of freedom, and use the discrete ellipse:

(D-1)j^2 = (x+y - (j+/-1))(x+y - (j+/-1))


(D-1)(uv)^2 = (x+y - (uv+/-1))(x+y - (uv+/-1))

and using f_1*f_2 = D-1 you can factor with:

(x+y - (uv+/-1)) = f_1*u


(x+y - (uv+/-1)) = f_2u*v^2

That's just one way to go, but notice I can now solve for u in terms of v (or vice versa but I think it's easier) and solve for x+y, and then solve for x and y directly as functions of v.

So how can I evaluate claims against this method using those equations above?

They go in BOTH directions.

So you can actually factor a composite D, get x and y, in rationals, and solve for j, and then have a choice of u and v.

What I've done is turn the factoring problem into a calculus problem of finding minima.

There are posters who unbelievably are making false claims about this research, but all you have to do is go backwards from Pell's Equation.

For instance, if x = r/c and y = s/c, then you can just let s=1, and use r+c = g_1, and r-c = g_2, where g_1*g_2 = D, and solve for r and c, and then find j and consider available u and v.

For some of you that may be the best way to approach this research. Pick some easy composites for your D. Find r and c, and then get j, and consider values for u and v that will work with it.

You will find minima are what work. Guaranteed.

Now the factoring problem is trivially solved. Posters arguing against it on newsgroups won't change that, and the equations will work no matter what any of you say.

But do any of you believe in mathematics at all?

I mean, a solution to the factoring problem using Pell's Equation, of all things, which brings in the calculus turning the problem into a minima problem is just very cool!

To hate this result because you don't like me or are jealous of me or want to protect your jobs or because you're just terribly dense is just sad. Very, very sad.

It doesn't get much bigger than this result. It's one of the hugest finds in all of mathematical history.

And there are people freaking lying about it??!!! What gives?

What's wrong with you people?

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