## Factoring problem trivially solved

We're now firmly in the danger zone. I have a simple solution to the factoring problem. I will give it in a moment but I need to emphasize to you all that there is no longer any room for playing games with this result nor for denial.

I've found in rationals that there is a simple relation connecting what is commonly called Pell's Equation to a discrete ellipse:

Given x^2 - Dy^2 = 1

it must be true that

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

where

j = ((x+Dy)-/+1)/D.

That allows you to solve for the variables by factoring:

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

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

so

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

and as you have j as a free variable, you can let j = mn, where m and n are rationals, f_1*f_2 = (D-1), where f_1 and f_2 are integers and generally solve with:

x+y + mn+/-1 = f_1*m

x+y - (mn +/-1) = f_2*mn^2

so you can, for instance, solve out m, and choose n to be some integer, and then solve for x+y, and then solve for x and y directly with

j = mn = ((x+Dy)-/+1)/D, so you have two simultaneous equations

which gives you a solution to

x^2 - Dy^2 = 1.

And you have available an infinity of such solutions, as many as you wish, simply by cycling through values for n. Or you could solve out n, and cycle through values for m.

Let D equal a target composite T to be factored, then it is shown that you can trivially find an infinite number of solutions in rationals to

x^2 - Ty^2 = 1.

Letting x=r/c and y=s/c, you have then a simple technique for finding integers r, s and c, such that

(r-c)(r+c) = Ts^2

which strongly indicates the factoring problem is simply solved.

The technique allows you to easily, casually, with trivial algebra, generate as many solutions to

x^2 = y^2 mod T

as you wish! Without regard or even consideration to the size of the target composite T.

You DO not have the choice to just play games with this result.

Unfortunately mathematicians thought factoring is a hard problem when it is a trivial one.

Factoring is a trivial problem, solved with easy algebra.