### Sunday, February 12, 2006

## Factoring problem, solution and proof

This post outlines a solution to the factoring problem with working equations and a proof that it is a solution.

Given a composite T to be factored, a factorization algorithm follows from the system of equations:

T = (x-n+vz)(vz-x)

when

k_2 z^2 + nx - x^2 = T

and

z = nv/(v^2 - k_2)

where v does not equal sqrt(k_2) to prevent division by zero.

>From the second equation in the system you have

x^2 - nx + T - k_2 z^2 = 0

and solving for x gives

x = (n +/- sqrt(n^2 - 4(T - k_2 z^2)))/2

and now you pick some nonzero integers k_2 and z, for instance

k_2 = z = 1

and then factor

4(T - k_2 z^2)

to get integer n, where with g_1 and g_2 its integer factors, that is

triival as

n = g_1 + g_2

where

g_1 g_2 = 4(T - k_2 z^2).

Given integer n, you have integer x or 2x is an integer, and you next solve for v, using the third equation in the system, which gives

v = (n +/- sqrt(n^2 + 4z^2 k_2))/2z

which may be non-rational for a given g_1 and g_2, but provably will be rational for some g_1 and g_2.

Given v, you now have the factorization as

f_1 = x-n+vz

and

f_2 = vz-x.

The proof that the system must factor is trivial. Note that for any integer factors f_1 and f_2, given any integer value for z, there must exist rationals x, v and n that will satisfy those equations for f_1 and f_2.

But, any integer value of n, must be a solution of the square root in

x = (n +/- sqrt(n^2 - 4(T - k_2 z^2)))/2

or x could not be an integer.

Therefore, given that T is composite, and an integer n and x must exist, some value of g_1 and g_2 must give that n, and that x, which will give integer v, so a factoring solution must be determinable by this method.

Note that

f_1 + f_2 = 2vz + n

and

f_1 - f_2 = 2x - n

so again, given just z, there must exist rationals x, v and n for f_1 and f_2, completing the proof.

For example, if you pick z=1, with T = 15, and f_1 = 3, and f_2 = 5, you have

3 = x - n + v and 5 = v - x

so

8 = 2v + n and 5 = 2x - n

and rational solutions can readily be found. Notice there are an infinity of them, but your choice of factors g_1 and g_2 from

g_1 g_2 = 4(T - k_2 z^2).

narrows the set down to a finite one.

The proof is done, but the social consequences haven't started yet. I do wonder why reason failed here.

Notice that it has already been some time since I started explaining what I'd do, and began the implementation, but so far you still have no movement from the mathematical community.

The evidence clearly shows that they are going to push this to the limit, and wait until society comes in and forces them to face the truth.

I doubt it'll take long. My decision is, again, for the future. Discoverers who stop to let the truth lose for some small group of people are putting the future in jeopardy for the comfort of a few, against the welfare of all those yet to be born.

The good of the many, outweighs the needs of the few.

I do hope there are many more to be born, but some of you may disagree.

However, it was my decision.

Good night, and good luck.

Given a composite T to be factored, a factorization algorithm follows from the system of equations:

T = (x-n+vz)(vz-x)

when

k_2 z^2 + nx - x^2 = T

and

z = nv/(v^2 - k_2)

where v does not equal sqrt(k_2) to prevent division by zero.

>From the second equation in the system you have

x^2 - nx + T - k_2 z^2 = 0

and solving for x gives

x = (n +/- sqrt(n^2 - 4(T - k_2 z^2)))/2

and now you pick some nonzero integers k_2 and z, for instance

k_2 = z = 1

and then factor

4(T - k_2 z^2)

to get integer n, where with g_1 and g_2 its integer factors, that is

triival as

n = g_1 + g_2

where

g_1 g_2 = 4(T - k_2 z^2).

Given integer n, you have integer x or 2x is an integer, and you next solve for v, using the third equation in the system, which gives

v = (n +/- sqrt(n^2 + 4z^2 k_2))/2z

which may be non-rational for a given g_1 and g_2, but provably will be rational for some g_1 and g_2.

Given v, you now have the factorization as

f_1 = x-n+vz

and

f_2 = vz-x.

The proof that the system must factor is trivial. Note that for any integer factors f_1 and f_2, given any integer value for z, there must exist rationals x, v and n that will satisfy those equations for f_1 and f_2.

But, any integer value of n, must be a solution of the square root in

x = (n +/- sqrt(n^2 - 4(T - k_2 z^2)))/2

or x could not be an integer.

Therefore, given that T is composite, and an integer n and x must exist, some value of g_1 and g_2 must give that n, and that x, which will give integer v, so a factoring solution must be determinable by this method.

Note that

f_1 + f_2 = 2vz + n

and

f_1 - f_2 = 2x - n

so again, given just z, there must exist rationals x, v and n for f_1 and f_2, completing the proof.

For example, if you pick z=1, with T = 15, and f_1 = 3, and f_2 = 5, you have

3 = x - n + v and 5 = v - x

so

8 = 2v + n and 5 = 2x - n

and rational solutions can readily be found. Notice there are an infinity of them, but your choice of factors g_1 and g_2 from

g_1 g_2 = 4(T - k_2 z^2).

narrows the set down to a finite one.

The proof is done, but the social consequences haven't started yet. I do wonder why reason failed here.

Notice that it has already been some time since I started explaining what I'd do, and began the implementation, but so far you still have no movement from the mathematical community.

The evidence clearly shows that they are going to push this to the limit, and wait until society comes in and forces them to face the truth.

I doubt it'll take long. My decision is, again, for the future. Discoverers who stop to let the truth lose for some small group of people are putting the future in jeopardy for the comfort of a few, against the welfare of all those yet to be born.

The good of the many, outweighs the needs of the few.

I do hope there are many more to be born, but some of you may disagree.

However, it was my decision.

Good night, and good luck.