### Sunday, February 12, 2006

## Latest surrogate factoring, complete

I just hate this, so I guess I still hedged a bit, but now, finally, here's the full thing. I feel forced into giving this theory.

I did keep working on surrogate factoring after the earlier failures on this newsgroup, but just didn't post about it here, while I put some things on my blog. It turns out that I found a simple approach where hey, I can see reason to doubt, but I'll put it up anyway.

I have concluded that I cannot justify withholding this research, as it relies on advanced mathematical techniques which should be of crucial value in the sciences, while economic reasons may seem important today, but in the big scheme of things, scientific advancement is most important.

With T the target integer to be factored I found the amazingly simple system:

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 a problem with division by zero.

Of course the problem then is, how do you pick all those variables so that you can non-trivially factor T?

It's easy, as from the second equation in the system you have

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

so you can just solve for x:

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

and substitute out k_2 using the last equation, and then solve to get

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

and you just pick some nonzero integer v and z, like v = z = 1, plug them in and factor the surrogate to get n, which gives you rational x, and now from the first equation in the system, you have

f_1 = x-n+vz

and

f_2 = vz-x

and that's it. Of course if T is prime then you'll just get T back, but otherwise, I don't see why it shouldn't work, and notice the algorithm that follows is lightweight and easily implemented.

That it's this easy is not actually a surprise as an Anatoly Plotnikov wrote a peer reviewed and published paper back in the late 1990's showing that P=NP, which was mostly ignored by the mathematical community.

I don't know why.

I did keep working on surrogate factoring after the earlier failures on this newsgroup, but just didn't post about it here, while I put some things on my blog. It turns out that I found a simple approach where hey, I can see reason to doubt, but I'll put it up anyway.

I have concluded that I cannot justify withholding this research, as it relies on advanced mathematical techniques which should be of crucial value in the sciences, while economic reasons may seem important today, but in the big scheme of things, scientific advancement is most important.

With T the target integer to be factored I found the amazingly simple system:

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 a problem with division by zero.

Of course the problem then is, how do you pick all those variables so that you can non-trivially factor T?

It's easy, as from the second equation in the system you have

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

so you can just solve for x:

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

and substitute out k_2 using the last equation, and then solve to get

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

and you just pick some nonzero integer v and z, like v = z = 1, plug them in and factor the surrogate to get n, which gives you rational x, and now from the first equation in the system, you have

f_1 = x-n+vz

and

f_2 = vz-x

and that's it. Of course if T is prime then you'll just get T back, but otherwise, I don't see why it shouldn't work, and notice the algorithm that follows is lightweight and easily implemented.

That it's this easy is not actually a surprise as an Anatoly Plotnikov wrote a peer reviewed and published paper back in the late 1990's showing that P=NP, which was mostly ignored by the mathematical community.

I don't know why.