### Monday, December 31, 2007

## JSH: Necessity and discovery

There is a significance I think in my finding some of the most incredible relations in number theory as 2007 closes, as I hope at this point in time that 2008 will open a new and much better chapter on a saga that has tested my ability to answer denial with discovery, as I've been forced to find ways to get around the mathematical community's refusal to acknowledge my research.

Necessity has been the mother of discovery, and at this time I have to admit that I just sit and stare at the fundamental factoring relations almost in disbelief. So yes, it more and more feels worth it, as who knows? Would I have ever really put everything into the factoring problem if I didn't really, really, really need it?

But that is how history goes and I guess why it can be so strange. Now I know I had the ability to make these discoveries because I've made them, but if mathematicians had been an honest and dedicated crew that didn't turn to insults and questioning people's sanity, would I ever have reached my potential?

Regardless, the lost time still kind of bugs me. It has been over five years since I found a proof of Fermat's Last Theorem and the prime counting function, and a bit of time since I put up the prime gap equation. I could have been resting on my laurels years ago.

But in that time I thought out a way for Google to finally monetize YouTube and came up with DMESE, a way to allow people to make legal copies of their DVD's, among a few other things harder to explain in a quick sentence.

Still with all that I have accomplished it is sobering that there is that wait as I look to see if your community will try to do what it did for so many years before now.

Will denial continue? And can it still work with simple mathematics that could if not addressed open the door to all the world's secrets? Will governments continue to BELIEVE and use mathematical ideas that a smart kid and a desktop could now conceivably breach?

Is security really just about people believing it exists? Or will a practical solution force, finally the math community to tell the truth about my work?

I am not very confident so I prepare for the worst and years of more wins for people who don't care at all about what is true. My faith in humanity is at a low.

So I prepare for the worst and if it happens what else could I do? I don't know. But I do know that it will change the world even more whatever it might be.

And maybe that's the point, as maybe that it is my true destiny, to be one of the most influential human beings ever—because people force me.

Necessity has been the mother of discovery, and at this time I have to admit that I just sit and stare at the fundamental factoring relations almost in disbelief. So yes, it more and more feels worth it, as who knows? Would I have ever really put everything into the factoring problem if I didn't really, really, really need it?

But that is how history goes and I guess why it can be so strange. Now I know I had the ability to make these discoveries because I've made them, but if mathematicians had been an honest and dedicated crew that didn't turn to insults and questioning people's sanity, would I ever have reached my potential?

Regardless, the lost time still kind of bugs me. It has been over five years since I found a proof of Fermat's Last Theorem and the prime counting function, and a bit of time since I put up the prime gap equation. I could have been resting on my laurels years ago.

But in that time I thought out a way for Google to finally monetize YouTube and came up with DMESE, a way to allow people to make legal copies of their DVD's, among a few other things harder to explain in a quick sentence.

Still with all that I have accomplished it is sobering that there is that wait as I look to see if your community will try to do what it did for so many years before now.

Will denial continue? And can it still work with simple mathematics that could if not addressed open the door to all the world's secrets? Will governments continue to BELIEVE and use mathematical ideas that a smart kid and a desktop could now conceivably breach?

Is security really just about people believing it exists? Or will a practical solution force, finally the math community to tell the truth about my work?

I am not very confident so I prepare for the worst and years of more wins for people who don't care at all about what is true. My faith in humanity is at a low.

So I prepare for the worst and if it happens what else could I do? I don't know. But I do know that it will change the world even more whatever it might be.

And maybe that's the point, as maybe that it is my true destiny, to be one of the most influential human beings ever—because people force me.

## Fundamental factoring relations

Given

z^2 = y^2 + nT

where T is the target composite to be factored, n is some nonzero integer, and z and y are rational you can solve for z modulo any odd prime p coprime to 3, y, z and nT, with the following crucial congruences:

z = 2^{-1}(3k) mod p

and

k^2 = 2^{-1}(nT) mod p

And those are the fundamental factoring relations.

Those of you who are honest will, once you check and verify that they always work, as they do, that RSA encryption is over. But if you wish to lie to the world and deny that fact then I hope you are brave enough to handle the consequences that I now assure you will be there.

The reason this information makes RSA encryption obsolete is that you can just pick p to be some prime greater than sqrt(nT) and solve for y and z mod p, to solve for factors mod p, and get the smaller factor directly as a residue.

The necessity of shifting n if 2^{-1}(nT) mod p is not a quadratic residue should be fixable with more generalized equations.

Also, it seems to me that you can solve for y directly using -nT, as then you'd have

y^2 = z^2 + (-nT)

and why should the math care if you switch z out from those congruences with y? If so then you can use these congruences to solve for y and z without ever needing to solve a quadratic residue.

z^2 = y^2 + nT

where T is the target composite to be factored, n is some nonzero integer, and z and y are rational you can solve for z modulo any odd prime p coprime to 3, y, z and nT, with the following crucial congruences:

z = 2^{-1}(3k) mod p

and

k^2 = 2^{-1}(nT) mod p

And those are the fundamental factoring relations.

Those of you who are honest will, once you check and verify that they always work, as they do, that RSA encryption is over. But if you wish to lie to the world and deny that fact then I hope you are brave enough to handle the consequences that I now assure you will be there.

The reason this information makes RSA encryption obsolete is that you can just pick p to be some prime greater than sqrt(nT) and solve for y and z mod p, to solve for factors mod p, and get the smaller factor directly as a residue.

The necessity of shifting n if 2^{-1}(nT) mod p is not a quadratic residue should be fixable with more generalized equations.

Also, it seems to me that you can solve for y directly using -nT, as then you'd have

y^2 = z^2 + (-nT)

and why should the math care if you switch z out from those congruences with y? If so then you can use these congruences to solve for y and z without ever needing to solve a quadratic residue.