Harristotelian Logic

I think the problem of the modern mathematical community is that people within it were satisfied with the appearance of success versus understanding that the accolades for achievements are secondary, but should come because what has been accomplished is actually difficult.

People give you some respect for having accomplished something very hard to do.

But math society replaced that with faking mathematics and getting the rewards, without the actual success, and then compounded the problem by going after people who actually DID get the accomplishments, as if all that mattered was the appearance of success.

What I've had to do is very difficult. First I had to actually find a proof of Fermat's Last Theorem, which forced me to move into an entirely new area of number theory that I designated non-polynomial factorization, and then I had to deal with a math society that didn't follow its own rules.

Publication didn't even matter, and looking at an entire math journal go under and nothing happen I was gripped with the fear that I might completely fail in getting acknowledgment for my research just because some people had managed to get themselves in a powerful position where they could lie about it for their own selfish reward.

And then I even could poke holes in research claimed to be top of the line.

Moving to factoring was about desperation as much as anything as my other mathematical research didn't have an impact and I realized that a deep corruption had taken over the mathematical field and realized it expanded beyond that into other areas of academics as I could look at the story of Dr. Halton Arp, and compare my situation with a LEADER in his field who got the crackpot label, who had compelling evidence to support his case, but he couldn't get past a group decision to just ignore the evidence. And this guy was an assistant to Hubble.

To keep doing his research he had to leave this country, his own country, to go to Germany so that he could get telescope time. A leader in his field.

Watching the world I could see the rise of George W. Bush who came forward with supposedly easy answers that boiled down to go to war, even if you had to lie to do it, and bankrupt the U.S. Treasury if you could, but just keep smiling, and I understood that my world had a problem.

Winning is hard.

Declaring success like George W. Bush did in Iraq, years ago, is a lot easier.

And if you look at people with the idea that what matters is what you can get from them, and think that it would be nice to, say, get treated like an Olympic athlete even if you cannot make the accomplishments, then you can do things like modern mathematicians have done and actually fail but congratulate yourself on getting the accouterments of success, without actually having accomplished anything.

But at the end of the day, the real judgment may be on the human race itself as with so many massive failures in our world, and with it harder and harder to see a good ending, maybe George W. Bush and mathematicians and cosmologists are just the outward sign of our coming failure as a species to have what it takes to survive, long-term.

The next step for humanity clearly is to push out into the greatest frontier, if it has what it takes, and make no mistake, sure, you may hear that faster than light travel is impossible, or that there is no way that our descendants could travel to other galaxies but your ancestors didn't believe men would ever fly.

The decision point is coming for our species, and it looks close. I can easily see a steady drumbeat of failure to a muddled extinction and if it happens how can I say it's not completely fair?

We ARE failing as a species. More and more that is clear. We are failing.

Winning is hard.

If our species loses, it dies. Others will someday take our place as we took the place of others.

It's that simple.

There will be no acting then. No faking success as survival is success. And if that happens at least this planet saw centuries of real accomplishment by people who did their best because for them there was no other way and I deeply believe that counts for something, somewhere.

As at the end, if there is no way to stop these people who fake it so viciously, then that may be all we have at the end.

Winning is hard. Losing is eternal.

Thought I should mention that examples I give from my research program testing surrogate factoring may be confusing, as you will see results that seem to defy what I say is the theory!!!

Like a prime p too large by the rules I've given will work, or f_1 mod p and f_2 mod p won't match what the theory says they should be for rational solutions.

And that's where the confusion can come in, as I solved out the rational theory of surrogate factoring, but the equations can factor with non-rationals, which behave, non-rationally!!!

For those who need an objective feel of that versus me just babbling, the full set of equations are:

x^2 = y^2 mod p

z^2 = y^2 + nT

2ax = k

and

z = x + ak

and the rational solutions I look for will satisfy that full set of equations with all integers. But the non-rational solutions can give you integers z and y, and factor T, while x, a, and k are non-rational, though their RESIDUES modulo p will be integers.

(The curious can just get some composite T, and try to find integer solutions that satisfy all those equations.)

My guess is that 50% of the solutions from surrogate factoring will come from non-rationals, which don't follow the rational rules, but I haven't tested that as I see it as irrelevant at this point to getting a practical algorithm going.

But it is a wild quirk of the mathematics. The non-rationals will factor for you, but they follow different rules in doing so from the rationals, so, um, you kind of have a crazy side to surrogate factoring as it's, well, not rational.

For those who missed it, a poster who has routinely critiqued my factoring research made a post noting that my latest research gave an algorithm better than Fermat's and I guess that was cool of that person to admit.

I am currently working on my own practical implementation of what I call surrogate factoring that follows from the latest theory, and it is pleasingly fast, while I still have to work at getting it to factor really big numbers to my satisfaction. Oh, my stated goal is the ability to factor an RSA encryption sized number in under 10 minutes on a standard desktop.

Now there is no doubt that I've discovered a new factoring method then. And that factoring method at its heart relies on the use of two congruences where mathematicians have traditionally used one:

x^2 = y^2 mod p

and

z^2 = y^2 mod T

where T is your target composite to factor and p is what I call a helper prime, as it is just there to help you factor, and otherwise does nothing. So it's just kind of like your buddy. There to help you, then kind of wander off, other things to do…

There are rules for p though, as, with T factored into two positive factors, p must be LESS than the smaller factor or p minus the smaller factors must be less than it, which will make more sense later with the congruence relations for those factors modulo p.

But incredibly, as in monster math that just kind of boggles the mind, surrogate factoring tells you that there is a deeper structure to z, where

z = (1 + 2α^2)k/(2α)

where

k^2 = (1 + α^2)^{-1}(nT) mod p

and 'n' is a helper variable allowing you to shift things around when needed, and α is found such that k exists.

And finally, with f_1*f_2 = nT:

f_1 = αk mod p

and

f_2 = α^{-1}(1 + α^2)k mod p.

So there is this whole zoo of new relations for the classical difference of squares that follow from just adding one more congruence!!! And it just gets better.

It turns out that for solutions in all integers, k will be near what I call k_0, where k_0 is the value of k such that

abs(nT - (1 + α^2)k_0)

is a minimum. Which is just one of the most beautiful pieces of mathematics that you will ever see, and it helps a lot.

So you can factor T, modulo p, where to find p you look for big primes that are not too big, and solve for k modulo p, and then find a k with that residue near k_0 and then you can look for the k that will factor modulo p.

Great!!! Easy!!! Factoring made easy!!! Pushing the easy button on factoring!!!

So why am I still babbling on newsgroups about this and math society isn't hollering to the public about it?

Because we are in an upside down situation.

I liken it to if Olympic runners faced races with people who were unethical actors who fantasized at being Olympic runners, who were in the stands, and were the judges and were everywhere, so that when the Olympic runner ran the race and won, they'd just say he didn't and award the gold medal to one of their own.

You see, I have other major discoveries. Math people just lie about them. And call me names. Nasty names, like crackpot and crank. And even NASTIER names.

They can get away with this as they have what I call, critical mass.

MOST mathematicians around the world are like those wannabe runners. So they can just as a group, lie, and who can challenge them then?

Only factoring can which is where this situation gets really, really, really strange, as hey, if the surrogate factoring theory is correct, then quite a few people around the world can now, um, probably, um, factor really huge numbers, but if math people ARE wannabes like I say, they have NO INCENTIVE to tell the truth, except being decent human beings, but I digress.

So game theory says they will lie and claim that nothing has changed, nothing is wrong, nothing to see here, and will explain any security breaches if they happen, away, and will do so indefinitely.

So game theory says they will do their best to collapse human civilization as we know it, and we are traveling down that scenario now.

End of this tale, may be that history is being ended now. The front story to troubles with stock markets will carry things so far, but eventually, oh, eventually civilization as we know it will just collapse, people will turn to endless wars, there will be mass starvation, disease, lots of nasty flies and things, Armageddon and all that stuff, but…

Yes, there is a but. I didn't like that end to the story.

So I changed it.

Oh, but I had to lose two countries.

It is so tiring to chase these mathematical results. It is so easy to just stop and try to rest with something that is wrong. Arguing with people who just want you to be wrong no matter what helps. It helps you keep going. And it helps you push to a solution.

It is so tiring. So, so very tiring.

But mathematics is nothing if it's not right.

But human nature is such that you may convince yourself and convince others because it feels good to be right.

So toss all of that out the window by having everyone that is critiquing you, hate you.

And argue.

I don't give a damn who I convince if I'm wrong.

But if I'm right, I mostly wish to convince myself.

Congress is looking at excessive executive pay packages where failures in terms of how well their company does are getting TONS of money in the subprime mess, when they leave, as in CEO's getting mega pay to fail.

Wow! What a system. Fail and make out like a bandit. Gee, do you really think they EARNED that money?

If you do, want some swampland in Florida?

They rigged the system so that they would be paid even with failure.

So succeed or fail, they get paid.

Academics have the same system—succeed or fail, get paid.

I want to change that and have always needed your help. I see many of you as fakes in a fake system who teach fakery to students, where the good students leave as they figure it out, leaving duds.

So I can squeeze that system most effectively by convincing policy makers to remove your incentive—taking away your funding.

Then my analysis indicates the problem can begin fixing itself—fakes will have no incentive to do fakery for no gain, leaving room for real researchers who can prove themselves.

But there is always a risk in such an approach as you may kill valid research as well, so part of what I do is kind of a triage to make certain that doesn't happen and I'm confident, for instance, that you can zero fund in number theory and have no negative impact, while, I'd say, keep topology fully funded, but I wouldn't worry if there were sharp reductions there as well, as I think that science and engineering are the major drivers.

Ultimately the "pure math" debacle will be ended, and that argument that uselessness means something is good will die as it should.

The problem is, from a game theory perspective, that if you make a system where people need to convince others that their research is correct as there is no other measure in the real world, then the optimal path is to become convincing versus correct, which is seen in math society by the emphasis on style.

The natural evolutionary path of such a system is learning to convince, and being right is actually somewhat negative as a strategy as it takes so much energy.

So humanity came up with a system that preferentially selected people who can fake "pure research" and tend to be wrong, which is kind of funny in a way. But you see, our Universe has a sense of humor.

It's just not a human one.

I want to do a little experiment which is relevant to the discussions on these pages as my analysis indicates that many of you seem to think the world listens to you more than it listens to me, which I find curious.

It seems to me that many of you believe that most people would not think that mathematicians can lie repeatedly and dangerously about mathematics and that the proof is in the world around you seeming to remain the same with respect to mathematicians to YOUR eyes, as incremental changes escape you. Like the proverbial frog in the water slowly coming to a boil.

So here is the experiment—

Go to Google wherever you are in the world and do a search on the following where you do not need to use quotes:

Superman plot idea

Do the experiment, and reply with comments here. Does the world listen to you more than me? And by you, I mean you as a GROUP.

One group, versus one man who understand the Internet, I think, far better than any of you can possibly come close to comprehending, with, your, I'm afraid, um, limited intellects.

I am curious about the effectiveness of the approach I have taken in communicating mathematical ideas and research I feel is important, so I am making this post to ask your opinion!

You can answer freely but I will give a few questions that reflect areas that are of great interest to me, so responses to those question would be appreciated:

1. Do you believe that I may have valuable mathematical ideas?


2. Do you believe that it is at all possible that mathematicians are fraudulently and deliberately blocking those ideas?


3. Can important ideas in factoring be presented loudly without anyone around the world noticing or caring, like governments or security agencies if those factoring ideas could lead to major breaches in security?


4. Have you heard me talk of a "z constraint"? Does that mean anything to you?


5. Do you think I should just shut-up, or should I post as much as I want—like anyone else?


6. Why do you think so many posters reply to me?


7. Do you trust mathematicians of today to tell the truth as best they know about mathematical ideas and research regardless of the source?

Thank you.

I've focused on an incredibly simple result that reaches to the limits of what the modern mathematical world knows about factoring as it covers the difference of squares, as if T mod 3 = 2, and

z^2 = y^2 + T

then

z = (1 + 2a^2)k/(2a)

for some integers k and 'a'.

Now the T mod 3 = 2 means that trivially the equation is always satisfied when a=1 and k is even, but there are plenty of examples that you can find where 'a' is greater than 1 and z dutifully has 1+2a^2 as a factor, and no cases where it does not work.

So z will have 9 as a factor or 33, or 51 because there is this mathematical rule that I like to call the z constraint, which makes it so.

To make the result more like what most math people learn you can have

x^2 = y^2 + N

and

x = (1+2a^2)k/(2a)

and the same result.

It gets more fun though as if you play with z you will find that, if f_1*f_2 = T, and p is some odd prime that you pick where k exists such that

k^2 = (1 + a^2)^{-1}(T) mod p then, you get your factors mod p:

f_1 = ak mod p

and

f_2 = a^{-1}(1 + a^2)k mod p

as long as p is less than the smaller factor which I'll call f_1, or p - f_1 is less than f_1.

And you people fought my research because you thought you could get away with it.

Stupid.

You destroyed a math journal and kept that out of the news but later I can use that information to show that there had to be a conspiracy on multiple levels reaching all the way into the most prestigious institutions like the editorial board of Princeton University.

You tried to end human progress in mathematics and create a welfare system where mathematicians get paid to not only do nothing of value, but to try and destroy people who can.

And you kept at it as the drumbeats grew louder of a solution to the factoring problem, even though millions of people around the world could lose a LOT of money because you are con artists and con artists do not quit until the police are coming.

You always hang on for that last little bit before folding, which is what I expected as you never outsmarted me.

You just set yourselves up for easy convictions.

My research takes you beyond what was previously known now in factoring as it did in other mathematical areas.

There simply is no way known outside of what I call surrogate factoring to get a general constraint on z, when

z^2 = y^2 mod N

to use the more conventional usage, where N is the target to be factored, where you also get the factors mod p, where p is some picked odd prime.

The knowledge simply transcends what mathematicians previously thought possible.

Now then, you can all pretend to not see and go about your lives, pay your bills and act as if the end is not in sight to this silly little game of deception that many of you have played on the world, but then you cannot believe in anything nor anybody.

And I think many of you do not.

But then, what do you do when they come for you, and you're sitting in a little box trying desperately to figure a way out, and you do not believe in anything nor anybody?

You walked into this trap. The mathematics laid it for you.

And now She will finish you.

While writing a long post about the current state of the art on surrogate factoring, I found myself talking about an equation that constrains a variable I call z, where

z^2 = y^2 + nT

and T is a target to be factored, while n is a free integer variable.

Seemed like a minor thing to me at the time but soon I found myself consumed with properly defining what I called the z constraint, and just figured out the full theory today, as the motivation has been to have a complete surrogate factoring theory.

That constraint is that for any factorizations of T into two positive integers f_1 and f_2, where there exists an odd prime p less than both, or if f_1 is the smaller factor p - f_1 is less than f_1, where also exists an α such that

k^2 = (1 + α^2)^{-1}(nT) mod p

then

z = (1 + 2α^2)k/(2α)

in general when nT mod 3 = 2, but conditionally if not, as then z is coprime to 3, and the z constraint applies only if also x exists as an integer where

x^2 = y^2 mod p

and z = x + αk.

The requirement that p be less than the smaller factor f_1, or that p - f_1 be less than f_1 comes from solutions for the factors modulo p, as if you have positive integer factors f_1 and f_2 where f_1*f_2 = nT:

f_1 = αk mod p

and

f_2 = α^{-1}(1 + α^2)k mod p.

(Oh yeah, that's cool in and of itself. If you play with these equations at all you'll find that with z divisible by 3, you can always find an odd prime p for which those hold.)

As an example of the conditional case when z is not divisible by 3, consider, n=1, T = 23(29)=667, p=17, so I can directly calculate tha= t z = (23 + 29)/2 = 26.

Notice that k^2 = (1 + α^2)^{-1}(667) mod 17 = (1 + α^2)^{-1}(4) mo= d 17

so α = 1 works to give k^2 = 2 mod 17, so k = 6 mod 17 is a solution= .

And notice then that f_1 = αk mod p = 6 mod 17, corresponding to the solution 23.

And f_2 = α^{-1}(1 + α^2)k mod p = 12 mod 17, giving the residue modulo 17 of 29.

But intriguingly enough

z = (1 + 2α^2)k/(2α)

is not valid for any integers k and α.

And if you dig deeply into the surrogate factoring equations you find that x cannot exist as an integer where

x^2 = y^2 mod 17

and z = x + αk. Note that y is given by (29-23)/2, so y=3, for this example.

As x is part of the full derivation it follows that x is not rational, and further as no integers k and α can give z, it follows that both of them are non-rational as well. because

k^2 = (1 + α^2)^{-1}(nT) mod p

so α = 1 mod 17, is what is actually happening, but it's not rational.

So you factored with the help of non-rationals, as well as the helper prime 17.

For more mundane examples and to see the z constraint in action, either use T = 2 mod 3, or if T = 1 mod 3, let n=5 or n=11 to force = nT mod 3 = 2, or any other n that you choose though I do stay away from n=2, as I stay away from even nT.

Now it seems to me that figuring out a previously unknown rule that constrains z, where z^2 = y^2 + nT, would be convincing, but it occurs to me that finding that last bit of surrogate factoring theory may have little to no impression on most of you.

But why not?

It's such a weird situation to be able to prove in multiple ways that people pretending to be real mathematicians are lying, and proof have no meaning with any of you people.

So that the only thing that will work is to collapse the security system of the Internet world by factoring an RSA number or a big enough number that it's clear that it is reachable, as otherwise you people will just keep going on like nothing is happening?

It's so odd.

Yes, I can analyze the psychology. I can figure out how modern mathematicians are like fiction writers, and figure out how the collapse of the power of past religions created pressures for people who in the past would have been priests to become "mathematicians" in "pure math" areas where amazingly enough uselessness was a prize, but why are NONE of you capable of breaking out of this spell?

What will I do with you later?

Where do I put you in this world?

Where can I place you?

You lack the ability to be convinced by reason alone. You follow along with people who parasitically use you, or you are parasites yourselves being trained to lie to others and use other people.

And you make it your business to block the gain of knowledge for the entire human race.

At one point I seriously considered that an alien force was working here to end humanity by stopping its progress mathematically and scientifically but now I think I'm just dealing with people who are doing this for minor, meager selfish gain.

So what do I do with you after? In the aftermath?

Where do you fit in this world that will be?

Now I finally figured out that modern math society is a society of fiction writers partly by thinking about "equals" as in the "=" sign as it is supposed to mean that what is on the left is equal to what is on the right.

So you can practically define any full mathematical statement as one where if you put in numbers and simplify you will end up with an identity:

e.g. x^2 + y^2 = z^2, try x=3, y =4, z = 5, and you get

9 + 16 = 25

25 = 25

and, yes, mathematical consistency is saved!!! As the equals actually means equal.

And you can define mathematical consistency that way.

If you end up with 3 = 4, then that is not a valid mathematical statement that you started with as, you can say, the books didn't balance.

Now then if it's that simple why would academics pretend that there is this massive issue with proving mathematical consistency?

Because then they have jobs. They can write papers. They can teach students and act like they're being very profound…

To see the truly bizarre though, go to logicians and claim that the equals should mean equal, as guess what?

You can define logical consistency in the same way.

If a logical statement is valid then it reduces to a truth mapping to itself.

The equals is equal.

And then you find that all supposed paradoxes or contradictions must be resolvable but then also you can kind of get rid of most of the "logicians" in the world as there's nothing else for them to do. Maybe they could move over to being mathematicians…

Academics work to stay in business as academics so the game theory for their jobs is not to be correct, but to convince others that they are correct.

It's trivial game theory.

Their best plan if no one is actually checking is not to actually be correct, but to have people convinced they are correct, if they cannot actually discover, and do not have someone around to spoil the game.

All that happened now is that I'm the next of a long, line of people who come in and figure things out, as I discover things.

So I talk about the game, but it doesn't necessarily end that game.

Many of you waste your time following invalid things because of the game theory that works for you as well, as you cannot convince yourselves that you are doing valuable work with the truth, but with the lies you can.

My analysis indicates that in previous human societies these people were priests. They pushed religion but religion diminished in power in human societies because with science people could ACTUALLY fly, versus hearing mythological tales of flying gods or demons.

And, of course, nuclear weapons are beyond the imaginings of just about any prior mythology, and any prior ideas about the wrath of God, so religion lost power, so these people moved to what had gained power.

So they write fictions with "pure math" which they pushed to escape the truth, but ran into the inevitable arrival of someone like me.

Those of you who deny the obvious truth, can continue if you wish pretending to be learning and pretending to be exerting energy in the pursuit of knowledge, but, you are liars like so many before you, and the end will be like it was for ALL those before you.

It's a logical tautology. The truth always wins, whether you believe it does or not, as truth is what actually happens. Not what you wish would happen, or pretend happened.

The equals IS equal.

I just really noticed and focused on a result I kind of thought of as minor that comes out of my surrogate factoring research which I'm now wondering if it might not be a breakthrough result.

With

z^2 = y^2 + nT

and nT NOT a perfect square, I have that

z = (1 + 2α^2)k/(2α)

where k is just some non-zero integer, the crucial result is that z must have 1 + 2α^2 as a factor for some non-zero α that is an integer.

I know it doesn't work if nT is a square—which is of course a trivial case but still—though I'm not sure why.

A key underlying equation is

(α^2+1)k^2 + p(r_1 + kr_2) = nT

where to get the result I'm setting r_2 = 0, so r_1 will exist as long as

(α^2+1)k^2 = nT mod p

where p is an odd prime.

But I know that won't happen regardless of the prime p, if nT is a perfect square as I have an easy counterexample:

5^2 = 4^2 + 3^2

if nT = 16 or nT = 9.

If that result holds, regardless of nT not being a square as that's a trivial case, then it is a constraint on ALL integer factorizations that rely on congruence of squares so it'd impact the Number Field Sieve.

(I'd guess that it conceivably could impact all factoring methods known or possible, except brute force.)

One tidbit result that has come out of the research into the concept I call surrogate factoring has been a constraint on z, where

z^2 = y^2 + nT

which I consider something of a weak constraint though it is an absolute one in a key way I'll explain is

z = (1 + 2α^2)k/(2α)

where k is just some non-zero integer, so it just boils down to z must have 1 + 2=E1^2 as a factor for some non-zero α that is an integer, which I consider kind of interesting.

If z has 3 as a factor, then of course, α = 1, works easily and the k can easily be found that will satisfy, so maybe it'd be more interesting as fun math oddity when z is NOT divisible by 3, and you try to find what α will work.

Theory as to why it's true is kind of neat, though trivially easy, unless I missed something! I'm fairly certain the result is true.

Somewhere around 4 years ago, as I think it was back in 2004 or maybe earlier I started wondering about factoring one number through the factorization of some other number that I eventually called the surrogate, so I called the entire concept surrogate factoring.

And I just kind of put that in the incubator as this concept that I'd work on from time to time while I did other things as I had a few other mathematical things going on at the time, including eventually the publication and the withdrawal by the SWJPAM editors of my paper introducing non-polynomial factorization. And there was also some prime research I was doing, still contemplating my prime counting function and later I presented the prime gap equation (search for "prime gap equation" in Google to get more info there).

So I had quite a few things going on mathematically as well as what I was doing outside of mathematics including my open source project Class Viewer (Google search again for more info) which I was fiddling with here and there during that period, and I'm digressing as the point is that I was working on surrogate factoring here and there between other things.

And at times I'd think I'd have something, brainstorm it a bit, including posting online and find out that what I had was crap, and it didn't help that I was giving dire warnings about the negative potentials of a simple solution to the factoring problem—as it scared me—and then having to just quiet down anyway when I didn't have same.

But the breakthrough came with simplification back in August 2007, as I'd done various things here and there as I tried to factor one number with another, and one day just sat down and said to myself that I needed to resolve this thing one way or the other, including maybe just giving up on the concept as just not practical, and at some point I started writing out something like:

x^2 = y^2 mod T

and

k = 2x mod T

and I realized that I was doing that JUST to get to a difference of squares which looked like

(x+k)^2 = y^2 + 2k^2 mod T

and I had a second difference of squares where with T the target composite I had connected a factorization of T at the start, with a factorization of 2k^2 mod T at the bottom—so an infinity of factorizations but it's easier to think of one—and that was surrogate factoring!!!

And there was this odd sense of the bizarre simplicity of the answer I'd been spending YEARS, literally years searching for, though as I said I was doing other things at the time as well, but it's kind of odd to get a simple answer when you've looked at something for years and puzzled and pondered, wondering exactly which way to go, and then it just leaps out at you.

And it took me MONTHS to go from there to what is currently the full surrogate factoring theory, where there was another surprise in store, as puzzling over those equations I found that primes stepped in, in a BIG WAY and would just help you out and then quietly exit, so here is the gist of what comes from the full surrogate factoring theory, leaving the next step being implementation as I switch gears from theoretical mode—which is like prospecting for gold—to the steady plodding of experimental mode:

Now there is

z^2 = y^2 + nT

as the primary factorization equation where T is the target and n is one of the many helper variables used to get to the factorization of T, where the latest research indicates that nT mod 3 = 2 is desired, so n is used to make that happen. It may have other uses which experiment will show.

My early idea of using 2x = k mod T, has evolved so that now the key equation is

2αx = k mod p

where p is an odd prime and now you also have alpha, which comes in with text in posts as "α", where usually I have the explicit with

2αx = k + pr_2

and I use r_2 for historical reasons as in the full theory there is also an r_1.

And then I have z = x + αk, and you can substitute out z, from

z^2 = y^2 + nT

and multiply everything out and simplify to get

x^2 = y^2 + nT - (1 + α^2)k^2 - kpr_2

and a truly remarkable result presents itself as low-hanging fruit at this point, and it is such a huge result as it allows you to get a good idea of where to find k, and you can get it easily enough by substituting with

k = k_0 + pj

where j is some integer and p is still your odd prime, as then you get:

x^2 = y^2 + nT - (1 + α^2)(k_0^2 + 2pjk_0 + p^2*j^2) - (k_0 + pj)pr_2.

And focusing on the behavior of

nT - (1 + α^2)(k_0^2 + 2pjk_0 + p^2*j^2)

as you increment or decrement k with j, you can just look to (k_0 + pj)pr_2 where you have j linear versus the quadratic j^2 in the previous so it's trivial to work out the behavior, which then indicates that k_0 will tend to be where you have the maximum k such that

abs(nT - (1 + α^2)k^2)

is a minima, which is just such a HUGE result, and an amazing one besides as it's such a powerful result for the ease with which it is gathered.

So one of the big surprises you can give yourself with this powerful new factoring theory is just to look at factorizations of small composite T's as they're easier to play with, and find out how close z is to the value given JUST be finding k such that abs(nT - (1 + α^2)k^2), and then checking with

z = (1 + 2α^2)k_0/(2α)

which is derived by using 2αx = k_0, and z = x + αk_0.

Notice that since z has 1 + 2α^2 as a factor, 3 is a factor unless α has 3 as a factor, so forcing z to have 3 as a factor is a good way to make things easier.

It's actually kind of weird to see how z skips around with various values of α focusing on easy cases as 1 + 2α^2 can be easier for the algebra to satisfy for certain values, like α=1, when it's equal to 3 and much harder for others, so it's about probability at the DEPTHS of factoring that was previously unknown to the mathematical world.

So all along values for z that solved z^2 = y^2 mod T, or as mathematicians traditionally put it, z^2 = y^2 mod N, were following these mathematical rules that preferred easy to harder cases when it was a necessity to satisfy:

z = (1 + 2α^2)k_0/(2α).

So the surrogate factoring approach to the factoring problem is really tackling finding how to get z, and all those variables are just helpers in that task.

They're like friendly neighbors trying to get you next to z.

And it can be shown that

k^2 = (1 + α^2)^{-1}(nT) mod p

so you can go looking for z by looking for k modulo a prime p that YOU GET TO CHOOSE, though not every choice will work, at least you get k about 50% of the time for your choice since p has (p-1)/2 quadratic residues.

The most preferred case still is with α=1, but that case may not give you k, for a particular odd prime p as explained by the theory, while another value may. And even when you get a k, it can be for a case where x, y and z are not all rational as the math it seems prefers to use k for the largest value of α that will work!

And so far that is something determined by experiment, as a result still just outside the reach of my understanding of the current theory, as a complex system is demanding an experimental approach which may be foreign to many mathematicians who aren't used to dealing with complex systems, unlike scientists in the real world who deal exclusively with complex systems, so they must rely on experiment.

Regardless there is this incredible result that in general

z = (1 + 2α^2)k_0/(2α)

which puts constraints on z itself which is a huge find, from simple algebra which affects any factoring idea out there, so I toss that in so that you know we are not in trivial territory here, and claims that we are or that I MUST factor an RSA number first are just specious, and defy reason.

And it is just incredible too that you have have prime numbers as helpers that disappear after helping you to factor, and you have a surprisingly simple result with a parabolic minima, and you get quadratic residues in there, and it is the factoring problem.

So why is there still primarily me just talking about my research as I begin switching gears to doing a factoring demonstration that will end any debate about the value of this research when already it can do things never done before, and show rules never known?

People go towards what attracts them—It's like a tautology.

The simple answer is that modern mathematicians are repulsed by the results.

The truth in this area is not what they like—not what attracts them— so they vote against it with their feet, as I like to say. They can't be interested, so far in the truth.

Showing democratically their disdain for how mathematics does the factoring problem.

That is the only answer that fits the behavior.

People go towards what they like. Math society is running from these results.

Ergo, math society dislikes them.