JSH: Theories finally accepted by AMS

Subject:	JSH: Theories finally accepted by AMS
Date:	Thu, 17 Dec 2009 12:26:07 -0600

[[JSH_Repostin_Service@WTF.RU? JSH posts from earlier this year, can you
spot a trend? limited topics perhaps? Has the Cheese slid off his cracker?
and we now provide "JSH MATH MUSIC" you can play while you read these posts,
Merry XMAS, RAMADON, KINISHAR, and Minioria (that Jewish thing) ]]

http://blogfiles.wfmu.org/LG/CUT/Church_Universal_and_Triumphant_-_06_-_Decree_10_05.mp3

<<<<<<<<<< o >>>>>>>>>>>>>>

Here it's worth reminding again that I DEFINED mathematical proof.

Google: define mathematical proof

Or you may even find it in Yahoo! as checking just now I found my
definition at #5.

When I say proof, it's a proof. After all, I wrote the definition!

I have proven a simple solution to the factoring problem using some
rather easy algebra where I've given the equations.

x^2 - Dy^2 = 1

where I SOLVE Pell's Equation, giving x and y explicitly as functions
of
a variable I call v, and factors of D-1:

y = [+/-2Dv/(f_1 - f_2*v^2 - 2v) +/- 1 -/+(f_1 + f_2*v^2)/(f_1 -
f_2*v^2 - 2v)]/(D-1)

and

x = +/-(f_1 + f_2*v^2)/(f_1 - f_2*v^2 - 2v) - [+/-2Dv/(f_1 - f_2*v^2 -
2v) +/- 1 -/+(f_1 + f_2*v^2)/(f_1 - f_2*v^2 - 2v)]/(D-1)

where f_1*f_2 = D-1.

Now stepping through the proof that those equations solve the
factoring problem trivially is easy, but those of you with some
mathematical sophistication can figure it out quickly on your own!

That's because if you have x = r(v)/t(v) and y = s(v)/t(v), and
multiply out you get

(r(v) - t(v))*(r(v) + t(v)) = D(s(v))^2

and can find rational v such that r, s and t are integers, non-zero
and at a MINIMUM, turning factoring into a calculus problem. Look
over x and y, and don't let all the f's fool you into thinking it's
very complicated--the independent variable is v which has a largest
exponent of 2.

Quadratics. You KNOW quadratics, or should.

Now I know most of you should know enough calculus to realize that
MUST BE TRUE, so opposition or denial about this mathematical find is
all about disbelief that the factoring problem could be trivially
solved.

You need to get out of disbelief mode and into rational adult mode.
You need to start THINKING versus just going by your freaking
feelings.

James Harris

For years I've been dealing with a problem where mathematical proofs
I've discovered are not being accepted by the mathematical community,
and tracing it back, I found that over a hundred years ago a
devastating error entered into the math field, which is so devastating
that it removes most of the research done in number theory over a
hundred plus years.

Seeing that error I was not terribly surprised that a simple solution
to the factoring problem might exist as the same practitioners who
came into the math field under the error, were making the claims about
the factoring problem being difficult.

But you see, factoring is like practical non "pure math" results in
that you can either factor a number or you can't.

So success requires math that works. While a lot of currently
established number theory is covered by the error so it does NOT work,
but that is obscured because the results are with non-rationals.

So modern mathematicians who came in under the error aren't actually
that good at mathematics, so to them, factoring was a hard problem,
but it's actually not.

Now I've found the simple solution to factoring and posted it, but the
people who are in charge in the math field are the people brought in
under the error!!!

And it's increasingly clear that they realize what they have invested
in the error: their careers.

So even with a simple proof of the factoring problem, with all that
entails, they are trying to just ignore it like they did with my proof
of the error and my other mathematical research including my proof of
Fermat's Last Theorem.

But the very reason I picked the factoring problem to work on was to
end the efficacy of that strategy.

However to my surprise there has been a delay that has gone on far
longer than I thought possible.

It is not clear that the simple solution is not being exploited. One
worst case scenario is that practitioners in various field related to
Internet security are simply making stuff up about breaches to hide
the fact that the current methods are no longer working.

If so, that can only go on for so long though. So we're in a delicate
and dangerous period where as far as I'm concerned the Internet no
longer has security, but is wide open for anyone with the factoring
solution, while the people who are supposed to alert the world are not
doing so as they realize that after the world knows the story about
factoring the full reality of the devastating error will be revealed,
ending the free ride that people have gotten with false mathematics.

Which would probably push a tremendous number of people out of the
mathematical field.

A hundred years with a math error of this type can wreak a lot of
havoc.

Here the entire field of number theory worldwide is full of people who
are, unfortunately, probably actually incompetent at mathematics.
They convinced the world that factoring was a hard problem on which to
base Internet security. I've proven an error in number theory, and
also solved the factoring problem trivially. These people are for the
moment not acknowledging the result.

It remains to be seen what the full consequences of their behavior
will be.

James Harris

Maybe some of you need an explanation as to how it is possible that
mathematicians as a group, around the world, in number theory can be
wrong, and even have it proven to them, and keep being wrong, and I
say it goes back to European history.

Feudal societies built first around powerful individuals but later
institutionalized worth by birth, so a noble could be someone not
noble at all!

In those societies people learned that some things were just about
position: some people regardless of ability just had these positions
and you accepted it for the good of society.

It's easy for us to look down on classed societies from a distance but
I doubt peasants spent most of their days thinking about overthrowing
their social order. Most probably thought of themselves as good and
loyal subjects who accepted things as they are.

While modern dominant societies claim to have escaped arbitrary
positional rules, repeatedly reality shows that they have not, like
the recent disastrous presidency of George W. Bush, where it is
difficult to understand how he ever even became president except for
him being a part of a dynastic family. He was, quite simply, born
into a family which allowed him to achieve high position.

Clearly, feudal realities still remain in modern societies!

Modern mathematicians have a SOCIAL position, and people are loathe to
undermine the social order unless they are pushed extremely hard, as
most people in modern societies I think see themselves as good and
loyal citizens--they support their societies.

And I think there is an unconscious continuation of feudal behavior,
which has put me in the difficult position of facing social structures
with mathematical proofs.

The factoring problem unfortunately represents the one area in
mathematics where there is the potential to shock people out of the
feudal behavior, forcing them to see that there is a major problem in
the mathematical field, just like financial woes have been necessary
to force people to accept problems with the financial system.

Note that a feudal class had started to evolve in finance as well,
where huge amounts of wealth were had by people who not only we later
found were not doing anything commensurate with their level of
compensation, but worse, they were actually a net negative!

We paid people billions of dollars to wreck our financial systems.

We REWARDED them for failing, which I see as feudal vestiges--
unconscious but powerful.

The rich were the nobles: with positions just because, given special
rights and dispensations just because.

Feudal behavior in modern societies.

The proof of the easy solution to the factoring problem will
indubitably have a massive impact on the world, and it will in time
force what other mathematical proofs could not--a shift in the view of
mathematicians and probably academics in general, just as there is an
ongoing shift of the view of people in the finance industry.

The sad thing is that catastrophe or near catastrophe is the only way
to change social inertia on this scale.

We are approaching potential catastrophe with Internet security, as
the only way to remove more feudal behavior, and force an upgrade of
the academic system--worldwide.

So what finance people are going through now, academics will face
next.

James Harris
For this game I need you to as they say in literature, movies or other
fictional settings: suspend disbelief

For the sake of argument, let's say I'm right: I have solved the
factoring problem. I found a proof of Fermat's Last Theorem years
ago. I found a devastating error in number theory.

But the world has not recognized any of it. Why?

Is there ANY explanation--beyond the obvious that I'm wrong, remember
the beginning of the game is the suspension of disbelief--for that
happening?

I mean, we have a world of people who should be excited about any one
of those things, how could someone have incredible discoveries on such
a scale and not have them accepted?

Is it possible that the human species is devolving? Or could there be
an alien influence?

It's brainstorming time. Any and all ideas within the parameters of
the game are welcome.

Is the human race getting stupider on a worldwide scale?

Or are possibly aliens insuring the demise of the species by acting to
prevent its growth in science and technology by blocking continued
development of mathematics?

Or something else?

James Harris

The following post steps through the mathematical underpinnings of a
remarkably simple factoring method which relies on dual
factorizations. It steps off from the familiar Pell's Equation--with
rationals--and through basic algebra proves a viable factoring
technique.

In rationals starting with Pell's equation:

x^2 - Dy^2 = 1

I have proven that

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

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

Here the +/- indicates that one variation will work so it is an OR and
not an AND. Either plus OR minus will give a valid j.

It suffices to substitute out j, and simplify, which will give Pell's
Equation again, showing the relations are valid.

Dual factorizations:

The utility of the expressions comes from dual factorizations:

(x-1)(x+1) = Dy^2

gives an opportunity to factor D, which I will declare to be an odd
composite.

While

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

gives the opportunity factor D-1.

Focusing on the second factorization, note to generalize I need
additional variables.

Introducing u and v, where j = uv, I further need factors f_1 and f_2
where f_1*f_2 = D-1.

Then I have for generality

(x+y - (uv+/-1)) = u*f_1

and

(x+y + (uv+/-1)) = u*v^2*f_2

Note that f_1 and f_2 can be declared to be integers, and further that
x and y uniquely determine u and v, as consider:

(x+y + (uv+/-1)) = (uv)*v*f_2

where I remind that j = ((x+Dy) -/+1)/D = uv, so given any x and y in
rationals, I can simply separate off all non-unit non-zero integer
factors in common with D-1, for f_1 in the first, and for f_2 in the
second, leaving u and v as rationals, and trivially solved and proven
to exist as rationals.

Therefore, it is proven that for every x and y that fulfill Pell's
Equation there is a u and v pair.

Given that there must exist x and y, such that (x-1)(x+1) = Dy^2, non-
trivially factors D, it is proven there exists a u and v pair unique
to any such cases.

That completes the underlying mathematical proof showing the dual
factorization and proving the existence of rational v, for a non-
trivial factorization of a composite D.

It is now possible to move further into the utilities of these
expressions and reveal a direct factoring method.

There are then three equation available:

(x+y - (uv+/-1) = u*f_1

(x+y + (uv+/-1) = u*v^2*f_2

((x+Dy) -/+1)/D = uv

Considering x and y to be unknowns it follows then that x and y can be
determined as functions of v. Doing so gives:

y = [+/-2Dv/(f_1 - f_2*v^2 - 2v) +/- 1 -/+(f_1 + f_2*v^2)/(f_1 -
f_2*v^2 - 2v)]/(D-1)

and

x = +/-(f_1 + f_2*v^2)/(f_1 - f_2*v^2 - 2v) - [+/-2Dv/(f_1 - f_2*v^2 -
2v) +/- 1 -/+(f_1 + f_2*v^2)/(f_1 - f_2*v^2 - 2v)]/(D-1)

Given that y and x are now expressed as functions of v, f_1 and f_2,
it is now possible to elucidate a fairly simple factoring method.
Said method requires the addition of new functions, so let x = r(v)/t
(v), and y = s(v)/t(v), where r(v) is the numerator of the x function,
s(v) is the numerator of the y function, and t(v) is the denominator.

Note that t(v) is the easiest given as:

t(v) = f_1 - f_2*v^2 - 2v

Then making the substitutions into the factorization of Pell's
Equation: (x-1)(x+1)=Dy^2, and simplifying slightly, I now have

(r(v) - t(v))(r(v) + t(v)) = D(s(v))^2

Note that it suffices if r(v) and t(v) are integers for

abs(r(v) - t(v)) < D and abs(r(v) + t(v)) < D

to guarantee a non-trivial factorization. The absolute values may at
first seem problematic, but note that if D is positive then the right
side must be positive, forcing the two expressions to match by sign.
So it is possible to simply add them, giving the result:

abs(2r(v)) < D

So remarkably, it is proven that rational v must be found such that r
(v) is an integer and abs(r(v)) < D/2, and you are guaranteed to non-
trivially factor D.

That completes the proof. The factoring problem is then, solved.

Comments or questions are appreciated.

Thank you for your time and attention.

James Harris

For me it is wacky how this story has worked out and such a weird
feeling to be commenting after my post of a fairly straightforward and
to the point stepping through of a solution to the factoring problem.

First off, why was it necessary?

Well I've had major mathematical discoveries for years now and people
in math society have lied about them. So I threatened to solve the
factoring problem to find something they couldn't lie about, and got
lucky and somehow did it after somewhere around 5 years of searching.

The lying I've seen has been so weird, and problematic.

It left me few options in considering how I should proceed in this
case because I have in the past made efforts to work through say, the
NSA. I have made contacts with or tried to make contacts with people
in the intelligence community both here and abroad.

But scarily, I've also faced the fear that hey, maybe mathematicians
had no limits, and like, the NSA is FULL of mathematicians.

What might they do if cornered?

STILL I've been talking about this solution in various ways as I
worked through it for three weeks now, and I find it incredible that
no intelligence services around the world picked up on it, despite my
efforts.

Like using words and phrasings that I'd think would be picked up. I
was so desperate I recently started naming Britain specifically and
talking about disaster.

But I also hoped someone would step up, some other person out there in
the world, who would understand the importance here, and make the
contacts, and I've been waiting now for weeks for someone from some
government agency to contact me.

None have.

With a situation completely beyond anything I expected, the best
solution was to go forward. Later inquiries can find out why
intelligence services around the world failed so badly. And inquiries
can determine why people lied about this result. And deeper inquiries
can determine why they lied about my prior research.

For me the weirdest thing about this situation has been that lying as
it seems to me that no one who expected to ever get caught would put
themselves in the position these people have--so they never expected
to get caught.

And THAT is scary, as my mathematical finds are, well, as time goes
forward that may be best left to others to fully explain, but at a
minimum they are worth the world knowing, and properly acknowledging.

Time is of the essence people. Your delusions about what can happen
are no match for reality.

Your imaginations are not big enough. I assure you that waiting and
hoping no one will notice is not a strategy.

It is suicidal.

James HarrisIf any of you have looked over my solution to the factoring
problem,
you may find yourselves very conflicted as it's easy algebra with a
rather basic concept. I just found a way to factor one number by
connecting it to a factorization of that number minus one.

Once you see the equations, it's obvious that has been achieved.

Easy algebra.

But how can math people lie about it?

Because they HAVE lied about my research for years. They are simply
following the same strategy as before, except this time I've found a
problem where that strategy can have real world consequences.

So, shouldn't I just implement it?

Even if I had the skill and the time, and could get something together
fast enough--before some critical breach of some security system--I'd
have a major problem after, as math society would simply then accept
the result, and claim they don't lie about anything else!!!

They'd lie about lying.

I have SEEN that behavior with my prime counting function, as after
all, it does count primes correctly! They lie about lying and shift
as needed when facts force their hand.

The situation here is nearly impossible. A major field got
corrupted. Given the weight we give to major fields and experts in
those fields it is nearly impossible to prove such a thing.

But you can look at a very clever and very correct solution to the
factoring problem, and read through replies to me where math people
are lying. And guess what? I found someone at RSA with an email
address public and emailed her yesterday.

I AM working to try and prove this result exists!!!

The problem is the math people are fighting me.

My fear is that the impasse may be broken by something terrible
happening.

And no, she hasn't emailed me back yet. I may have not even got past
her spam filter. It is far more difficult than you imagine to inform
people about even such a result as solving the factoring problem.

I've been trying now for over three weeks.

James Harris
With a simple proof of a solution to the factoring problem there is
just no more room for denial about the reality of people lying about
my research, while one remaining question is, why?

x^2 - Dy^2 = 1

requires that

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

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

And it's easy algebra to go from there and factor D with factors of
D-1, but I've watched posters lie for days and posters CONTINUE lying
about the simple reality here.

I used mathematical objects I call tautological spaces against Pell's
Equation just to see what they might do, through what I call my
Quadratic Diophantine Theorem.

Similar mathematical objects to what I used to prove Fermat's Last
Theorem. But in this case I generally solved binary quadratic
Diophantine equations, and part of the bonus of that accomplishment
was a solution to the factoring problem.

Easy algebra.

Mathematical society to date has to my knowledge ignored the general
solution to binary quadratic Diophantine equations and is currently
trying to ignore the solution to the factoring problem.

I imagine researchers around the world still trying to squeeze out
improvements with things like the Number Field Sieve--and probably
relying on public funds in many cases--despite the problem being
solved.

There is a willfulness in that I believe. A desire to be wrong as
long as they maintain their flawed view of the mathematics.

At this time it's not clear what the full impact will be, but it is
increasingly clear that mathematics as a discipline took a wrong turn
over a hundred years ago. Current practitioners are quite simply,
possibly, fakes.

The fate of the human race lies in correct mathematics. We're running
on fumes mathematically now mostly using mathematics that is over one
hundred years old, with mathematicians doing bogus research.

It seems impossible for greater advancement of science to occur
without the correct mathematics. It is unclear how bad the impact is
on things we think we know, but it may be huge.

Our reality itself may be very different than people imagine because
there has been this massive error corrupting things.

Our physics might be a hundred years further than it is now, without
the error. Question is, will we recover from this massive blow?

What will be the full impact on the survival potential of the human
species?

James Harris

After weeks of debate on the mathematical proof showing I've solved
the factoring problem using Pell's Equation, I've finally gone ahead
and worked out the simple equations that result from it, which solve
the factoring problem.

A critical equation that posters have bugged me about has only two
variants, where one is:

r(v) = (D+1)*f_2*v^2 - 2(D-1)v + (D-1)*f_1

If D is the target to be factored, and f_1 and f_2 are integers where
f_1*f_2 = D-1, I have proven that if rational v can be found such that
r(v) is an integer and abs(r(v)) < D, then D MUST be non-trivially
factored. The non-trivial factorization is then forced, as in a
mathematical absolute!

Finding v is as simple as solving for r(v) < D, with the quadratic
formula. That gives you v:

v < ((D-1) +/- sqrt((D-1)^2 - [(D-1)*f_1 - D][(D+1)*f_2]))/(D+1)

Using calculus it's easy to see the lowest bound for v is:

v >= (D-1)/[(D+1)*f_2]

To get the non-trivial factorization you just need one more equation
called t(v), where:

t(v) = (D-1)*(f_1 - f_2*v^2 - 2v)

as non-trivial factors of D MUST be available from r(v) + t(v), or r
(v) - t(v), by the gcd.

ALL of the arguing over this research is needless, pointless and dumb,
as if I'm correct, I just gave you enough algebra to factor an RSA
public key. Just make it D.

I really didn't want to just give the damn thing, but I hoped by now
that the NSA would have contacted me, as the algebra is easy.

They have not. No one in the US Government or any other friendly or
hostile or any government has contacted me.

All has been scarily and depressingly silent. Our intelligence
services all failed.

I don't quite know why posters argued about this result as if it were
so complicated, why others on the newsgroups let them, nor why no one
just used the easy equations to factor.

The mathematical proof says they do work, though I have one more hedge
as there is one more equation for r(v) as there are two variants, just
in case the one above doesn't work.

James Harris

For weeks I've been talking about having a proof that solves the
factoring problem by using Pell's Equation and posters have argued
with me. Well it turns out they were arguing over details of
implementation, and ignoring the proof.

I've carefully figured out the answer to the implementation detail
which they criticized:

r(v) = [(D+1)*v^2 - 2f_1*v + f_1^2]/2^k

and

t(v) = [f_1*(f_1 - f_2*v^2 - 2v)]/2^k

where D is the target composite to be factored, f_1 and f_2 are
integer factors of D-1, where f_1*f_2 = D-1, and k is a natural
number, so it cannot be non-zero, which is the critical element that
was missing, which is necessary for *implementation* of the proof.

The proof notes that with r(v) and t(v) coprime to each other, and abs
(r(v))or r(v) - t(v) MUST factor D non-trivially, and in fact, if D is
prime, those conditions cannot be met, but the proof shows that if it
is not, they can ALWAYS be met. Which is why the proof solves the
factoring problem.

Now others may disagree saying that the proof should tell you how get r
(v) and t(v), but it doesn't.

But I say that's like Einstein's theories don't tell you how to build
a nuclear bomb.

Implementation is separate from theory.

The proof DOES give you the route to finding r(v) and t(v) by giving
you the conditions they must meet, and as I've been enamored with the
proof I've ignored appeals to give r(v) and t(v), and then gave them
wrong repeatedly as I didn't bother to figure them out, but after
enough arguing I finally decided to settle down and figure them out.

I know that a Patricia Shanahan keeps asking for pseudo-code, but in
this case the answer is easy--once you have r(v) and t(v).

Finding them involves using whatever rules you know to get k above,
which is just about making the expressions coprime with respect to 2,
with the caveat that if there is a way to do that with k=0 that will
NOT work, so k has to be a natural number, so it is nonzero.

What makes this story kind of sad is that now, yes, the factoring
problem is solved, and yes, I had a mathematical proof all along.

If posters had focused on the proof then they'd have realized it HAD
to be correct, and that arguments were over an implementation detail.

So if intelligence services listened to them, or their own analysts
looked the argument over and agreed with their objection, then they
failed the most basic test of mathematical understanding: believe the
proof first.

If you believed in mathematical proof, then you knew there had to be
an answer and could find it, as I did.

I did write the definition of mathematical proof. Just Google: define
mathematical proof

The factoring problem is solved. Posters ignoring mathematical proof
focused on one issue looking for me to be wrong, but they were focused
on an implementation detail. Now with that detail removed ANYONE who
wishes can factor, including factoring RSA public keys.

RSA encryption is, as I've said for a while now, obsolete.

James Harris

Some of you may have noticed a very big discussion over the last few
weeks where I've claimed to have solved the factoring problem and
various posters have said I haven't and put up issues with my claims.

So here's a quick synopsis to get you up to speed on what happened:

1. I noticed that you can use Pell's Equation to do dual
factorizations to attack factoring a composite target D by factoring
D-1 instead:

In rationals, x^2 - Dy^2 = 1

requires that

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

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

Ok, so I know you've seen that but the issue was how to actually get a
key variable called v, where j=uv, and I had an explicit solution for
x that looked like:

x = +/-(f_1 + f_2*v^2)/(f_1 - f_2*v^2 - 2v) - [+/-2Dv/(f_1 - f_2*v^2 -
2v) +/- 1 -/+(f_1 + f_2*v^2)/(f_1 - f_2*v^2 - 2v)]/(D-1)

where f_1*f_2 = D-1, and the f's are integers, while v is a free
variable, which is a mess I didn't want to deal with, so I didn't try
to simplify it, and just hoped others with math software would do so,
but I talked about factoring D non-trivially with its numerator and
denominator, as with functions I call r(v) and t(v), and

x=r(v)/t(v)

it is trivial that if they are integers and abs(r(v) - t(v)) < D, and
abs(r(v) + t(v)) < D, then you non-trivially factor D.

And I'd proven the existence of integer solutions so I thought I was
done.

But posters started claiming they couldn't work, and some put up:

r(v) + t(v) = 2Dv^2

which to me implied that v had to have a non-trivial factor in common
with D before you could use it, but hey, that couldn't work as v is
squared, so you'd always have a fraction! So I figured they were
lying, and said so repeatedly as I didn't want to bother with that
monster expression I had for x, and kept thinking that all those plus
or minuses meant a lot of variations.

But I was wrong. I finally simplified things further and found a
simple equation for x:

x = [(D+1)v^2 - 2f_1*v + f_1^2]/[f1(f_1 - f_2*v^2 - 2v)]

And lo and behold if adding the numerator to the denominator didn't
give what I'd claimed had to be wrong!!!

Turned out the math was more subtle than I'd imagined and was slapping
a freaking non-trivial factor of D on the numerator and denominator.
That divides off easily enough--if you know what it is--but when you
add them, you get 2Dv^2.

But NOT a big deal as instead of taking the gcd of r(v) + t(v) with D
as I'd originally said, you take it with r(v). No worries, I thought.

But then I noticed that I could NOT get

r(v) = (D+1)v^2 - 2f_1*v + f_1^2

and

t(v) = f_1*(f_1 - f_2*v^2 - 2v)

to both be integers with a rational v. It was a what the f-moment.

Turns out the algebra had one more trick up its sleeve as both have
been divided by 2.

They have to both be integers to force a non-trivial solution with abs
(r(v)) < D--oh yeah, I found that as a primary condition--so I ended
up needing to multiply both by 2, and that's it.

However, a month of me working through these issues, especially weeks
of me refusing to simplify x left the door open to posters who spread
confusion about the result! One funny part though was when they
specifically claimed it couldn't factor 15. But I had an example of a
factorization of 15, so I knew they were wrong.

And that's the story, of how there can have been so much debate about
a solution to the factoring problem for weeks.

Synopsis: I didn't feel like simplifying a key equation. The algebra
had some subtleties that opened the door to confusion if you DID
simplify the damn thing. And even after that point you needed to
figure out one more wacky thing as in the functions had been divided
by 2, and then everything works just fine.

Factoring problem is solved. No more room for confusion about the
result. All issues are resolved.

James Harris

Ok, here's a posting of some code which provides the proof of concept
for my mathematical proof which solves the factoring problem. The
code factors numbers of the form 2^n + 1 because it factors D by
factoring D-1, so if D = 2^n + 1, then it's trivial to factor D-1 =
2^n, and also trivial to loop through those factors. Further it is
proven that ANY number of that form can be non-trivially factored in
at most n+1 iterations! Which I think brings Fermat numbers into
reach.

I was going to only post the code to comp.theory as it seems more
appropriate but I've been making the case to the physics community
that I have actually solved the factoring problem, so I'm including
them as well, as with physics people, yeah, a demonstration can be
very effective and I'm sure plenty of them can run Java.

The factoring problem IS solved. This code isn't meant to allow you
to go factor anything but to demonstrate a very beautiful proof that
relies on dual factorizations through Pell's Equation.

James Harris
_____________________________


import java.lang.Math;
import java.math.BigInteger;
import java.util.ArrayList;

//Code utilizes my surrogate factoring algorithm where the underlying
equations rely on
//Pell's Equation and a dual factorization.  Go to mymath.blogspot.com
for more info.

//To use input a natural number less than 32, like 5, and the code
will, for instance, factor 2^5+1


/**
 * The FermatNumber class implements an application that
 * attempts to factor a number of the form 2^n + 1.
 */
public class FermatNumber {

    static int n;


    public static void main(String[] args) {


        try{

n = Integer.parseInt(args[0]);
if (n>29){
System.out.println("Number too large.  Input has to be a positive
integer less than 30.");
System.exit(0);
}

}
catch (NumberFormatException e){
System.out.println("Not a number.");
}

FermatNumber Factorer= new FermatNumber();

Factorer.process();

    }

private long D, f_1, h, mult;
private ArrayList g_2 = new ArrayList();

private BigInteger BigD;

private double v, max_range, min_range;
private double temp;

private int max_iter;
private int trivial_factors=0;


public FermatNumber(){
}

private void process(){

long temp_long;

//D is of the form 2^n + 1
D = 1<D++;
System.out.println("D="+D);
System.out.println("");

if (checkForSquare(D)){
System.out.println("D is a perfect square.");
System.out.println("sqrt("+D+")="+(long)Math.sqrt(D));
return;
}

BigD = BigInteger.valueOf(D);

f_1 = 1;
mult = 1;

max_iter = n;

for (int i=0; i
f_1 = mult;


//code for the integer v case
h = 1;
for (int j=0; j

temp_long = setIntRad(f_1, h);

if (temp_long>0){

setRange(f_1, temp_long);
searchInt_v(j+1);

}

h = h<<1;

}

//negative case
h = 1;
f_1=-f_1;
for (int j=0; j

temp_long = setIntRad(f_1, h);

if (temp_long>0){

setRange(f_1, temp_long);
searchInt_v(j+1);

}

h = h<<1;

}

//need to make f_1 positive again, may make neater later
f_1 = -f_1;


//Searching for the fractional v case
temp_long = setFracRad(f_1);
if (temp_long<0) break;

setRange(f_1, temp_long);
searchFrac_v();

//now negative values for f_1

f_1 = -f_1;

setRange(f_1, temp_long);
searchFrac_v();

mult = mult<<1;

}


System.out.println("");
System.out.println("");

if (!g_2.isEmpty()){

System.out.println("D="+D);
System.out.println("");

Object[] factors = g_2.toArray();

if (trivial_factors>=factors.length) System.out.println(D+" is
prime.");
else{
for (int i=0; i
System.out.println("g_1="+(D/(Long)factors[i])+", "+"g_2="+(Long)
factors[i]);
System.out.println("");

}
}

}
else System.out.println(D+" is prime.");
if (trivial_factors!=0) System.out.println("Trivial factorizations
found: "+trivial_factors);

}

private long r_c(double v_in){

temp = (D+1)*v*v - 2*f_1*v + f_1*f_1;


return (long)(2*temp);

}

private void setRange(long f_in, long other_in){

max_range = (f_in + Math.sqrt(other_in))/(D+1);

min_range = (f_in - Math.sqrt(other_in))/(D+1);


}

private boolean checkGCD(long in_value){


long check_value = BigD.gcd(BigInteger.valueOf(in_value)).longValue
();

if (check_value!=1){

Long add_value = new Long(check_value);

if (!g_2.contains(add_value)){

g_2.add(add_value);

}

if (check_value==D) trivial_factors++;


return true;
}

return false;

}

private boolean checkForSquare(long D){

long temp_long = (long)Math.sqrt(D);
if (temp_long*temp_long == D) return true;
else return false;

}


private long setFracRad(long in_f_1){

return (f_1*f_1 - (f_1*f_1 - D/2)*(D+1));

}

private long setIntRad(long in_f_1, long in_h){

return (f_1*f_1 - (f_1*f_1 - in_h*in_h*D)*(D+1));

}

private void searchFrac_v(){

v = (double)((int)(2*min_range))/2;

if (v
do{

checkGCD(r_c(v));

v = v+ 0.5;

}while(v}

}


private void searchInt_v(int in_count){

int c = 1;

for (int k=0; k
temp = min_range/c;

if (temp>-1 && temp<1) continue;


v = ((int)temp)*c;

if (v
do{

checkGCD(r_c(v));
v = v + c;


}while(v}

c = c<<1;

}

}


}

Hey it's way past time for niceness as I solved the factoring
problem. I have a mathematical proof using rather basic algebra that
shows that clearly, and I've put up test code showing a basic
demonstration for a specialized case.

Yet math people put up headlines around the world for quantum
computing when it factored 15.

They follow their own wacky rules!

And one major rule they have is to protect their system, block out
outsiders, and maintain that major mathematical proof is only the
province of people with math Ph.D's.

To them, it's simple: protect their turf.

To you, your actions on the Internet may be transparent. Your
precious public key may be hacked in milliseconds by hostiles using
the mathematics that the established mathematicians are trying to
ignore.

Civilizations are destroyed over stupidity like is happening now.

The mathematicians do not wish to accept merit. They have a class
system. In that class system they do not recognize people like me, so
they refuse to recognize major research results from me.

ALL they can see is later having to clap or applaud or pretend to like
my research. They have nightmares about the ignominy of seeing me get
a math prize and they will not have it.

They are in their own little world.

You are in this one.

Why can't Iran use my mathematical research? Or North Korea? Why
can't the US government for that matter? Or Russia? Or China?

What happens to you when everything you do on the Internet--
EVERYTHING--is open and transparent because no one will accept that
the system is broken so the people with the power are those who are
simply using the mathematics anyway?

Mathematicians are changing your life here and now.

They refuse to follow rules. Re-think the evidence now. Remember I
was published in a peer reviewed mathematical journal, with other very
important research that just so happens to not have the potential of
crashing the world economy.

They PULLED my paper after publication. The journal SHUT-DOWN. The
hosting university, Cameron University, SCRUBBED ALL MENTION of the
journal from its website though it had been around for ten years.

These people are not who you think. And right now your life depends
on getting a clue.

Google: SWJPAM

for the journal story.

The mathematicians are acting with one goal in mind: stop me from
getting rewards for my research or getting elevation in their society.

That's it. They've made up their minds on social class issues.

You need to make up your mind, on the facts.

James Harris