## Puzzling over latest surrogate factoring

Recently I started considering a reverse approach with what I call surrogate factoring and have been puzzling over it. It seems that there are factorizations with small test numbers that indicate some mathematical requirement but I haven't been able to prove it!

In any event, here's the approach I am puzzling over.

With a target composite S, where

(x+k)^2 = y^2 + S

consider a non-trivial factorization, x+k-y = f_1 and x+k+y = f_2, where S = f_1*f_2, so k is some chosen non-zero integer, so x and y are then determined by the factorization.

Now expanding gives

x^2 + 2xk + k^2 = y^2 + S

and if you assume ahead of time some non-zero integer T, such that x^2 = y^2 + n_1*T, then

2xk + k^2 = S + n_1*T

and if you subtract 2k^2 from both sides then you have

2xk - k^2 = S - 2k^2 + n_1*T

and NOW if you assume that S - 2k^2 = n_2*T, you have

2xk - k^2 = (n_2 + n_1)* T, so

n_1 + n_2 = (2xk - k^2)/T

where now you have that T needs to be a factor of 2xk - k^2, and if you look at factors coprime to k, the real focus is on 2x - k, and I've checked with some test factorizations and it always has been such that n_1 and n_2 are integers.

If that were a rigid requirement then there would always exist some T a factor of S - 2k^2, such that

x = (k+T)/2 if k is odd, or

x = k/2 + T

if k is even, assuming that S is odd, as why have an even S to factor.

The weird thing to me is that it has worked with dinky test numbers and I don't know why, as couldn't it be true that no prime factors were in common between S - 2k^2, x^2 - y^2, and 2x - k, given

x+k-y = f_1 and x+k+y = f_2

where S = f_1*f_2?

After all, k is then the only choice as x and y are then forced by a non-trivial factorization, for instance,

2(x+k) = f_1 + f_2.

Oh, there is one remarkable thing as well in that trivial factorizations are blocked!!!

It's not possible for this technique to pull S itself as f_1 or f_2, where there is an easy little proof of that fact.

## Surrogate factoring algorithm

Simple relations connect every integer factorization to an infinity of others as I have shown with simple congruences:

In the ring of integers given

x^2 = y^2 mod T and k = 2x mod T

it must be true that

(x+k)^2 = y^2 + S

where S = 2k^2 mod T

so one congruence of squares is connected to an infinity of others.

That can be used to factor in either direction, but I will focus on letting S be the target composite to be factored in this post.

If S has non-unit factors f_1 and f_2, such that S = f_1*f_2, then a non-trivial factorization is given by

x + k - y = f_1

and

x + k + y = f_2

so

2(x+k) = f_1 + f_2 and

all you need do is determine x and y, and choose k.

So the algorithm is that given a target S to be factored, choose a non-integer k, and find T, from

S - 2k^2 = 0 mod T, so if factor_pool = S - 2k^2, then you factor factor_pool, and loop through combinations of factors of factor_pool, using each combination as T.

Then you find x from x = 2^{-1} k mod T, and can then check

sqrt((x+k)^2 - 4S)

to see if it is an integer and if it is then

f_1 = (-(x+k) + sqrt((x+k)^2 - 4S))/2

but how do you pick k? Well it can be shown that

3k - f_1 - f_2 = 0 mod T

so if S is odd, then k cannot be even, unless you get lucky and 3k - f_1 - f_2 = 0. So k should be odd. Also, if k has 3 as a factor, and f_1 + f_2 does as well, then it is blocked from working unless S has 3 as a factor, so k should be coprime to 3.

Other than that any k can be chosen so it would make sense to choose k such that S - 2k^2 is a minimum with k even and not divisible by 3.

Oddly enough this may be a perfect factoring algorithm which would end considering the factoring problem being a hard problem, but that depends on there not being some other conditions which would stop a particular k from working, or some other reason that looping through the combination of factors of the factor_pool would not give a non-trivial factorization.

This method is more specific than my previous surrogate factoring methods going the other way where T is a target as with those it was not possible to prove that only a single k with some minimal qualifications should work.

## Primes conjecture with surrogate factoring

Oddly enough looking at going in a different direction with what I call the surrogate factoring congruence lead me to a fascinating little result!

Given a composite S, it is not possible for abs(S - 2k^2) to be prime for any non-zero k, unless S is a product of two primes, and they have to be differing primes.

It is kind of an odd little result. Here's an example of when a prime is allowed, S = 77, k = 6, then

77 - 2*36 = 5

but in contrast, say if S = 105, which has 3 prime factors, by this conjecture no non-zero k exists such that

abs(105 - 2k^2)

is a prime number.

The proof is oddly easy but I'll call it a conjecture in case I made a mistake in the argument which can be found at my math blog.

## JSH: Mixed result?

I felt really good about my latest twist on surrogate factoring but thinking about it more I realize it may work best when prime factors are all roughly the same size and it is unknown to me how it would behave if one prime factor is very much greater than the others.

Also, oddly enough, I think it may work best if you do NOT factor S - 2k^2 at all, but simply take it as it is, maybe even needing it to be rather large, especially if one prime factor is very much greater than sqrt(S).

So the picture is muddled to me now when a little while before I was ready yet again to trumpet the demise of the impasse where mathematicians worldwide have so far managed to avoid the truth that my research is revolutionary and correct while they maintain the status quo—including teaching wrong info to trusting students.

With that said, it is a new direction that I came up with hours ago, and experience shows that as time goes on the picture clears.

So then, it's still about time. I pursue the demonstration that will end the impasse.

And the math community lies.

Eventually though I feel I will catch you, and the world will just see one more scandal to add to all the others.

Most people probably won't even yawn, let alone be surprised to hear that the mathematical community is riddled with lying and corruption, with a lot of fake math on which plenty of professors depend, so they lied as long as they could.

Too many liars in our world these days for most people to get too excited I'd think.

You people will be just another soap opera.

## JSH: So now what?

You people pushed and pushed and pushed, so I sat down and figured out what I increasingly think is a polynomial time factoring algorithm.

So now what?

Since it recurses so well my guess is that it can be made to factor an RSA sized number on a desktop PC in minutes. Maybe even seconds.

So now what?

More taunts? More stupid arguments?

Hurry up people. I will get around to implementing this so that I figure I can post an RSA challenge number factorization in a couple of days, but meantime, if you sit on it and it gets exploited I think some of you should face treason charges in your respective countries.

The math is not hard. There is no way any math person with even a modicum of knowledge could have doubted that there might be a way to do this, but instead most of you sat and waited, while the rest of you just lied.

There is no defense for that behavior. At least, no legal defense against a treason charge.

[A reply to someone who asked James to factor some stuff in polynomial time.]

Will do. Soon enough. And then part of my charge against people like you and the rest of the mathematical community that participated on these forums will be bullying as well as lying, while playing with the futures of others because there is something inhuman about you people.

And now I can prove it so that the entire world will be paying attention looking at everything that was said, and tracking down all of you, no matter how anonymous you thought you were.

Here will be a case where bullying online cost the world a truly steep price.

## Polynomial time factoring algorithm?

Given

x^2 = y^2 mod T

where T is a non-zero integer, introducing k = 2x mod T, you can easily solve to find

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

so with S = 2k^2 mod T, you have a second difference of squares.

But what if S is your target composite to be factored?

Then S - 2k^2 = 0 mod T, so by some choice of k, you have T as a factor of S - 2k^2.

Then x = 2^{-1} k mod T, gives you x.

It can be shown that only for certain cases then will y exist as an integer, such that all congruences are satisfied and that if S = p_1*p_2 that case is equivalent to

2(x+k) = p_1 + p_2 or 2(x+k) = p_1 - p_2.

And then it is trivial to factor S now through a difference of squares.

Here is an example. Let the target composite to be factored be 77, so S = 77.

Looking for the smallest S - 2k^2, I take floor(sqrt(77/2)) = 6, so k = 6 gives me a minimum, so

S - 2k^2 = 77 - 72 = 5.

So T = 5. Now x = 2^{1}*6 mod 5 = 3*6 mod 5 = 3.

Then x+k = 9, and p_1 + p_2 = 18, so 2(x+k) = p_1 + p_2, and you can non-trivially factor by solving the quadratic formula as

m^2 + 18 + 77 = (m+7)(m+11).

So the algorithm that follows naturally is with a target S, let k = floor(sqrt(S/2)), then you factor

S - 2(floor(sqrt(S/2))^2

to get T, looping through its factors, and solve for x, using x = 2^{-1} k mod T, and then check

sqrt(((x+k)/2)^2 + 4*S)

to see if it is an integer and if it is, check to see if you have a non-trivial factorization of your target composite S.

Notice this technique can be recursive, as for a MAXIMUM for T you have sqrt(S/2), while it can be much smaller than that, so you have progressively smaller numbers to factor than your target.

There must exist an x for the given k that will work as

x = (p_1+p_2)/2 - k

while the only question then that remains is, how often will x < T, work, as it did with my example above, or can you have cases where abs(x)>T?

If so that could be a problem with this method, otherwise it is guaranteed to non-trivially factor.

I've noted that every integer factorization can be connected to some other factorization with simple congruences, and have talked about a method for factoring by going in one direction with that connection, but why not go the other?

Given

x^2 = y^2 mod T

where T is a non-zero integer, introducing k = 2x mod T, you can easily solve to find

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

so with S = 2k^2 mod T, you have a second difference of squares.

But what if S is your target composite to be factored?

Then S - 2k^2 = 0 mod T, so by some choice of k, you have T as a factor of S - 2k^2.

Then x = 2^{-1} k mod T, gives you x.

It can be shown that only for certain cases then will y exist as an integer, such that all congruences are satisfied and that if S = p_1*p_2 that case is equivalent to

2(x+k) = p_1 + p_2 or 2(x+k) = p_1 - p_2.

And then it is trivial to factor S now through a difference of squares.

Here is an example. Let the target composite to be factored be 77, so S = 77.

Looking for the smallest S - 2k^2, I take floor(sqrt(77/2)) = 6, so k = 6 gives me a minimum, so

S - 2k^2 = 77 - 72 = 5.

So T = 5. Now x = 2^{1}*6 mod 5 = 3*6 mod 5 = 3.

Then x+k = 9, and p_1 + p_2 = 18, so 2(x+k) = p_1 + p_2, and you can non-trivially factor by solving the quadratic formula as

m^2 + 18 + 77 = (m+7)(m+11).

So the algorithm that follows naturally is with a target S, let k = floor(sqrt(S/2)), then you factor

S - 2(floor(sqrt(S/2))^2

to get T, looping through its factors, and solve for x, using x = 2^{-1} k mod T, and then check

sqrt(((x+k)/2)^2 + 4*S)

to see if it is an integer and if it is, check to see if you have a non-trivial factorization of your target composite S.

Notice this technique can be recursive, as for a MAXIMUM for T you have sqrt(S/2), while it can be much smaller than that, so you have progressively smaller numbers to factor than your target.

There must exist an x for the given k that will work as

x = (p_1+p_2)/2 - k

while the only question then that remains is, how often will x < T, work, as it did with my example above, or can you have cases where abs(x)>T?

If so that could be a problem with this method, otherwise it is guaranteed to non-trivially factor.

And that extension of surrogate factoring comes because the connection goes in BOTH DIRECTIONS, where I've focused on one direction until now.

## Congruence relations and algebraic residues

The start of my major mathematical successes came in December 1999, when in pursuit of more variables to chase after a short proof of Fermat's Last Theorem I first wrote:

x+y+vz = 0 (mod x+y+vz)

where I fiddled with various forms, before deciding to stick my new variable v next to z.

Introducing the concept on math newsgroups to howls of derision, one of the more remarkable objections was that it was an invalid expression despite my explaining that is equivalent to

x+y+vz = x+y+vz

so it's an identity. So my big idea was to start with an identity, but use it with congruences.

So how do you use it? Well you subtract out some expression from an expression derived from it, and analyze the residue, which I now call the algebraic residue.

So for instance, say you want to analyze x^2 + y^2 = z^2, to keep things familiar.

x+y+vz = 0(mod x+y+vz), so

x+y=-vz(mod x+y+vz)

and squaring both sides gives

x^2 + 2xy + y^2 = v^2 z^2(mod x+y+vz)

and now you can finally subtract out what I call the conditional expression as

x^2 + 2xy + y^2 = v^2 z^2(mod x+y+vz)

is still an identity so it is always true, and you get

2xy + (v^2-1)z^2(mod x+y+vz), which is

(v^2 - 1)z^2 - 2xy = 0(mod x+y+vz).

And that is the algebraic residue. It is only true if x^2 + y^2 = z^2, so for instance, with

x=3, y=4, and z=5

(v^2 - 1)(25) - 2(3)(4) = 0(mod 7+5v)

and you can trivially prove that is always true without regard to the value of v.

But that is a trivial example while you can do more with more complex examples and in fact all the controversy generated by my paper that went to the now defunct journal SWJPAM, was over an argument using

x^2 + y^2 + vz^2 = 0(mod x^2 + y^2 + vz^2)

with a variant of the FLT equation with p=3, as I think I used something like x^3 - xy + y^2 = z^3, and subtracted to get a residue on which I did some analysis and the rest is history.

Now then at the fundamental level of developing core mathematics what I've done is extend Gauss's idea of using congruence relations in a rather natural way, as of course, if you put in numbers for the variables you just get regular congruence relations. Where to me one of the more remarkable things is that I've introduced using identities that are in ways more complex than the other expressions used with
them.

Using identities of course is common as I showed the familiar completing of the square with my post on factoring where with

x^2 + 2xk = y^2 + k^2 mod T

a more traditional congruence where I put things slightly differently as I put the () around my more complex congruence relations as I think it looks neater—you can of course use k^2 = k^2 to complete the square:

x^2 + 2xk + k^2 = y^2 + k^2 + k^2 mod T

and that is a traditional way to use identities where the identity is LESS complex than the expression it is used with as you have one variable versus four—x, y, k and T.

In contrast with x+y+vz = 0(mod x+y+vz) you have MORE complexity with the identity as you have 4 variables, x, y, z, and v, while the expressions to be subtracted out that I use have only 3, x, y and z.

I say it like that as I don't see why you couldn't subtract out expressions with more variables or whatever as it's new research that I've pioneered and I just use it a certain way. More brilliant minds may find other ways.

The more astute of you who know your math history, know about Gauss's research on congruences, and what he introduced, as well as what is done in modern math in this area may realize that I have taken the position as a successor by extending the concept of using congruences in this way, and there really is nowhere else to go with congruences. So it is finishing research completing what Gauss started.

Now the algebraic residue is interesting as it is only true when what I call the conditional is true, and it must be true if the conditional is true.

And what happened was years ago feeling a bit down about my failed attempts at proving Fermat's Last Theorem, I thought to myself that what I really needed were more variables!

I'd been playing with x, y and z for years and had done every approach I could think of, and in desperation deliberately set out to introduce more variables, so I invented what I now call non-polynomial factorization as a direct result of necessity.

Oh yeah, I call x+y+vz=0(mod x+y+vz) a tautological space. It is an identity so it is always true. And I call x^2 + y^2 = z^2, a conditional, as it is not always true but is conditionally true depending on the values of x, y and z.

So I linked logic, calling identities tautological, to mathematics, and that is a still unexplored branch of this approach.

Now then I have research that covers a lot of territory, but this area is the biggest as any equation or expression available by traditional mathematics can be analyzed by subtracting it from a tautological space and poking at the algebraic residue.

So it encompasses all prior known mathematical research, and extends it.

More than enough for massive resistance. More than enough for a dead math journal.

More than enough to mean that human progress is stalled while the resistance continues no matter what anyone else says or how much they disagree.

Quite simply, while the impasse continues, humanity is not progressing in basic number theory research at all.

## JSH: United States, key here?

Thinking about my own post going over just how much evidence has to be ignored for mathematicians to keep avoiding my research, looking for answers keeps bringing me back to the US and it's odd history in certain areas.

Having a major discoverer from one of the disadvantaged races in this country could help create a shift to education as key—versus hoping for a long-shot in entertainment industries. And would remove the silent argument that many in certain groups here use amongst themselves that "blacks" in this country are actually inferior which explains their current difficulties, even if it's not politically correct to say so.

If you don't think that certain racial groups are inferior then problems they are having today can only be partly explained by historical imbalances, which goes to that other argument that racism was a problem of the past, so it has nothing to do with today's members of advantaged groups, but what if they are lying?

I still like to point out that story where I had the highest SAT of my graduating class but the school changed the rules to give the award for that achievement to a white kid, as an example of how cheating in certain areas can lead to unfair advantages.

Without that award, I couldn't put it on my college application, while that kid could, and he could have that prominently displayed to this day. He can show off his award—society's nod to him for my achievement.

It didn't matter to some extent as I went to Vanderbilt University—but whites made it a point to claim that was about affirmative action.

Kind of see how it works? How there is a lot of psychological warfare involved?

If you can convince someone they are inferior or that nothing they can do will matter to change their social status, then you can more easily dominate them.

Really nasty kind of evil, but one I understand which is why I've understood so well the rather vicious personal attacks many of you have leveled at me.

You see, others were there before you. I know your kind from past experience.

I don't think the issue is really about race though but about class.

I think some British coming to this country instilled in their children a wish to be royalty in their own neo-Britain over here, and got a lot wrong about how class works.

So they dream of being nobles who don't have to work, don't even really have to learn anything, but you just go to parties mostly and rule. Look at George W. Bush as an example of the dream fulfilled.

To them it is only about blood.

So merit does not matter to these neo-Brits in America, who can take one kid's award to give to a white kid who lost, but can then claim to have won.

And hey that does send a message, like your community's refusal to acknowledge proof after proof after proof and I think the US connection is that some of you think that is being brilliant on your part. You think that it'd be nice if we lived in a fair world, and lying is just how things really work, and it's better to be on top than on the bottom.

So you justify inhuman actions by seeing virtue in ruthlessness, and honor in lying.

And who cares really if you actually know math? Or, maybe later someone will be able to steal my results anyway, right? And just take credit for them and you can suddenly gain a renewed appreciation for mathematical truth!!!

Of course that kind of behavior is about weakness and using group processes to overcome personal ineptness.

The reality of needing a majority in the US to do really dumb things can be seen with Republicans and George W. Bush.

The naive belief is that if you have a group of powerful people you can stand against all others, live by no law except your own promotion and that of your own, and that is the way the world really works.

But that is actually the way a lot of people get killed.

The Russian and French communities ended dominance of royalty in their countries the way it is often ended, and the dark side of the group against others is a need for violence, and a fear of it.

In contrast middle class values are about doing your best and winning by merit, so that your win stands.

The middle class is about feeling secure and escaping the need for violence.

Some of you may wonder about that kid, who would now be grown up, who got that award that I won by a rule change.

Will he someday be on the news with a profile asking him about the event?

i know it is a group process many of you are relying on, and I know that you naively think that you can win if you just stick together and follow through with the plan, like what George W. Bush keeps saying should be done with Iraq.

Get the message here? That what you're doing is not hidden nor is it brilliant? That others like politicians do the same thing?

So I know what you're doing. I know it's about your personal weakness and your belief that lying and hiding certain things while relying on the group is all you need because you're not smart enough to know that there is no final solution.

If there were we'd live in one huge empire by now with one indomitable ruling class.

But we don't.

## JSH: Understanding the conundrum

By the rules I can easily show the value of my mathematical research. But the math community is not following its own rules.

So the conundrum is that mathematical proof is proving to be useless with the math community!

That leaves me pushing for the truth in what venues I can where yes, I contact mathematicians directly, yes, I send papers to journals for peer review, and yes I have had others with contacts in academic circles try to get straight answers on my research.

Kind of funny as with one person in California when his colleague asked him about my prime counting function, he went on six month sabbatical!

When he came back he said he didn't remember ever being asked and now would have no comment.

Years ago I asked the sci.math newsgroup if backup from a NASA scientist would matter, and posters ripped on NASA, and yes I had in mind a person at NASA who might have potentially tried to help me.

When I was in an ultra high IQ group, some of its members dared post support for my research, and were ripped on ceaselessly by posters.

Yet repeatedly posters will STILL claim that I'm all alone in believing my mathematical ideas are valuable, and I really find it remarkable when some will say that NO mathematician in the world supports my ideas.

Especially considering that brief publication in the journal SWJPAM, where you people may have destroyed some editorial careers.

Get the gist of the situation?

If sci.math posters can claim that no one supports my research after I've had that brief publication, have had highly intelligent people arguing in support of my work, who will rip NASA when I ask if comments from a scientist there would help, then the actual position is an absolute one.

With my prime counting function I contacted leading researchers in the field and Odlyzko claimed it was not of interest, while Lagarias would just suggest I get published!

With non-polynomial factorization and an early version of the paper that did get briefly published by SWJPAM, I got feedback from Barry Mazur. I sent a draft to Andrew Granville for publication in the New York Journal of Mathematics, and he deferred to the chief editor, saying it was out of his area, and the chief editor said the paper was too small. As in, not long enough in size, not in value as no one claimed it was wrong.

What do posters reply to such facts? They claim that all those people were just being polite! That none of them thought anything of my research but didn't wish to say that in reply, but I should be glad that such important people would even bother replying to me in the first place!

Now there is factoring, and the bar moves so that beyond inventing a new factoring method I also must mature the research to an applied level great enough to tackle very large numbers.

Seem reasonable? Like attacking NASA? Ignoring publication? Ignoring defense of my research? Ignoring the growing volume of that research?

To me Iraq is the best example of denial in another area to give some perspective on why people lie on this scale and why being able to get away with it, is such an important part.

While George W. Bush's party dominated all three major branches of government they could do whatever, knowing there was no authority around to hold them to task, and changes in policy only started here when Republicans lost control of Congress.

Even now Bush can feel safe from impeachment because his party has enough members that they can still block a real investigation.

People lie. And they lie to advance their agenda.

Lying about math isn't even hard today in "pure math" areas as all modern mathematicians have to do to block an idea is say nothing. Do nothing.

And I can't even get a comment from most of them, which is another reason to post on newsgroups as people do reply, and since they're fighting mathematical proof and overwhelming evidence, their replies betray their disregard for the truth.

While the mathematical constituency leaves mathematicians free to lie and devalue mathematical proof, like with Bush and the American people, nothing will change.

So yeah, evidence is something I have lots of, and it doesn't matter if you are dealing with people who feel invulnerable to the truth.

Like Bush, modern number theorists can sit back confident that their constituency will protect them no matter what to insulate them from responsibility to their own jobs.

Which is why often in the real world, proof is not enough.

## Factoring and identities

Much of my research is about using identities and with my factoring research I have a completely traditional usage in contrast with the very complex identities of non-polynomial factorization. I'm going to step through a derivation once again and highlight the identity more this time as the situation is sort of puzzling to me.

For ANY composite T, x and y can be found such that

x^2 = y^2 mod T

and x+y and x-y reveal factors of T, which is just the familiar congruence of squares relation.

I was puzzling over that a bit over a year ago back in August of last year as I began a serious assessment of a concept I called surrogate factoring, wondering if there were any hope for the idea as I had a string of failures trying to get it to work mathematically, and I thought about identities.

To see what occurred to me in a simple way that I think makes it more obvious as it's not exactly how I figured it out over a year ago, start with

y^2 = x^2 mod T

and introduce k, where k = 2x mod T, so it must exist, right? You're just doubling x modulo T, so given x, k MUST exist, so now I have

k = 2x mod T

and do something clever, which is multiply both sides by k, to get

k^2 = 2xk mod T

and with

y^2 = x^2 mod T

it kind of jumps out at you what's needed next as you just add one to the other to get

y^2 + k^2 = x^2 + 2xk mod T

which just begs for a completing of the square and now we have use of an identity!

y^2 + 2k^2 = x^2 + 2xk + k^2 mod T

as k^2 is added to both sides, so I have

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

and that is

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

and you have a new congruence result, which exists for ANY solution of a congruence of squares.

That is the absolute which is easily derived and it must exist, so there isn't a doubt about whether or not it is there.

So what happened over a year ago when I was puzzling over the concept of surrogate factoring trying to mathematicize it, I realized that completion of the square was key, so I deliberately looked to complete the square against x, which is why I needed 2kx.

But why hasn't anyone thought of that before? I don't know, but I have use of identities where I use far more complicated ones than y^2 = y^2, and subtract out equations from them to analyze the residue, and why hasn't anyone thought of that before? Who knows.

How does anyone figure anything out? And why one person and not another? And why now?

Using the derived result requires an explicit equation, so I toss in an integer n to get

(x+k)^2 = y^2 + 2k^2 + nT

and let S = 2k^2 + nT, and you have

(x+k)^2 = y^2 + S

and another congruence of squares! Where now S is potentially factored, or you can factor S, to get x and y, and maybe factor T.

So every congruence of squares

x^2 = y^2 mod T

is connected to another where

(x+k)^2 = y^2 + S

and S = 2k^2 + nT.

And that is then a fundamental result in number theory showing a deep underlying connection between any given factorization and some other number, where I found it because I wanted to complete the square on x^2 = y^2 mod T, in order to implement a concept I call surrogate factoring.

The research then is on picking k and n in the optimal way, which goes to the practical matter of making this into a powerful technique, but regardless of that practical aspect, it is fascinating number theory.

I'll end with a factorization that I like to use so it's scattered around on my web pages as my demonstration example:

T = 732367903, k=floor(T/30) = 24412263, n= -2

S = 2k^2 + nT = 1191915704826532 = ( 2^2 )( 7 )( 73 )( 583129014103 )

y=-170273672118069/2 and x=170273623293557/2

so, x+y=-24412256, which has 223 as a factor.

T = 732367903 = (223)(3284161).

One thing you may note is that the factorization actually works to pull just one prime factor of T, so that you do not have

x^2 = y^2 mod T

but have x^2 = y^2 mod p, where p is a prime factor, which of course, works just as well, and the derivation is hardly different except you have p instead of T.

For me it is a wonderful sense of satisfaction in applying simple ideas and techniques against a well-known problem to find a new approach, and it's kind of odd thinking back that it was about a concept first, and then a deliberate attempt to get to a place where I could use an identity, by completing the square.

On the political side, factoring represents a way for me to demonstrate the power of techniques I call extreme mathematics, as well as highlight how obvious it is the mathematical community is not obeying its own rules.

So a fundamental number theory result is ignored—unless I or someone else rapidly figures out the applied mathematics that would follow from the theory to produce a real world and powerful application.

For me it's just another result where I can see what I saw years ago, when I found my prime counting function, back when I still had some faith in the math community, which is that my research methods work.

Now I test factoring with what I call surrogate factoring, like before I watched my screen fill with prime numbers as the math declared to me the arrival of my prime counting function, and with those absolutes I know absolutely that lesser people who could not, who now will not even be bothered to acknowledge one who could—and did, are liars.

## JSH: Unfair burden

Surrogate factoring is my answer to a math community that does not follow its own rules, which has to date ignored proof after proof after proof that challenges closely held beliefs and the current status quo.

So it is a way to force the situation and with force comes consequences that bother me.

Mathematicians are still fighting to ignore my research which pushes me to bring surrogate factoring to rapid maturation, which would not only collapse confidence in the mathematical community, but it could destabilize the world economy.

And I can state that math people can laugh at it like it's nonsense and the world trusts them, but I know and smart people know it's true.

It can happen because I now have years of looking at this behavior and studying it to understand why mathematicians would rather destroy the world than admit that they are not as smart as they have presented themselves to be or believed themselves to be.

Publication in a math journal is not a minor feat. But publication followed by a once in history attempt at retraction AFTER publication and the death of the journal is the kind of sign that clears that shows how big this situation is. Search on "SWJPAM".

I have stepped through my research answering all mathematical objections and it has not mattered to a math community that refuses to accept mathematical proof.

IN answer I get lies about even the most basic mathematics so that the distributive property is disputed and then claims immediately follow that is is not, against logic, against the evidence, against reality.

With my prime counting research there simply is no other way in all of previous human history to count prime numbers that relies on summing a partial difference equation. That is significant as it immediately leads to a partial differential equation and an integral, unlike any other research previously known in this area.

There is no comparable research in that area which is about prime numbers. None.

There is NO WAY the mathematical community does not understand that, and NO WAY leading researchers in the field do not know what they are doing.

What they are doing is making me the first major researcher in history who is tasked with destroying his world to get even the most simply proven research accepted, and the mathematics gives me the tools.

What I want you people to understand though is that this is not an "oops" situation. It is not a situation where you get to acknowledge defeat if forced on you and just apologize to the world.

You are willfully ignoring a new way to factor.

It HAS to be fundamental as I just included k = 2x mod T, with the already well known and very much studied x^2 = y^2 mod T.

Such a seemingly minor addition is at a fundamental level but despite that I took a year presenting it giving your community an opportunity to show it a trivial addition and your community could not.

It is increasingly clear that mathematicians have taken an absolute position of denying the value of my research putting me in the position of breaking that denial with the only tool that will work, which can also break this country and others.

Forcing me to choose.

The best explanation for denial of my research worldwide is that what I call non-polynomial factorization challenges very closely held beliefs of mathematicians worldwide and forces them to re-think how intelligent they are in certain areas.

The best explanation for denial in the United States is a need for a class view that puts "black people" at the bottom of the pile in terms of intelligence which does not want a major intellectual accomplishment changing the rules for people who want a two-tiered society with their own children on top as some kind of new nobility.

So I live in a world of powerful, willful fools who have put on me the responsibility of destroying that world for its stupdies lies against its own interests.

There is no "white " superiority. People in the United States lie about, well, just about everything.

Lies about Iraq are not out of the norm. White lies are the norm in this country.

This country lie, and lies about lying. It is dedicated to lying.

Like with that story about when I had the highest SAT of my graduating class but whites changed the rules to block me from an award to give it to a white kid who was so, so, proud—whites here cheat, lose, but claim victory in a bizarre need to believe they are the descendants of royal blood as I've explained as a throwback to British lower class backgrounds.

They are social climbers. Royal wannabes. Who cannot really deliver, but will claim otherwise as if lying makes you a winner.

And I am presented with ending the current world order to stop the lies.

What should I choose?

Back people who make it their mission in life to pretend versus do?

Help people who will kill like in Iraq just to pretend to have something they can never achieve?

To pretend to be royalty?

Why should I?

Proof after proof after proof has been denied. Publication in a mathematical journal has been tossed to the side.

Irony demands an ending that is full of death, misery and blood, and I do not want that burden.

Stupid people have died in stupid ways throughout human history.

But I do not want to pull the trigger.

You fight for whites in the US to play at being royals and you die and maybe I die when all I really did was figure out some basic mathematics?

That is not fair.

Your children die, and our future dies because one group of people cannot face the truth?

Maybe China and Russia think they can weather the storm and emerge as the new dominant world powers but WE HAVE NUCLEAR WEAPONS.

What makes them think that people here pushed to the max by reality will care about MAD?

The truth is the answer. Sooner not later.

Or you have your hopes on rationality from irrational people shocked into accepting that they are just human and not royal blood.

You have the potential of a nightmare that does not end because you were stupid enough to think that logic or rationality mattered in an irrational situation.

Forget the plans.

Look at Hiroshima and Nagasaki as they were on those fateful days to see what can happen.

## JSH: State of the research

Whether you realize it or not the preamble with any factoring research that I do is looking for some way to show it's trivial.

Like with my first attempts at factoring algorithms back about 5 years ago, I was just working on extensions of ideas used by Fermat.

Later I had the concept of surrogate factoring as an idea from a question: could you factor one number using another?

And method after method after method failed as I'd figure out or others would figure out that it was something trivial where the underlying relations were often about random or some kind of sieving that would not be earth-shattering in terms of impact on the problem.

So the first year of the life of the latest surrogate factoring research where after over 3 years of searching I realized I only needed to add one variable k, where k = 2x mod T, where T is the target to factor, was really about finding some way to trivialize the research.

And it survived that year plus.

Now it is clear that what I call surrogate factoring IS a new way to factor and is as fundamental in mathematics as methods related to congruence of squares, and there is no way to show it is just trivial, meaning that like methods before it, there is the possibility of a growing body of research that continually improves it.

Except I somewhat accidentally discovered how rapidly it can be improved while typing in some numbers when I watched it factor a 100+ bit number, so already it is far beyond methods based on congruence of squares at this point in its life, and has behavior more like Dixon's.

Surrogate factoring preferentially yanks out small prime factors. And it doesn't seem to care much how big the number is when it does that yanking.

The mathematics is very fundamental as I've only added k = 2x mod T, to x^2 = y^2 mod T, so you have a basis in very rudimentary equations, and now after a year I am certain that it cannot be shown to be trivial.

So it's about time, effort and the natural maturation of an idea.

Previous factoring methods took hundreds of years to reach maturity, but that was without modern computing technology, modern mathematical technique, modern problem solving technique, and trillions of dollars flowing behind an encryption standard that could be made obsolete motivating highly intelligent people to work very hard.

Past history with my prior research indicates that modern mathematicians have taken an absolute position of holding against my research in denial—no matter what.

Even the destruction of a modern electronic mathematical journal had little if any impact, as you can see by searching on "SWJPAM".

But factoring research has the potential of breaking them like people before who have often been broken by taking absolute positions against more powerful forces. Consider Chinese in the Boxer Rebellion who thought that painting themselves "magically" could stop bullets.

And in this case, breaking the absolutism of the mathematical community is unlikely to happen without changing the economic landscape of the entire world.

There is no way that I can see that mathematicians ignoring this research and waiting until it matures is helpful for my own country, currently the dominant world power.

But it is the decision of the modern mathematical and cryptography community that holds sway here as the world trust you.

They trust you, so that is how you have the power to decide the fate of the world.

They sure don't trust me. I'm the "crackpot" with yet another idea claiming it's important.

If you all say it's not, and you're wrong, then the maturation process of surrogate factoring can happen mostly in the dark, and the world instead of facing a relatively new idea that has a distance to go in order to be as powerful as it can be, can instead face a fully matured factoring method—known because it is unleashed.

The issue here is ignorance in the now, and full realization later.

Or an end to the absolute position taken by the world's mathematical community against my research now, versus later.

Fight me on this and you can wake up in a few years to a totally changed world order where you helped create it, dashing the now dominant countries to the ground on their fateful and naive trust in your honesty about your discipline.

History shows that in these situations, your choice is usually against your own best interest, which is why history is so interesting, as empires fall not just on the decisions of the world leaders, but on the seemingly minor ones of people at the fulcrum point.

And this time, in this history to be, the lever is surrogate factoring, and I assure you that with years of research under my belt I now even more firmly believe that it can move the world.

The challenge to me is to balance the needs of the many against the wants of the few.

As I consider the livelihoods of mathematicians around the world, and the savings of people around the world, including in my own country, against the survival of the human race depending on ever forward progress in our knowledge, science and technology.

And I have no choice but to sacrifice the few against the needs of the many. I will sacrifice your actual lives against the needs of the future, against the children yet to be born.

As far as I'm concerned if you make the wrong choice, you simply killed yourselves.

## JSH: Really weird story, not your faults

There really is NO reason at this point for surrogate factoring to not be taken seriously as potentially a powerful factoring technique.

And I don't rely on Usenet alone to get my ideas out.

I HAVE contacted the NSA repeatedly and now am too scared to keep bugging them so I haven't sent them the latest, but still…

I have contacted mathematicians around the world and some people that I have reason to believe are connected with the intelligence community.

There is the Bulletin of the AMS which I've discussed and there are others.

Sometimes I think that there are people who really, really, really want me to post factorizations of RSA Challenge numbers—which I have not yet achieved—and collapse confidence overnight in the current system, but why?

Or do they really think I can't do it?

Where are the people who are supposed to protect us?

Why would world leaders sit back and let this happen? Why? Is our world really this stupid?

## JSH: Let's recap

Years ago believing Andrew Wiles had found a proof of Fermat's Last Theorem I began wondering if there wasn't a simpler answer out there as what he had was so convoluted that he himself said it was impossible to explain simply to laypeople.

To myself I said that maybe with modern problem solving techniques not available over the centuries of searching in the past, and believing that maybe there was a simpler answer when mathematicians for the most part thought there was none, I might stumble across something, and so I began.

Eventually I ended up posting on math newsgroups versus bugging mathematicians in person or at journals which I'd done before, and entered a hellworld of insults and taunts from people who claimed to be mathematically adept.

Still I persevered and in December 1999 I introduced the technique I now call non-polynomial factorization where I use a complex identity and subtract an equation from it—just one step in concept from the normal use of identities.

Four more years later using that breakthrough I found a short proof of Fermat's Last Theorem and expected that I could convince serious members of the math community so I discussed it expecting that with time disagreement would die down as people realized I was right.

During that initial time I was arguing about my proof when I started thinking about counting prime numbers for some reason and within a few weeks had discovered a somewhat complex form of my prime counting function which over the months I simplified.

It was from my prime counting function that I realized that there was something seriously wrong with the math community as I stepped through ways it was like no other discovery in this area and mathematicians I contacted behaved oddly, while on newsgroups people just lied, repeatedly denying even the most obvious differences from what was previously known.

Persevering and hoping still I went back to my proof of Fermat's Last Theorem and isolated out one part of the argument and expanded it into a paper which I sent to a mathematical journal which formally peer reviewed and published it. I thought it was over and finally my research would be acknowledged.

But members of the sci.math newsgroup mounted an email campaign against the paper and convinced the editors who yanked it after publication. A few months later after one more edition the journal itself died.

The story is like some horror movie with one constant: a peanut gallery behaving demoniacally, with taunts and jeers.

I began to realize I was facing a monumental problem unlike any I could have imagined, and an world that did not behave as it should, as there were parallels in the rest of the world:
1. Enron collapsed dramatically after capturing the hopes of many with promises that turned out to be lies. So finance was involved.

2. The Catholic Church finally was forced to acknowledge pedophilia among its priests after years of denial. So religion was involved.

3. The US invaded Iraq under false pretenses relying on the thirst of the American people for revenge for the 9–11 terrorist attacks and kept up its invasion to this day despite much of the truth coming out. So politics was involved.
The pattern was clear: powerful and entrenched people when faced with a truth they did not like, would simply deny it until broken by it.

I began working on the factoring problem.

Now with an invention of surrogate factoring and a rapid development unlike any other in history where first there was a concept from a question, then there were mathematical equations, and then analysis, to find the best practical approach and now there is implementation closing in on a final solution, all within four years.

In contrast the congruence of squares approach with its penultimate level reached with the number field sieve took HUNDREDS of years to develop.

Meanwhile demonic taunts and insults continue from the math newsgroups as the world is on the brink.

In concert with my progress the financial world is crumbling and the US is on the brink of recession, and the taunts continue with one clear demand: give a factorization of an RSA Challenge number.

IF the worst case scenario occurs after that demand is met, the financial world may collapse, and the world as we know it now will be a thing of the past, so why is this happening?

Who could put together such an extraordinary sequence of events?

So I hold yet a little while longer while I ponder these questions.

What intelligence or intelligences, could be great enough to have created this situation? And why?

Why bother?

What does the human species matter anyway?

So I hold.

Who else is out there pulling the strings on all these puppets?

Dare you finally answer me clearly yet? Or must this sad, sick game continue still further?

## JSH: Triumph of stupidity

So I'm stuck now having concluded that no matter what I can prove by mathematical proof alone, math people will just ignore the truth so they can feel comfortable with lies, which leaves me chasing the factoring problem as if I can factor some HUGE number then I can bring their world crashing down.

And that's a long-shot so we're looking at the potential of a triumph of stupidity over mathematical proof.

Math people are just such liars. It amazes me.

It's like with Iraq where George W. Bush came into the White House with a plan to invade Iraq maybe because it was personal to him and he used 9-11, as an excuse.

We all know it and it does not matter. Idiots rule the world and you know why?

Because it's easier to be dumb.

I have proven so many things and even managed to get published but some damn fools claim publication doesn't matter or it was a mistake and hey, go figure! That's all it takes!

It's a world of fools, so who should be surprised with global warming?

The physics isn't even hard but the irony is so damn amazing which is why I talk about boiling frogs.

Frogs in water slowly coming to a boil.

At the end of it all the reality may be that the human race never had a chance to evolve fast enough to keep from boiling itself to death and that is just reality so it is just.

It is fair.

We're just too damn dumb as a species to deserve to survive.

But in the meantime, at least we can have fun. I personally will drink some more, as, at least I have that.

Thank God for alcohol.

## JSH: SF Algorithm

Oh well, enough bravado on my part as I'm not certain this will work and waiting for Wednesday just seems silly. The Bulletin of the AMS did reject. Why would they change their minds between now and then?

The expert opinion is noted. Here is what my research says, which presumably then will not work, but I do not know why it would not.

Given a target composite T, from theory using x^2 = y^2 mod T and k = 2x mod T, it can be proven that

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

must be true for any solution of a difference of squares.

Explicitly to solve you need solutions for

(x+k)^2 = y^2 + 2k^2 + nT.

The algorithm picks x directly, choosing x = floor(sqrt(T)), so k = 2x, and then ranges for the n's from

n_max = floor(((x+k)^2 - 2k^2)/T)

and

n_min = floor((4(x+k-1) - 2k^2)/T)

which with my program has meant roughly 32 surrogates to factor.

By the theory, if you can fully factor all 32 surrogates for any target T, then you will non-trivially factor T.

If you cannot factor all 32 with the given x, you can increment it by 1 and try again, indefinitely.

Note that you can also use x = floor(sqrt(2T)) to have about 64 surrogates and much greater odds but I'm not clear how that works exactly and besides if you can factor 32 with the first one then you have the target in hand.

It is so weirdly simple and I think the theory is correct, but I guess I could be wrong.

I have tried to implement with my own programs but as I pointed out in a previous post, I use recursion and with big numbers fewer and fewer of the surrogates get factored, so it craps out.

I am not confident that I can work that problem out so what I said earlier was bravado on my part.

[A reply to someone who wrote that (x + k)² − 2k² − nT will not, in general, be a perfect square.]

Wo, hey, I didn't think of that until you mentioned it, but it is possible that my latest theory would REQUIRE that one of the n's from n_max to n_min give you a perfect square with that x and k, which would also be required to give a non-trivial factorization of T.

The governing limits are different as they are

-(k+2xT)/T < m_1 < -(k + 2xp_1)/T

where with m_1 all you really need to know is that its absolute value should be greater than or equal to 1, but you don't know p_1 of course as that is a prime factor of T, so you guess at it.

You DO know that one of the prime factors of T MUST be less than sqrt(T) which is why I picked

x = floor(sqrt(T))

to force x larger than the smallest prime factor with the limit:

-(k+2xp)/p < m < -(k + 2x)/p

assuming p is the smallest prime factor. But if you are within the previous limit with m_1 big enough then yeah, theory says you should non-trivially factor T by looping through possible n's with that k and x, looking for a perfect square solution to

(x+k)^2 - 2k^2 - nT

as that would be your y.

So then, with x = floor(sqrt(T)), k = 2x,

n_min = floor((4(x+k-1) - 2k^2)/T)

and

n_max = floor(((x+k)^2 - 2k^2)/T) ,

you could loop through y = sqrt((x+k)^2 - 2k^2 - nT), looking for a perfect square and the theory says there must be at least one, if for a prime factor p_1 of T with

-(k+2xT)/T < m_1 < -(k + 2xp_1)/T

the absolute value of m_1 is 1 or greater.

Oh, hey, and the theory says it MUST non-trivially factor T, so it is an absolute as in 100% probability of factoring T non-trivially.

And that is a definitive test of the theory!

Thanks Marcus!!!

[A reply to someone who wrote that surrogate factoring has the big advantage of getting James some sitting time with God and that he would wager that that is probably not something people get with Fermat's method.]

You people are so damn dumb.

Marcus pointed out a problem with my earlier algorithm as it turns out I have an assumption that

x^2>y^2

so x=floor(sqrt(T)) is too small, so add one.

Here's the corrected algorithm:

Kind of odd really, but fascinating to contemplate as by the theory the following algorithm should be applicable against an RSA sized number factoring it in a maximum of 32 trials:

Note, still with x^2 = y^2 mod T, and k = 2x mod T.

So then, with x = floor(sqrt(T))+1, k = 2x,

n_min = floor((4(x+k-1) - 2k^2)/T)

and

n_max = floor(((x+k)^2 - 2k^2)/T) ,

you loop through y = sqrt((x+k)^2 - 2k^2 - nT), with n's from n_min to n_max, looking for a perfect square and the theory says there must be at least one, if for
a prime factor p_1 of T with

-(k+2xT)/T < m_1 < -(k + 2xp_1)/T

the absolute value of m_1 is 1 or greater.

Guess I should add that now x+y must have one prime factor of T, and x- y must have another, if you find an integer y, for at LEAST ONE of the n's so you take a gcd with T.

So the theory says it will give you a solution to a difference of squares, AND that for at least one of the n's that solution must non-trivially factor T.

That's the theory rejected by the Bulletin of the AMS.

That tests my theory here and intriguingly enough in case some peoplethink there are a lot of n's, there are 32.

32 n's.

So if that theory is correct then you could factor an RSA sized number within 32 trials.

Math people are not very bright. They are PRETEND bright.

Pretend smart, but not real smart.
Yuck. You're slimey. I'm sure you're smiling to yourself as you read this.

Maybe I will never crack the factoring problem.

Maybe you people will win with lies, but win what?

Satisfaction at convincing people too stupid to care about the truth?

So many others have done that and done it better.

But all you win is the death of the human species, and if that's what you're after, then fine.

I'm beaten. Let them die. I can do other things than continue to care.

Somehow I lost when I didn't think that possible which is why it happened. Maybe it's just a learning experience for me. Training for bigger battles down the line when it matters.

The worst can now begin. And the real pain for the planet can now start.

I have stalled it as long as I can and now I'm just tired.

## JSH: One other option

I have completed the theory on surrogate factoring but not posted it or put it in public view.

I wrote a paper and sent it first to the Bulletin of the AMS which rejected it.

I think that gives me permission to put up the factoring algorithm that follows from the paper and if it IS viable, blame the editors of the Bulletin.

But I will not do that until Sept. 12th.

And I will continue working on my own implementation.

That completes my options where again if this research is not viable then there can be no harm from me posting the correct algorithm that follows from it.

And given expert opinion against it by the editors of the Bulletin of the AMS I can justify releasing the full theory on all of my webpages and in posts following their expert opinion as shown by their rejection of the paper.

But will not do so until Wednesday, just in case, those editors decide to change their minds.

Say this is more mindless ravings from a lunatic with no real math ability? Fine. Then just wait until Wednesday.

It is a coincidence. Besides I decided I was being silly talking about waiting till Wednesday as if that would make a difference and went ahead and posted what I have from the latest theory.

And I have been forthcoming about the rejection by the Bulletin of the AMS which was a big disappointment.

I'm running out of room for hope with this research and the expert opinion of one of the world's leading math journals is against it.

So no reason not to just release what I have and hope.

I picked factoring because ultimately I don't need any of you to do your jobs.

I don't need you to tell the truth.

IN fact, it is just funny if you keep lying.

All I have to do is factor an RSA challenge number and the theory I now have says that with time I can do it.

Now laugh at that all you want. Show your continuing bravado against mathematical proof.

But that just makes it that much sweeter.

You people do not know mathematics which is why you are being beaten in this way.

And the point now is that you are not mathematicians, you lie stupidly, and there was no other way to get the truth out so you stand against human progress and the only way to prove it was to put you against a wall and crush you with the truth.

[A reply to someone who reminded James that his algorithm runs about as slow as a random GCD algorithm.]

Clearly I'm testing you people to see if you're still confident.

Sounds like you are.

If you know your math and what you know says I can't do it, then why worry about it?

BUT if your community has been lying about my research hoping I'd never find a way to prove that with some super dramatic discovery that's almost yanked out of the clear blue because I am a great discoverer then yeah, maybe you should worry.

It would seem reasonable to suppose that I can't yet do it, or I'd stuff it in your faces.

The question is, do I have the potential?

If you know that I don't then why worry?

But if you know that I do then make no mistake, the warnings are nearly over and when I'm done I'll make certain that no one ever tries what you people did, again.

## JSH: Surrogate factoring, implementation

It is increasingly clear to me that mathematicians STILL will not follow mathematical proof so I'm working now on the implementation of surrogate factoring to give the demonstration that so many of you clamor for.

That is slow going though as my solution is to have my surrogate factoring code call itself to factor its surrogates that are being used to factor the target.

But even that can do ok, as here's some output where I'm just kind of typing numbers in at random:

Factors:
( 131 )( 2531 )( 10957 )( 40357 )( 682739052250888729441 )
Product: 100098765678976543215788865435214249

In coming is 100098765678976543215788865435214249

Surrogate factorization data for target:

Surrogates factored : 36
Surrogates not factored : 106
Factored fuel percentage: 25%

Data about all surrogates including those from recursions:

Factored fuel : 116306
Fuel not factored: 206
Factored fuel percentage: 99%

Processing time: 62172
Number of digits: 36
bitLength=117

------------------------------------

So with a really big number most of the surrogates aren't getting factored as the program just kind of tosses them and moves on if it can't factor the surrogate, but still it took only 142 surrogates total, though luckily for me there were small prime factors with one big one, so it's not that grand of a demo but I'm getting there.

With recursive calls trying to factor surrogates the program factored 116,306 but was unable to factor 206, which includes the 106 of the surrogates for the target T.

If I can get the program to factor more of its own surrogates when the size increases then I can put in an RSA challenge number.

Slow going but given time I can probably get a program together that ends all debate and simply post a factorization of an RSA challenge number.

And then watch how the math community worldwide reacts.

Posters on sci.math have begged for a demonstration against RSA and it looks like it is coming.

Maybe irony demands that I post the solution if I achieve that goal, on this newsgroup, where so many of you have begged for it.

I think the newsgroup deserves it and the aftermath.

[A reply to someone who suggested that James might want to use the Factor software.]

My invention.

My math.

Your funeral if I win this thing.

No more debates. No more stupid taunts from people like you.

I shut you down. Period.

The editors of the Bulletin of the AMS agreed with you.

So there is no way anyone can blame me for releasing the full theory, correct?

I will do so, Wednesday.

[A reply to his own assertion that if he can get the program to factor more of its own surrogates when the size increases, then he can put in an RSA challenge number.]

Maybe. Here's current output where I'm looking at increasing k, as there is some pattern to this where increasing k is part of the solution.

Begin program output:

n_min=-4095
n_max=511
n_max-n_min=4606

Surrogate factorization: factored? Yes.
( 227961623 )
Product: 227961623
Surrogate factorization: factored? Yes.
( 2 )( 228404723 )
Product: 456809446
Surrogate factorization: factored? Yes.
( 3^2 )( 11^2 )( 277 )( 2273 )
Product: 685657269
Surrogate factorization: factored? Yes.
( 2^2 )( 228626273 )
Product: 914505092
Surrogate factorization: factored? Yes.
( 5 )( 1237 )( 184859 )
Product: 1143352915
Surrogate factorization: factored? Yes.
( 2 )( 3 )( 439 )( 520957 )
Product: 1372200738
Surrogate factorization: factored? Yes.
( 7 )( 23 )( 47 )( 211583 )
Product: 1601048561
Surrogate factorization: factored? Yes.
( 2^6 )( 19 )( 733 )( 2053 )
Product: 1829896384
Surrogate factorization: factored? Yes.
( 3 )( 13 )( 1151 )( 45863 )
Product: 2058744207
Surrogate factorization: factored? Yes.
( 2 )( 5 )( 181 )( 1263863 )
Product: 2287592030
Surrogate factorization: factored? Yes.
( 2516439853 )
Product: 2516439853
Surrogate factorization: factored? Yes.
( 2^2 )( 3^2 )( 41 )( 61 )( 30491 )
Product: 2745287676
Surrogate factorization: factored? Yes.
( 17 )( 1399 )( 125053 )
Product: 2974135499
Surrogate factorization: factored? Yes.
( 2 )( 7 )( 11 )( 20798593 )
Product: 3202983322
Surrogate factorization: factored? Yes.
( 3 )( 5 )( 29 )( 1021 )( 7727 )
Product: 3431831145
Surrogate factorization: factored? Yes.
( 2^3 )( 269 )( 1701059 )
Product: 3660678968
Surrogate factorization: factored? Yes.
( 3889526791 )
Product: 3889526791
Surrogate factorization: factored? Yes.
( 2 )( 3 )( 31 )( 37 )( 598427 )
Product: 4118374614
Surrogate factorization: factored? Yes.
( 521 )( 577 )( 14461 )
Product: 4347222437
Surrogate factorization: factored? Yes.
( 2^2 )( 5 )( 79 )( 2896247 )
Product: 4576070260
Surrogate factorization: factored? Yes.
( 3^3 )( 7 )( 43 )( 199 )( 2971 )
Product: 4804918083
Surrogate factorization: factored? Yes.
( 2 )( 13 )( 83 )( 2332607 )
Product: 5033765906
Surrogate factorization: factored? Yes.
( 11257 )( 467497 )
Product: 5262613729
Surrogate factorization: factored? Yes.
( 2^4 )( 3 )( 1069 )( 107021 )
Product: 5491461552

Number factored.

k=+/-684602
n=-4072
n_max=511

Total all combinations: 22518
Time: 293
Time/combination: 0.01301181277200462

Surrogate:
( 2^4 )( 3 )( 1069 )( 107021 )
Product: 5491461552

Surrogate combinations checked: 233
Initial Factorization:

f_1=12757
f_2=17939
Now checking its factors…
Success!
Factors:
( 3 )( 12757 )( 17939 )
Product: 686543469

In coming is 686543469

Surrogate factorization data for target:

Surrogates factored : 24
Surrogates not factored : 0
Factored fuel percentage: 100%

Data about all surrogates including those from recursions:

Factored fuel : 2177
Fuel not factored: 851
Factored fuel percentage: 71%

Processing time: 625
Number of digits: 9
bitLength=30

--------------------End program output-------------------------

What's interesting here is that all the surrogates were factored and it took 24 to factor with 233 combinations checked, which are what other posters call "probes".

The combinations are when the program takes a factorization of a particular surrogate and loops through every way to factor it into two factors and checks them with the equations derived from the surrogate factoring congruences.

Now I did that run after doing a previous run on the same number with a smaller range for the n's, and it took 1893 combinations, and 151 surrogates.

The difference here was using k = 2*floor(sqrt(9T)) for the first example versus k = 2*floor(sqrt(8T)) for the second.

Now I'd like to remind that I believe there is every evidence that this approach to factoring is worth considering but that the mathematical community to date is WILLFULLY choosing to act like this is nothing, and thereby saying to the world that it is research that should not concern world leaders, investors around the world and ordinary citizens who care about their countries and the way of life.

If they turn out to be wrong then they should necessarily face the consequences.

With expert status comes expert responsibility.

I say that because I have quite a bit of research now where I believe that narrow minded members of the math community choose to deliberately ignore it, in order to protect the current status quo.

If so, then they should have to pay a price if their strategy is hugely wrong.
Proof of concept.

And proof of contradiction from a mathematical community that pushed headlines when another factoring technique managed just barely to factor—15.

Your community if full of fools.

Now see if you can get a major news organization to care about what you do so easily as you did in the past.

You people come after me and you shred yourselves.

The news organizations of the world are learning that you decide on a whim what you choose to think is important, so one factoring technique manages to factor 15, and you promote it like it's a mega event.

And I demonstrate factoring much bigger numbers and you say it's nothing.

What will your community do when you are just talking to yourselves and no major news organization will report anything you claim?

I suspect you will do what you're doing now, continue to go insane.

As the mathematical community slowly goes insane and I watch, you think the rest of the world can't notice as well—because you're going insane.

## JSH: My math

Today like many other days I looked over what I call my toys, my mathematical discoveries. I counted some primes with my prime counting function, and factored some numbers with my invention, surrogate factoring, and it is always this weird feeling as with mathematics I know how much of this world is about lying.

People lie. So it's about time and history before other people may know what I have discovered or, better yet, admit what I've discovered, but I'm outside of time and can appreciate it now.

I can look over my non-polynomial factorization results and contemplate my proof of Fermat's Last Theorem.

I can read over my definition of mathematical proof, and think about the object ring.

I can contemplate the connection between math and logic, and wonder about where ideas come from, and what it means to seek truth.

At the end of the day, as I'm typing this at the end of a day, I have the knowledge of my humanity and the seeking of knowledge for its own sake, beyond what people say, beyond what it got me because from others it got me headaches, and insults and a lot of people trying to tell me I was wrong.

Knowing the truth does not mean that others know that you know. And maybe the pain of that is just part of understanding what truth is.

It is not about what people believe you know. Proof is not about belief.

What I got, what I have is the knowledge of rightness. The sense of a perfection in reality.

What I got was sitting time with God.

And at the end of the day, sitting as the sole human being in that place is not a negative, but a gut check.

For now, for this time, I am the sole human being in charge of some of the greatest knowledge that humanity has come across to date, and it is my destiny to know.

It was my destiny to know.

And for while that lasts, it's my math.

And in this place while I stand alone at the pinnacle of human knowledge, and I look out at infinity, and see it alone.

I appreciate it that much more.

I stand alone, and I know, and in knowing, with my place far beyond what most people can imagine, I touch infinity, and in knowing that small part that I do, I appreciate that no matter what anyone else thinks.

It is all—brilliant.

## JSH: Contradictory behavior, issue of math fraud

I have "pure math" research that I say is important, while mathematicians have in various ways denied that, even when I had backup, albeit brief, from mathematicians who published some of my research in the now defunct journal SWJPAM.

So I say my work is important, yet mathematicians say, in various ways, so that it can be a symbolic say, that it is not, which leaves me with a quandary, as it's my word against theirs.

So I went to the factoring problem.

If as I say mathematicians routinely lie about math, and I do come up with a viable factoring approach then it stands to reason that they would CONTINUE to lie, but other people might use the research anyway.

But then again, I might simply be unable to come up with a viable approach to the factoring problem.

However, if I come up with an approach then if it is NOT viable then mathematicians, supposedly brilliant, should be able to settle it, and simply proclaim me as just being the crackpot who has nothing—and prove it.

Otherwise they leave the world at the wrong end of the whip where BILLIONS of dollars US, as in yes, BILLIONS of dollars could swing in hours on what is the truth, and that is the lever.

Archimedes said, give me a lever long enough and a solid place to stand and I can move the world.

We live in a world that has learned to dismiss ideas, and believes that genius can be controlled.

The death of modern mathematics as a viable discipline so that science depends on the discoveries of past mathematicians is about attempts at controlling creativity squeezing the life out of the modern research world.

But these people are about politics, so they work to convince, and if they get something wrong then entire economies can fall.j

The entire planet of humanity can believe the world is flat and be wrong.

Belief can be just a way for you to get yourself killed. And LOTS of people believing the same thing can be just a way for you to get yourselves all killed.

What did the people of Hiroshima and Nagasaki believe?

Did that matter?

So for you there is the question on which your life savings can depend—the potential to lose everything you have worked your entire life for, literally, overnight, because some people you do not even know, lied.

Lose everything.

No retirement. No golden years looking back but working harder than ever knowing that everything you did before was lost because you trusted the wrong people.

Or there is nothing here and I'm just a loudmouth on Usenet babbling nonsense, and you can trust those math people you don't even know to keep you safe, and keep your retirement safe, and keep your family safe.

Or force mathematicians to settle the question:

Does surrogate factoring work or not? If not, what is the mathematical analysis that proves it does not?

Or sit and wait, and stay at the wrong end of the whip and see how long you get yanked around.

[A reply to someone who asked why is it that James keeps posting at the sci.math newsgroup.]

Because by posting here I can present ideas to the math community worldwide.

If later those ideas are shown to be viable and there is no possible way a math community that cares about mathematical research for real and has natural human curiosity could have ignored them, then I make my point that most math people today are con artists.

So you see, I have to put the information in a place where math people can get to it.

And I have to talk enough around it about basic human curiosity and evidence of truly valuing a subject to take away what is often called plausible deniability.

That is, I have to remove all other possibilities EXCEPT math people being con artists.

And that takes time and some careful maneuvering as well as multiple actions to ensure that mathematicians had every chance to do the right thing.

Challenging Santos to commit every dime is part of that action, as to con artists, what really is more important than money?

I challenge you as well to pledge every penny you own to anyone worldwide who loses money if surrogate factoring turns out to be what your community is saying it is not.

By pledging what you own to them you can give them the knowledge that you have determined to the best of your ability that there is nothing to what I say, and are willing to accept consequences if you are wrong.

As being an expert gives responsibility and part of responsibility is accepting the consequence of your actions.

So it's gut check time.

Put up all your money—after all, what does it mean to you anyway if math is what you care about—as a gift to anyone who loses money if you are wrong, as your personal compensation for their loss.

[A reply to someone who wrote that James has been perpetuating a con job on the sci.math newsgroup, always promising something and never having delivered on one promise.]

Except I HAVE delivered. Instead of just arguing with people over my proof of Fermat's Last Theorem I wrote a paper over a key results that followed from it and got it published.

Posters on sci.math then declared that the journal system was flawed and that math journals routinely publish false papers!!!

Others mounted an email campaign against the paper and convinced the journal editors it was false, so they yanked it, and later the journal shut down.

With my prime counting research I first found my prime counting function, and then proved how it was different from anything else previously known as to this day no one can give any other partial difference equation used to count prime numbers, and no other known that finds primes on its own.

Posters on sci.math when challenged with those points shift the definition of "difference equation" to a non-standard one, and ignore the second point about finding primes or just lie about it.

Repeatedly, by all normal standards, I achieve and posters deny in unreasonable ways all achievements while making dubious achievements of their own—like killing a math journal.

REASONABLE people who listen to me talk about the factoring problem can note that I'm making sense, while posters arguing with me, can't even be bothered to present a mathematical argument against my research or in support of their claims.

[A reply to someone who wrote that people at the sci.math newsgroup have repeatedly tried to implement James' many variants of surrogate factoring and that none of them is remotely close to competitive with existing methods for factoring integers.]

Sounds like you're ready to defend on every point, so your position is clear, and now if that position is refuted ultimately by the evidence, so that everything you said falls apart like a house of cards, what then?

Can you get what I'm emphasizing here? That the math community cannot have its cake and eat it too?

I want it clear that if you turn out to be wrong you lose the title of experts.

Period.

No if's and's or but's, but quite succinctly, you lose the title of experts in the field.

Get it? So it's not about me asking anything from you except clarity on this position.

I want you and your community to understand that if I force this and prove that I am correct, you lose the title "mathematicians".

[A reply to someone who wrote that people at the sci.math newsgroup would have a much higher probability of looking stupid if they spent a lot of time trying to get every crank's pet notions to work.]

And I think that you resist all evidence for class reasons, and a need to try and hold onto a nasty worldview that some people in the United States want because it justifies dominating other "races" based on the idea that they are inherently genetically inferior.

If you acknowledge that mathematicians made mistakes that I discovered, or missed simple mathematical ideas that I've found, you lose the genetic argument.

You can no longer claim that "whites" are born to rule, and that claims otherwise are about being politically correct, or to keep "minorities" from rioting.

It is a US position against the future of the world, where mathematical proof hasn't mattered because people like you hold that view as a security blanket believing that if you can manipulate and lie enough to control populations while hiding that from them, then that must mean you are smarter, better and deserve to survive when and if the time comes that a decision has to be made, who lives and who dies.

So to you lying about lying and getting away with it is just proof that you are smarter, as how else can intelligent people ignore proofs? How could they ignore publication in a math journal unless they were too stupid to understand why people like you do those kinds of tests to be more certain of your control.

The US invaded Iraq on lies and pretenses and gets away with it partly to test that control.

The US has a peculiar position of living by contrasts and contradictions.

The real point I'm making is how far some of you will go to hold on to some very perverted views about the human species.

You think you are born better so that you feel more secure.

And I tell you that you are just another human being and what you are is about merit, not race.

## JSH: What is surrogate factoring? Once more.

IN arguing about research I call surrogate factoring, I bump into this weird thing where posters seem to be lost on what is actually going on, so I thought I'd start a thread informing, yet again, what surrogate factoring is.

Years ago, while thinking about RSA encryption, I wondered to myself if instead of directly attacking a large number that you wanted to factor, you might instead factor some other number and in that way factor the target.

I termed the concept: surrogate factoring.

So, to repeat, years ago, as in about four years ago I think it was, I was just kind of wondering about factoring because I was thinking about RSA encryption, and I wondered if you might go after a large composite that was otherwise hard to factor, by instead factoring some other number.

To me a good name for the concept was "surrogate factoring", so it was called surrogate factoring.

Now years later I have finally settled in my own mind that mathematically the concept reduces to considering

x^2 = y^2 mod T

and

k = 2x mod T

and equations that result from those two basic congruences, where T is the target, which took me about three years to figure out.

With those two relations I found that my surrogate to factor is given by deriving

(x+k)^2 = y^2 + 2k^2 + nT

as then the surrogate S, is

S = 2k^2 + nT

and the big question is, how do you pick k and n?

For those of you who wonder how it works from there, it's trivial algebra that if you let

4f_1*f_2 = 2k^2 + nT

then

x+y = 2f_1 - k and x-y = 2f_2 - k

so once you factor the surrogate S, you just loop through solutions for x+y and x-y, by going through the various possible values for f_1 and f_2, and check the gcd with T.

So, to recap, about four years ago I was wondering whether or not you could go after an RSA sized number to try and factor it by instead factoring another number. For years I tried various approaches and last year I boiled down the idea to two congruence relations, which lead through some simple algebra to a way to factor the target T, by factoring the surrogate S.

So I had an idea, and after four years I have the math that implements the idea.

That is the "pure math" aspect of it all where a person just pursues a mathematical problem for the hell of it, you might say, while, of course, I had practical reasons for picking the factoring problem.

Now then, from the realm of mathematical curiosity to a world changing idea requires that surrogate factoring be a way to actually factor a large composite faster than the other known methods, which is where the arguing comes in with people who want to be certain that no one believes it can be, or who are getting on my case for declaring it is, and then not delivering by factoring some large number.

But that is secondary, as it is a practical matter that can move stock markets and scare people because if surrogate factoring makes factoring easy, then a lot of industries around the world would be impacted.

But what real mathematician cares about practical crap anyway?

And to the extent that math people act more like business people who care about the practical side than math people who would care about the curiosity side, I point out a contradiction!

Maybe they are not math people after all, eh?

As there is the practical and political reality of possibly changing the world with a simple concept.

Understand surrogate factoring now?

Oh yeah, so recently I came up with a detailed analysis of when and why surrogate factoring works, which has some very complicated looking equations in it, so it is a massive puzzle.

A MASSIVE puzzle.

Some have done experiments where they claim that surrogate factoring works worse than random!

And the world hangs in the balance on the answer, or maybe not, if it's just a crap idea, but for some reason, supposedly brilliant mathematicians have not settled the question so that the stock markets can rest easy.

And your fate may depend on the answer, so the math world cannot keep looking, so Google and Yahoo! search results move accordingly, as if this concept is viable, then it ends the modern math world as it currently operates.

But, on the other hand, it is also just a 'pure math' idea in a lot of ways.

Two ways of looking at it, and entire economies can be destroyed if people do not do the right thing here, and guess wrong.