### Monday, November 30, 2009

## JSH: Ranting aside, a remarkable error

When people refuse to acknowledge even the most basic mathematics it can be amazingly hard to get your point across where with this particular error, a hundred years plus of research built on the error gives some people (unfortunately) a lot of motivation to just fail as mathematicians in the face of it.

In a lot of ways it's a trivial demonstration, given

7(175x^2 - 15x + 2) = (5a_1(x) + 7)(5a_2(x)+ 7)

where the a's are roots of

a^2 - (7x-1)a + (49x^2 - 14x) = 0

the mathematics does not allow that 7 to be controlled by the factorization of what it is multiplying, so you have with a simple example:

7(x^2 + 3x + 2) = (7x + 7)(x + 2)

a CHOICE.

That's because it could be: 7(x^2 + 3x + 2) = (x + 1)(7x + 14).

Now I've pointed this out many times over the years as I've repeatedly explained this error!

No person thinking rationally would suppose that 7 is being told how to multiply by mysterious forces. I used to say, years ago, that the tail does not wag the dog.

So, ok, a non-standard factorization reveals a bizarre problem with the ring of algebraic integers. The construct creatively FORCES the roots of monic polynomial with integer coefficients when x is an integer i.e.

a^2 - (7x-1)a + (49x^2 - 14x) = 0

to be part of a factorization:

7(175x^2 - 15x + 2) = (5a_1(x) + 7)(5a_2(x)+ 7)

so yes you can solve for the a's with the quadratic formula:

a_1(x) = ((7x-1)+/- sqrt((7x-1)^2 - 4(49x^2 - 14x)))/2,

a_2(x) = ((7x-1) -/+ sqrt((7x-1)^2 - 4(49x^2 - 14x)))/2

And you can make the substitutions and see that it all works. If Dedekind had been given this demonstration over a hundred years ago, you would not be learning the ring of algebraic integers today, except maybe as an oddity: a fairly useless ring which has a fatal flaw.

But a hundred years of error built up around this error.

It takes away the usefulness of Galois Theory. It removes the ring of algebraic integers as a useful ring.

Ok, since I found this error students have continued to be indoctrinated into it. That is a lot of human waste.

Professors who keep feeding it to students may in some cases be functionally insane due to the enormity of the problem.

Or they may just be cynically determined not to face consequences of it. Who knows.

But if you learn it, you are training your brain useless knowledge which does not work.

The math done with it is pure—purely wrong.

So yeah, they'll teach you style. It's style WITHOUT substance.

They force style because the math itself is wrong. Style crap also helps to keep out outsiders like me!

Rigor in error is still error. You are just rigorously wrong.

The problem is a remarkable error. It has taken over a hundred years to be revealed. Puzzle over it. Take your time.

Ask yourself: why can't I divide that 7 off?

There is only one answer because it's mathematics.

Remember that word? "Mathematics"

There is an answer. The right one.

In a lot of ways it's a trivial demonstration, given

7(175x^2 - 15x + 2) = (5a_1(x) + 7)(5a_2(x)+ 7)

where the a's are roots of

a^2 - (7x-1)a + (49x^2 - 14x) = 0

the mathematics does not allow that 7 to be controlled by the factorization of what it is multiplying, so you have with a simple example:

7(x^2 + 3x + 2) = (7x + 7)(x + 2)

a CHOICE.

That's because it could be: 7(x^2 + 3x + 2) = (x + 1)(7x + 14).

Now I've pointed this out many times over the years as I've repeatedly explained this error!

No person thinking rationally would suppose that 7 is being told how to multiply by mysterious forces. I used to say, years ago, that the tail does not wag the dog.

So, ok, a non-standard factorization reveals a bizarre problem with the ring of algebraic integers. The construct creatively FORCES the roots of monic polynomial with integer coefficients when x is an integer i.e.

a^2 - (7x-1)a + (49x^2 - 14x) = 0

to be part of a factorization:

7(175x^2 - 15x + 2) = (5a_1(x) + 7)(5a_2(x)+ 7)

so yes you can solve for the a's with the quadratic formula:

a_1(x) = ((7x-1)+/- sqrt((7x-1)^2 - 4(49x^2 - 14x)))/2,

a_2(x) = ((7x-1) -/+ sqrt((7x-1)^2 - 4(49x^2 - 14x)))/2

And you can make the substitutions and see that it all works. If Dedekind had been given this demonstration over a hundred years ago, you would not be learning the ring of algebraic integers today, except maybe as an oddity: a fairly useless ring which has a fatal flaw.

But a hundred years of error built up around this error.

It takes away the usefulness of Galois Theory. It removes the ring of algebraic integers as a useful ring.

Ok, since I found this error students have continued to be indoctrinated into it. That is a lot of human waste.

Professors who keep feeding it to students may in some cases be functionally insane due to the enormity of the problem.

Or they may just be cynically determined not to face consequences of it. Who knows.

But if you learn it, you are training your brain useless knowledge which does not work.

The math done with it is pure—purely wrong.

So yeah, they'll teach you style. It's style WITHOUT substance.

They force style because the math itself is wrong. Style crap also helps to keep out outsiders like me!

Rigor in error is still error. You are just rigorously wrong.

The problem is a remarkable error. It has taken over a hundred years to be revealed. Puzzle over it. Take your time.

Ask yourself: why can't I divide that 7 off?

There is only one answer because it's mathematics.

Remember that word? "Mathematics"

There is an answer. The right one.

## Trashing Galois Theory

Turns out you can destroy the underpinnings of modern number theory with a simple mathematical example:

7(175x^2 - 15x + 2) = (5a_1(x) + 7)(5a_2(x)+ 7)

where the a's are roots of

a^2 - (7x-1)a + (49x^2 - 14x) = 0.

The 7 on the left of the equals with the first expression cannot in general be removed on the right hand side IN THE RING OF ALGEBRAIC INTEGERS.

The a's are the roots of the second expression, so you can use the quadratic formula to solve for them:

a_1(x) = ((7x-1)+/- sqrt((7x-1)^2 - 4(49x^2 - 14x)))/2, a_2(x) = ((7x-1) -/+ sqrt((7x-1)^2 - 4(49x^2 - 14x)))/2

I've brought this issue up time and time again, and one objection I've heard repeatedly is that you CAN divide the 7 off for INDIVIDUAL CASES, like for x=1, you can do some convoluted crap using Galois Theory to find the "factors" of 7 in the ring of algebraic integers that would divide off FOR THAT SPECIAL CASE.

But hey, isn't this algebra? Why don't you have special case for:

7(x^2 + 3x + 2) = (7x + 7)(x + 2)?

Why can I just divide THAT 7 off to get: x^2 + 3x + 2 = (x + 1)(x + 2)?

The answer is: the ring of algebraic integers is CRAP. It was crap years ago when I found this problem and it's still crap now though YOUR STUPID PROFESSORS STILL TEACH YOU CRAP.

Yes, they are stupid for teaching you crap and you are stupid if you keep learning it after this example.

Or are you such a moron that you don't know that if you multiply something times say, 7, you should be able to divide the freaking 7 back off?

I don't care if your professor is at Harvard. If he's teaching you stupid crap it's still stupid crap.

It doesn't matter if she is at Oxford. If she is teaching you stupid crap it's still stupid crap.

And you're still stupid for learning it.

7(175x^2 - 15x + 2) = (5a_1(x) + 7)(5a_2(x)+ 7)

where the a's are roots of

a^2 - (7x-1)a + (49x^2 - 14x) = 0.

The 7 on the left of the equals with the first expression cannot in general be removed on the right hand side IN THE RING OF ALGEBRAIC INTEGERS.

The a's are the roots of the second expression, so you can use the quadratic formula to solve for them:

a_1(x) = ((7x-1)+/- sqrt((7x-1)^2 - 4(49x^2 - 14x)))/2, a_2(x) = ((7x-1) -/+ sqrt((7x-1)^2 - 4(49x^2 - 14x)))/2

I've brought this issue up time and time again, and one objection I've heard repeatedly is that you CAN divide the 7 off for INDIVIDUAL CASES, like for x=1, you can do some convoluted crap using Galois Theory to find the "factors" of 7 in the ring of algebraic integers that would divide off FOR THAT SPECIAL CASE.

But hey, isn't this algebra? Why don't you have special case for:

7(x^2 + 3x + 2) = (7x + 7)(x + 2)?

Why can I just divide THAT 7 off to get: x^2 + 3x + 2 = (x + 1)(x + 2)?

The answer is: the ring of algebraic integers is CRAP. It was crap years ago when I found this problem and it's still crap now though YOUR STUPID PROFESSORS STILL TEACH YOU CRAP.

Yes, they are stupid for teaching you crap and you are stupid if you keep learning it after this example.

Or are you such a moron that you don't know that if you multiply something times say, 7, you should be able to divide the freaking 7 back off?

I don't care if your professor is at Harvard. If he's teaching you stupid crap it's still stupid crap.

It doesn't matter if she is at Oxford. If she is teaching you stupid crap it's still stupid crap.

And you're still stupid for learning it.

### Friday, November 27, 2009

## JSH: Authority from Google search results?

One of the more fascinating things that I've faced recently is the question of why some of my own amateur research like on my math blog comes up highly in certain Google search results. But one thing is clear, when

What is the current accepted criteria for research authority is citation of published papers in formally peer reviewed journals where an additional layer of highly regarded formally peer reviewed journals is often given as well.

But web search offers the opportunity for research to just be used, regardless of whether it goes through some academic process or not.

And presumably search engines deliver to people search results they find useful.

When it comes to research then, someone like myself would like to present web search results as an indication of interest in my results, while the establishment might consider it a threat to the current peer review journal system.

That's because it IS.

If web search engines can objectively push up research based on world interest then they can simply remove human opinion from the process entire.

That could plunge formally peer reviewed journals into irrelevancy. Remove power from editorial boards at those journals. And even challenge the academic system by forcing professors at even top universities around the world to potentially compete with non-professionals doing their own research.

The question then is, does web search offer the possibility of an

Is the modern journal system doomed by the Internet?

For the curious here are some Internet searches where I appear to dominate in Google, around the world:

define mathematical proof

solving binary quadratic Diophantine equations

Class Viewer

prime number compression

All those cases are WITHOUT quotes. Those should give #1. I have two results that show up highly in Yahoo! as well:

Class Viewer

solving quadratic residues

For results that just show up in the top 10:

optimal path algorithm

tautological spaces

core error

object ring

I have surmised that Google search results may be a leading indicator to research trends otherwise invisible, as some of you may be aware that often new ideas take a convoluted path to acceptance, and can seem unknown despite being around for years before some critical event occurs and they take over.

For instance, the leading indication from Google about the definition of mathematical proof may indicate that at some point in that future, my definition will be the established one, which will then be in dictionaries around the world, which is of course a self-serving notion!

I welcome disagreement. The data is fairly easy to check: you just do the web searches.

The implications could change our world, and shatter the academic hierarchy, forcing academics to not just compete with each other, or well-funded groups at private companies, but even with self-funded laypeople—from around the world.

The hotbed of intellectual activity that could result is hard to imagine. It could be a renaissance beyond anything the world has ever seen, or given a lot of the crap that is on the Internet, one might argue, it could be a recipe for intellectual anarchy.

What is the answer?

**newsgroups**are presented with the information, posters immediately make efforts to discount such results, which is actually kind of interesting in and of itself.What is the current accepted criteria for research authority is citation of published papers in formally peer reviewed journals where an additional layer of highly regarded formally peer reviewed journals is often given as well.

But web search offers the opportunity for research to just be used, regardless of whether it goes through some academic process or not.

And presumably search engines deliver to people search results they find useful.

When it comes to research then, someone like myself would like to present web search results as an indication of interest in my results, while the establishment might consider it a threat to the current peer review journal system.

That's because it IS.

If web search engines can objectively push up research based on world interest then they can simply remove human opinion from the process entire.

That could plunge formally peer reviewed journals into irrelevancy. Remove power from editorial boards at those journals. And even challenge the academic system by forcing professors at even top universities around the world to potentially compete with non-professionals doing their own research.

The question then is, does web search offer the possibility of an

**objective**"peer review" of sorts that is simply more effective than the current academic journal system? Can the "wisdom of crowds" just out-do editors at journals around the world?Is the modern journal system doomed by the Internet?

For the curious here are some Internet searches where I appear to dominate in Google, around the world:

define mathematical proof

solving binary quadratic Diophantine equations

Class Viewer

prime number compression

All those cases are WITHOUT quotes. Those should give #1. I have two results that show up highly in Yahoo! as well:

Class Viewer

solving quadratic residues

For results that just show up in the top 10:

optimal path algorithm

tautological spaces

core error

object ring

I have surmised that Google search results may be a leading indicator to research trends otherwise invisible, as some of you may be aware that often new ideas take a convoluted path to acceptance, and can seem unknown despite being around for years before some critical event occurs and they take over.

For instance, the leading indication from Google about the definition of mathematical proof may indicate that at some point in that future, my definition will be the established one, which will then be in dictionaries around the world, which is of course a self-serving notion!

I welcome disagreement. The data is fairly easy to check: you just do the web searches.

The implications could change our world, and shatter the academic hierarchy, forcing academics to not just compete with each other, or well-funded groups at private companies, but even with self-funded laypeople—from around the world.

The hotbed of intellectual activity that could result is hard to imagine. It could be a renaissance beyond anything the world has ever seen, or given a lot of the crap that is on the Internet, one might argue, it could be a recipe for intellectual anarchy.

What is the answer?

## JSH: Right answer on research funding?

Recent news has the government of Japan considering what science funding is worthwhile, and my recent conclusion has been that sharp cuts in research funding could seemingly paradoxically create a boom in useful research results, so it raises the question of, what is the right answer on research funding?

The louder voices from the scientific community would have the world believe that there is no limit, and that science research should be funded to the maximum possible, but humanity got along for quite a long time without ANY systematic research funding, and it's an open question on how best to motivate good results as revealed by the bonus situation in the financial sector.

Big bonuses for financial people led to widespread systemic risk, and a near meltdown of the world's financial system.

Open ended funding for anything and everything "science" could have unintended negative consequences as well.

Also recent research (sorry no citations but it is readily available) has indicated that human beings behave oddly with financial reward anyway, and that it can be counter-productive to give monetary benefit when a social good is what is desired.

And, of course, the most gifted researchers will find a way.

The urge to know is a drive within them which they will try to fulfill come what may. With limited funding they may simply get more creative, and make discoveries they might not make if funding were easy, as necessity drives invention.

My own fear is that the current system can hide people who can play at the academic game of looking like they're doing something, while doing nothing of value—and knowing it. Who unfortunately can go all the way to being completely wrong, yet fight evidence that they are wrong as that can impact their funding!!!

So then society funds itself into error.

What is happening in Japan is of great interest I'd think worldwide and I complement the Japanese government on taking this important effort, leading the way as they have in other areas, like confronting global climate change.

My hope is that they do what is necessary, including slashing funds deeply, despite loud protests from the academic community.

I don't care how many Nobel prize winners protest. The issue may be beyond their areas of expertise!

It's probable they themselves don't fully understand HOW they got their own research results, or we as a species would have the mystery of discovery solved.

I would call on world governments to consider slashing funding for science as an option to see if a paradoxical increase in valuable research results occurs as an effect. Doomsday cries from the academic community when histrionic are simply to be ignored.

After all, if there are objective reasons those should be presented—not crying and whining. Adults should be able to do better.

The louder voices from the scientific community would have the world believe that there is no limit, and that science research should be funded to the maximum possible, but humanity got along for quite a long time without ANY systematic research funding, and it's an open question on how best to motivate good results as revealed by the bonus situation in the financial sector.

Big bonuses for financial people led to widespread systemic risk, and a near meltdown of the world's financial system.

Open ended funding for anything and everything "science" could have unintended negative consequences as well.

Also recent research (sorry no citations but it is readily available) has indicated that human beings behave oddly with financial reward anyway, and that it can be counter-productive to give monetary benefit when a social good is what is desired.

And, of course, the most gifted researchers will find a way.

The urge to know is a drive within them which they will try to fulfill come what may. With limited funding they may simply get more creative, and make discoveries they might not make if funding were easy, as necessity drives invention.

My own fear is that the current system can hide people who can play at the academic game of looking like they're doing something, while doing nothing of value—and knowing it. Who unfortunately can go all the way to being completely wrong, yet fight evidence that they are wrong as that can impact their funding!!!

So then society funds itself into error.

What is happening in Japan is of great interest I'd think worldwide and I complement the Japanese government on taking this important effort, leading the way as they have in other areas, like confronting global climate change.

My hope is that they do what is necessary, including slashing funds deeply, despite loud protests from the academic community.

I don't care how many Nobel prize winners protest. The issue may be beyond their areas of expertise!

It's probable they themselves don't fully understand HOW they got their own research results, or we as a species would have the mystery of discovery solved.

I would call on world governments to consider slashing funding for science as an option to see if a paradoxical increase in valuable research results occurs as an effect. Doomsday cries from the academic community when histrionic are simply to be ignored.

After all, if there are objective reasons those should be presented—not crying and whining. Adults should be able to do better.

### Thursday, November 26, 2009

## JSH: My sphere packing idea

I noted a little while back that back around 1996 I came up with what I think is a solution to the sphere packing problem, which relied on only basic algebra. At that time I wrote a paper and sent it to a math journal where the editor told me it was too simple, and it was not published.

I lost that paper (I tend to lose things as part of an on-going experiment to see what historical impact it might have) but in recollecting my ideas recently I posted what I THINK the solution was on my math blog, where now that blog post comes up for me with a Google search in the top 10. But the Google search (has to be Google) is:

spherical packing problem

I posted about this before and the newsgroup subject I made took over and dropped the blog page out of the top 10 for a while so I've been a little more creative with the thread subject title this time so that doesn't happen.

Now why is any of this significant to a physics newsgroup?

Well if the approach I found IS viable then in materials science it could allow researchers to determine the properties of mixing different things together by computer more easily than previously realized and even tell them things like if they'd have a glass, or a crystal, and if a crystal what its minimal configuration would be.

That is, the result could be HUGE for any nation that used it instead of whatever the hell is the supposed best in the world to advance its own materials science program.

But it could be junk.

History can turn on JUST this result, while I have lots of these type results so I can just chatter about them from time to time.

I am looking at times out for news of research results in materials science to see evidence of the use of the simplified mathematical approach.

If correct, a materials scientist could build materials with special properties that others could not ever build, as the computer could easily cycle through endless "experiments" that could check countless combinations faster than any other approach possibly could.

If someone is using this thing, they will quite simply, take over the field of materials science.

I'm just kind of waiting for that to happen and collecting responses with posts like this one in the interim: kind of a hobby.

I lost that paper (I tend to lose things as part of an on-going experiment to see what historical impact it might have) but in recollecting my ideas recently I posted what I THINK the solution was on my math blog, where now that blog post comes up for me with a Google search in the top 10. But the Google search (has to be Google) is:

spherical packing problem

I posted about this before and the newsgroup subject I made took over and dropped the blog page out of the top 10 for a while so I've been a little more creative with the thread subject title this time so that doesn't happen.

Now why is any of this significant to a physics newsgroup?

Well if the approach I found IS viable then in materials science it could allow researchers to determine the properties of mixing different things together by computer more easily than previously realized and even tell them things like if they'd have a glass, or a crystal, and if a crystal what its minimal configuration would be.

That is, the result could be HUGE for any nation that used it instead of whatever the hell is the supposed best in the world to advance its own materials science program.

But it could be junk.

History can turn on JUST this result, while I have lots of these type results so I can just chatter about them from time to time.

I am looking at times out for news of research results in materials science to see evidence of the use of the simplified mathematical approach.

If correct, a materials scientist could build materials with special properties that others could not ever build, as the computer could easily cycle through endless "experiments" that could check countless combinations faster than any other approach possibly could.

If someone is using this thing, they will quite simply, take over the field of materials science.

I'm just kind of waiting for that to happen and collecting responses with posts like this one in the interim: kind of a hobby.

## JSH: My optimal path idea, solution to TSP?

A while back I came up with an approach to the Traveling Salesman Problem which was resoundingly panned on this newsgroup, from one poster who proudly and repeatedly claimed it just did not work, to others who challenged the assertion that I'd even really given an approach!

I like that "wisdom of crowds" thing to test assertions versus endless arguing with one or two people, so I'm noting now that the algorithm is probably available to you anywhere in the world by a Google search (has to be Google):

optimal path algorithm

I just did that search and a page on my math blog comes up #2 behind some guy named Dijkstra, whoever he is. I guess he did something with a least times path, while I'm doing something that is supposed to solve the TSP.

Now I've noted my high search rankings before in newsgroups and run into some predictable rationalizations:

Well, one or two or even a dozen of you might proudly and loudly proclaim the idea is crap, but for Google search results to be so driven presumably a LOT more people around the world disagree, except then, why isn't this idea part of the status quo?

Turns out that new ideas routinely face an uphill climb so the Google search results I hypothesize could be a leading indicator.

Even the idea for sliced bread took years to catch on. See: http://en.wikipedia.org/wiki/Sliced_bread

The idea for escalators was from the late 1800's but didn't get widely implemented until the mid-twentieth century.

Lasers were to a large extent an early idea of Einstein's.

So why post here now?

Group effects interest me.

The posters who were loud and proud in decrying the idea join a long line of people who are hostile to new ideas and do so I think for status reasons. It is of interest to see their reaction to this post.

Presumably it will be defensive.

They are fighting for what they believe is their status in the hierarchy of this group. Their responses also reveal what role they

I am challenging their authority and intimating that they lead "their group" down the wrong path, so they are "bad people" who are also inferior as to their real knowledge and dangerous to people who trust them. That assertion should lead to a response.

I like that "wisdom of crowds" thing to test assertions versus endless arguing with one or two people, so I'm noting now that the algorithm is probably available to you anywhere in the world by a Google search (has to be Google):

optimal path algorithm

I just did that search and a page on my math blog comes up #2 behind some guy named Dijkstra, whoever he is. I guess he did something with a least times path, while I'm doing something that is supposed to solve the TSP.

Now I've noted my high search rankings before in newsgroups and run into some predictable rationalizations:

- Posters claim only I'm getting the particular search, and say Google has stuff on my computer to tell them what I want. Well that one's easily proven or disproven by YOU when you do the search yourself!!!
- Posters claim I post a lot of links all over the place to drive up search results for myself. Well, that one is harder to refute though I actually rarely if ever post links to MY research, partly because I've driven up search results for years and puzzle over why myself.

Well, one or two or even a dozen of you might proudly and loudly proclaim the idea is crap, but for Google search results to be so driven presumably a LOT more people around the world disagree, except then, why isn't this idea part of the status quo?

Turns out that new ideas routinely face an uphill climb so the Google search results I hypothesize could be a leading indicator.

Even the idea for sliced bread took years to catch on. See: http://en.wikipedia.org/wiki/Sliced_bread

The idea for escalators was from the late 1800's but didn't get widely implemented until the mid-twentieth century.

Lasers were to a large extent an early idea of Einstein's.

So why post here now?

Group effects interest me.

The posters who were loud and proud in decrying the idea join a long line of people who are hostile to new ideas and do so I think for status reasons. It is of interest to see their reaction to this post.

Presumably it will be defensive.

They are fighting for what they believe is their status in the hierarchy of this group. Their responses also reveal what role they

**think**they play in that hierarchy. It is a dominance test.I am challenging their authority and intimating that they lead "their group" down the wrong path, so they are "bad people" who are also inferior as to their real knowledge and dangerous to people who trust them. That assertion should lead to a response.

### Monday, November 23, 2009

## JSH: Generalizing quadratic residues solution idea

Earlier today I was pondering this weirdly simple idea for solving for quadratic residues I have and realized I should look at the generalization, so I wrote down: f_1 = ak mod p, and f_2 = bk mod p, and quickly realized I had:

k = (a+b)^{-1}(f_1 + f_2) mod p

with T = f_1*f_2 = abk^2 = abq mod p

to solve for k, when k^2 = q mod p.

But it's so weirdly simple and worse ab and a+b are unique!

So to the math when you have k = (a+b)^{-1}(f_1 + f_2) mod p, where f_1*f_2 = abq mod p, then you are TELLING it what 'a' and 'b' are, as that's forced. So the math knows what you're playing with as it's unique!!!

But then again, with p an odd prime, you can have a BUNCH of potential 'a' and 'b' that will work, so conceivably what will happen is that they'll keep kicking out the same k, unless a given choice is blocked for a particular T.

So how can a choice be "blocked" for T? Well, like if ak < p and bk < p, but T does not have k as a factor, then that T is blocked.

But if there are no blockings then this method should give k.

That is so weird. But even weirder if it works well and was missed!!!

One reason it could be missed is that math people learn that congruences are USELESS with factoring as, given say,

product = mn mod p

where m and n are residues modulo p, there are p-1 factorizations: that is, there is a factorization for EVERY nonzero residue!

Math people have it banged into their heads that for that reason residues and factoring don't mix.

But I've found that looking at just the product is wrong!!! You can also get a SUM and that SUM AND PRODUCT ARE UNIQUE.

Such a simple difference and so much history can change. Maybe if Gauss had noticed this thing we'd have never had public key encryption, eh?

IMAGINE how the history of our world would have changed! History turned on a simple miss.

k = (a+b)^{-1}(f_1 + f_2) mod p

with T = f_1*f_2 = abk^2 = abq mod p

to solve for k, when k^2 = q mod p.

But it's so weirdly simple and worse ab and a+b are unique!

So to the math when you have k = (a+b)^{-1}(f_1 + f_2) mod p, where f_1*f_2 = abq mod p, then you are TELLING it what 'a' and 'b' are, as that's forced. So the math knows what you're playing with as it's unique!!!

But then again, with p an odd prime, you can have a BUNCH of potential 'a' and 'b' that will work, so conceivably what will happen is that they'll keep kicking out the same k, unless a given choice is blocked for a particular T.

So how can a choice be "blocked" for T? Well, like if ak < p and bk < p, but T does not have k as a factor, then that T is blocked.

But if there are no blockings then this method should give k.

That is so weird. But even weirder if it works well and was missed!!!

One reason it could be missed is that math people learn that congruences are USELESS with factoring as, given say,

product = mn mod p

where m and n are residues modulo p, there are p-1 factorizations: that is, there is a factorization for EVERY nonzero residue!

Math people have it banged into their heads that for that reason residues and factoring don't mix.

But I've found that looking at just the product is wrong!!! You can also get a SUM and that SUM AND PRODUCT ARE UNIQUE.

Such a simple difference and so much history can change. Maybe if Gauss had noticed this thing we'd have never had public key encryption, eh?

IMAGINE how the history of our world would have changed! History turned on a simple miss.

## Unique factor key and quadratic residues

Quadratic residues and factoring are intimately related using some basic congruences. And there is a uniqueness property with residues which appears to have been missed by practitioners in various math disciplines that consider factoring.

This unique key is critical in determining quadratic residues by using factoring.

With k^2 = q mod p, where p is an odd prime, consider that with:

f_1 = ak mod p, and f_2 = bk mod p

where 'a' and 'b' are natural numbers, with T = f_1*f_2 = abk^2 mod p, you can simply add f_1 and f_2, to get:

f_1 + f_2 = (a+b)k mod p, and solve for k with:

k = (a+b)^{-1}(f_1 + f_2) mod p

assuming a+b is coprime to p. So you can find k, by factoring T, where since k^2 = q mod p, and T = abk^2 mod p, you have:

T = abq mod p.

But where is this unique key?

Well one reason some have wrongly assumed that congruences are useless with factoring is the product ab can have as its factors ANY non-zero residue of p; however, there is a sum involved here, which is a+b.

Now ask, with some other residues 'c' and 'd', can ab = cd mod p, and a+b = c+d mod p? That answer is, no.

Proof:

ab = cd mod p, and a+b = c+d mod p, so a = c+d-b mod p, and substituting:

b(c+d-b) = cd mod p, so b^2 = bc + bd - cd mod p.

But notice, b^2 - bc = bd - cd, gives b(b-c) = d(b-c) mod p, gives b = d mod p. (Choice comes from b-c, as if b=c, then a divide by zero error.)

So 'a' and 'b' are equal to 'c' and 'd'.

Then you have a UNIQUE KEY for 'a' and 'b'.

So, k = (a+b)^{-1}(f_1 + f_2) mod p, uniquely identifies f_1 = ak mod p, and f_2 = bk mod p, where T = abq mod p, when q is a quadratic residue modulo p.

Given that T = abq mod p is selected for particular factors by a+b and ab, one may wonder why this approach would ever fail to find k, but that is easily answered.

For instance if ak<p and bk < p, but abk^2 > p and q is small compared

to it, then T = abq mod p, may be too small at lower values for T, and only work for higher ones.

Also if ak < p and bk < p, then f_1 and f_2 would have k as a factor, so only when T had k as a factor could you have a solution.

So you'd have solutions with T = abk^2 + pk. So there are rules that will block solutions with certain T's.

Then remarkably there is a simple way to solve for quadratic residues using factoring, where a unique key is part of that solution.

Given a quadratic residue q, modulo p, you can find k, such that k^2 = q mod p, by finding T = abq mod p, where 'a' and 'b' are natural numbers you choose, from:

k = (a+b)^{-1}(f_1 + f_2) mod p

where f_1*f_2 = T.

Here that solution is uniquely associated with a particular factorization of T—despite congruences—by a+b and ab.

This unique key is critical in determining quadratic residues by using factoring.

With k^2 = q mod p, where p is an odd prime, consider that with:

f_1 = ak mod p, and f_2 = bk mod p

where 'a' and 'b' are natural numbers, with T = f_1*f_2 = abk^2 mod p, you can simply add f_1 and f_2, to get:

f_1 + f_2 = (a+b)k mod p, and solve for k with:

k = (a+b)^{-1}(f_1 + f_2) mod p

assuming a+b is coprime to p. So you can find k, by factoring T, where since k^2 = q mod p, and T = abk^2 mod p, you have:

T = abq mod p.

But where is this unique key?

Well one reason some have wrongly assumed that congruences are useless with factoring is the product ab can have as its factors ANY non-zero residue of p; however, there is a sum involved here, which is a+b.

Now ask, with some other residues 'c' and 'd', can ab = cd mod p, and a+b = c+d mod p? That answer is, no.

Proof:

ab = cd mod p, and a+b = c+d mod p, so a = c+d-b mod p, and substituting:

b(c+d-b) = cd mod p, so b^2 = bc + bd - cd mod p.

But notice, b^2 - bc = bd - cd, gives b(b-c) = d(b-c) mod p, gives b = d mod p. (Choice comes from b-c, as if b=c, then a divide by zero error.)

So 'a' and 'b' are equal to 'c' and 'd'.

Then you have a UNIQUE KEY for 'a' and 'b'.

So, k = (a+b)^{-1}(f_1 + f_2) mod p, uniquely identifies f_1 = ak mod p, and f_2 = bk mod p, where T = abq mod p, when q is a quadratic residue modulo p.

Given that T = abq mod p is selected for particular factors by a+b and ab, one may wonder why this approach would ever fail to find k, but that is easily answered.

For instance if ak<p and bk < p, but abk^2 > p and q is small compared

to it, then T = abq mod p, may be too small at lower values for T, and only work for higher ones.

Also if ak < p and bk < p, then f_1 and f_2 would have k as a factor, so only when T had k as a factor could you have a solution.

So you'd have solutions with T = abk^2 + pk. So there are rules that will block solutions with certain T's.

Then remarkably there is a simple way to solve for quadratic residues using factoring, where a unique key is part of that solution.

Given a quadratic residue q, modulo p, you can find k, such that k^2 = q mod p, by finding T = abq mod p, where 'a' and 'b' are natural numbers you choose, from:

k = (a+b)^{-1}(f_1 + f_2) mod p

where f_1*f_2 = T.

Here that solution is uniquely associated with a particular factorization of T—despite congruences—by a+b and ab.

### Saturday, November 21, 2009

## JSH: But is quadratic residue idea new?

Finding a result is one thing but often I wonder afterwards, why is this result new? Now I'm in pondering mode as I consider something so trivially simple that it seems weird if it's NOT previously known.

Because of the way congruences work, it's possible to do something interesting with quadratic residues, where I've simplified from the heavy machinery of the full idea, to make things understandable, and now it looks like I'm grabbing certain equations from a hat, but at least it should be easily understood.

Imagine you have k^2 = q mod p, where p is an odd prime.

And now take the equations:

f_1 = k mod p and f_2 = 2k mod p

And let T = f_1*f_2, so T = 2k^2 mod p, and I can see k^2 in there, so k^2 = 2^{-1}*T mod p, which means that

q = 2^{-1}*T mod p, and T = 2q mod p. Easy.

Now

z^2 - y^2 = T, then (z-y)(z+y) = T, so I have a factorization, and in fact I have simply enough:

z = (f_1 + f_2)/2, so I can substitute with f_1 and f_2 above, and get f_1 + f_2 = 3k mod p, so

z = 3k/2 mod p

and I have the implication that the factorization of T is connected with the value of k, and I have finally:

k = 3^{-1}(f_1 + f_2) mod p, and T = 2q mod p

when k^2 = q mod p.

So this simple approach has connected finding k, when k^2 = q mod p, with factoring T, where T = 2q mod p.

I'm guessing that the probability of it working is roughly 1/k.

But is this idea new? And what might be serious problems with it? Can it work well?

Because of the way congruences work, it's possible to do something interesting with quadratic residues, where I've simplified from the heavy machinery of the full idea, to make things understandable, and now it looks like I'm grabbing certain equations from a hat, but at least it should be easily understood.

Imagine you have k^2 = q mod p, where p is an odd prime.

And now take the equations:

f_1 = k mod p and f_2 = 2k mod p

And let T = f_1*f_2, so T = 2k^2 mod p, and I can see k^2 in there, so k^2 = 2^{-1}*T mod p, which means that

q = 2^{-1}*T mod p, and T = 2q mod p. Easy.

Now

z^2 - y^2 = T, then (z-y)(z+y) = T, so I have a factorization, and in fact I have simply enough:

z = (f_1 + f_2)/2, so I can substitute with f_1 and f_2 above, and get f_1 + f_2 = 3k mod p, so

z = 3k/2 mod p

and I have the implication that the factorization of T is connected with the value of k, and I have finally:

k = 3^{-1}(f_1 + f_2) mod p, and T = 2q mod p

when k^2 = q mod p.

So this simple approach has connected finding k, when k^2 = q mod p, with factoring T, where T = 2q mod p.

I'm guessing that the probability of it working is roughly 1/k.

But is this idea new? And what might be serious problems with it? Can it work well?

### Tuesday, November 17, 2009

## JSH: Understanding quadratic residues result

Using very simple mathematics I've found a fascinatingly simple connection between the solution for a quadratic residue, and a factorization, where it is unbelievable how simple the mathematics is, so explaining it is trivial.

Given a quadratic residue q, modulo N, where N is odd, I've found that given

k^2 = q mod N

you can solve for k, with

k = a*(1 + 2a^2)^{-1}*(f_1 + f_2) mod N

where T = f_1*f_2 = (1 + a^2)*q + jN,

where j is a nonzero integer, where 'a' is a Natural number. And selecting 'a' can be critical to getting an answer quickly.

The equations come from z^2 = y^2 + T, x^2 = y^2 mod p, z = x + ak mod p, k = 2ax mod p, where p is a prime factor of N.

What makes this approach of interest is simply that k^2 is given in one equation and k is given in another, where solutions exist for any choice of 'a'—I remind 'a' is a Natural number—but the SIZE of f_1*f_2 needed to find the solutions is what varies depending on what 'a' is chosen.

My reason for taking this approach path originally was an attempt at factoring T by using another number, for which p is the factor, but it's easier to factor T, and use the luck of k^2 and k being given to you, in order to solve for k through that factorization.

Notice that with z^2 = y^2 + T, and x^2 = y^2 mod p, there are an infinity of solutions for x, for a non-trivial factorization which gives you one z and y.

The remaining two congruences, z = x + ak mod p and k = 2ax mod p, narrow that infinity down, and in so doing, allow you to define 'a' and k in such a way as to get k, given a quadratic residue modulo N, by factoring T.

The jump to N from p is rather straightforward, though that is remarkable as well that is is possible to make that jump.

The math is easy. Solutions exist for any 'a' if q is a quadratic residue, but finding them can be harder or easier dependent on 'a', as T may be relatively large for one 'a', while much smaller for another.

Weirdly enough there are 4 ruling congruences which I also call constraints which decide when T will work:

Those 4 constraints are wild though because the mathematics is saying, hey, if you can fulfill these conditions, then ok, you get the right answer, if not, no.

So I came across this technique kind of by accident where there is just this oddity that the mathematics will solve for k^2 and k, so you can set k^2 = q mod N, and then go find k from factoring.

It can be wicked fast.

For example did a toy test bigger than I usually try, I picked p=91, and q=30, as 11^2 = 30 mod 91. With a=1, I have T = 60 mod 91.

My first try is with T=151, which is prime, and that didn't work (though trivial factorizations MAY work). If the first non-trivial factorization doesn't work then none will so I know not to try any others. Ok, next is:

T = 242, = 11(22), which works! k = 3^{-1}(11 + 22) mod 91 = 11 mod 91.

Here with ak less than p the math found an answer that fit the bill, though it can also use 91 - 11 that would be with a bigger T.

So these relationships are fundamental relationships of factoring and quadratic residues. They were always there.

It is so wild.

The answers are there it's just that where they show up depends on 'a' and the actual answer for k. It's kind of strange but you can watch the mathematics shifting things around to handle each issue presented by the 4 constraints when it gives the answer.

But yeah, it's mathematics, so it has infinite intelligence, so it's not like it actually DOES anything as these answers are logically built into numbers already.

Wild. Just wild.

Given a quadratic residue q, modulo N, where N is odd, I've found that given

k^2 = q mod N

you can solve for k, with

k = a*(1 + 2a^2)^{-1}*(f_1 + f_2) mod N

where T = f_1*f_2 = (1 + a^2)*q + jN,

where j is a nonzero integer, where 'a' is a Natural number. And selecting 'a' can be critical to getting an answer quickly.

The equations come from z^2 = y^2 + T, x^2 = y^2 mod p, z = x + ak mod p, k = 2ax mod p, where p is a prime factor of N.

What makes this approach of interest is simply that k^2 is given in one equation and k is given in another, where solutions exist for any choice of 'a'—I remind 'a' is a Natural number—but the SIZE of f_1*f_2 needed to find the solutions is what varies depending on what 'a' is chosen.

My reason for taking this approach path originally was an attempt at factoring T by using another number, for which p is the factor, but it's easier to factor T, and use the luck of k^2 and k being given to you, in order to solve for k through that factorization.

Notice that with z^2 = y^2 + T, and x^2 = y^2 mod p, there are an infinity of solutions for x, for a non-trivial factorization which gives you one z and y.

The remaining two congruences, z = x + ak mod p and k = 2ax mod p, narrow that infinity down, and in so doing, allow you to define 'a' and k in such a way as to get k, given a quadratic residue modulo N, by factoring T.

The jump to N from p is rather straightforward, though that is remarkable as well that is is possible to make that jump.

The math is easy. Solutions exist for any 'a' if q is a quadratic residue, but finding them can be harder or easier dependent on 'a', as T may be relatively large for one 'a', while much smaller for another.

Weirdly enough there are 4 ruling congruences which I also call constraints which decide when T will work:

- f_1 = ak mod p
- f_2 = a^{-1}*(1 + a^2)*k mod p
- z = (2a)^{-1}*(1 + 2a^2)*k mod p

and - y = (2a)^{-1}*k mod p

Those 4 constraints are wild though because the mathematics is saying, hey, if you can fulfill these conditions, then ok, you get the right answer, if not, no.

So I came across this technique kind of by accident where there is just this oddity that the mathematics will solve for k^2 and k, so you can set k^2 = q mod N, and then go find k from factoring.

It can be wicked fast.

For example did a toy test bigger than I usually try, I picked p=91, and q=30, as 11^2 = 30 mod 91. With a=1, I have T = 60 mod 91.

My first try is with T=151, which is prime, and that didn't work (though trivial factorizations MAY work). If the first non-trivial factorization doesn't work then none will so I know not to try any others. Ok, next is:

T = 242, = 11(22), which works! k = 3^{-1}(11 + 22) mod 91 = 11 mod 91.

Here with ak less than p the math found an answer that fit the bill, though it can also use 91 - 11 that would be with a bigger T.

So these relationships are fundamental relationships of factoring and quadratic residues. They were always there.

It is so wild.

The answers are there it's just that where they show up depends on 'a' and the actual answer for k. It's kind of strange but you can watch the mathematics shifting things around to handle each issue presented by the 4 constraints when it gives the answer.

But yeah, it's mathematics, so it has infinite intelligence, so it's not like it actually DOES anything as these answers are logically built into numbers already.

Wild. Just wild.

### Sunday, November 15, 2009

## JSH: Some history is the future

One of the weirder things that has emerged from my mathematical research is the possibility of continual transmission of information from the future to the past in order to CREATE the future, where key is what I call the optimal path algorithm.

Used against the Traveling Salesman Problem it gives you a traveler going backwards in time to meet himself, where the algorithm requires continual communication between the two travelers in order to get the optimal path.

If that is a routine part of nature then light takes the optimal path in that way, and it also gives an arrow to time—we think we're traveling forward in the future as we're the collapsed path, when actually we're traveling both forwards AND backwards in time.

The collapse to an optimal path gives us the illusion of only going forwards in time.

The arrow is the collapsed optimal path which appears to only go forward in time.

If so, then some of our "history" can be information transmitted to the past in order to create our future (and our present).

And that includes legends and mythologies, so yes, Revelations, for instance, seems to have future weapons in it. With this theory that's because it actually hasn't happened yet, but was information sent to the past about what will happen in order to MAKE it happen.

Parts of Genesis appear also to actually be stories about the future, not the past.

So the future is an active participant in creating itself.

Which means there is no way to know what knowledge is future knowledge transmitted to the past, unless that future wishes you to know, and then of course you CAN know. And there seem to be no limits. (So yes, conceivably you can get winning lottery numbers from the future, but ONLY if the future needs you to have that information to create itself.)

And if you can understand all of that you gotta be a singular type of human being as it is very confusing. But one thing is clear, if I'm talking about it, and I'm right, then the only reason I know to even tell it is because the future needs me to know and needs me to tell it for that future to exist.

It seems to me that such mind-numbing information would have a predictably large impact on a human population, which is why people haven't maybe really known it before; therefore, it seems likely some epic event is about to occur, which is allowing the information to be known.

For various reasons the idea is floating around that the "end of the world" is in 2012. My memory is that Sir Isaac Newton actually calculated the correct year and he got 2010, but I've seen no mention of that in the record, so I'm not sure why I have that number.

Used against the Traveling Salesman Problem it gives you a traveler going backwards in time to meet himself, where the algorithm requires continual communication between the two travelers in order to get the optimal path.

If that is a routine part of nature then light takes the optimal path in that way, and it also gives an arrow to time—we think we're traveling forward in the future as we're the collapsed path, when actually we're traveling both forwards AND backwards in time.

The collapse to an optimal path gives us the illusion of only going forwards in time.

The arrow is the collapsed optimal path which appears to only go forward in time.

If so, then some of our "history" can be information transmitted to the past in order to create our future (and our present).

And that includes legends and mythologies, so yes, Revelations, for instance, seems to have future weapons in it. With this theory that's because it actually hasn't happened yet, but was information sent to the past about what will happen in order to MAKE it happen.

Parts of Genesis appear also to actually be stories about the future, not the past.

So the future is an active participant in creating itself.

Which means there is no way to know what knowledge is future knowledge transmitted to the past, unless that future wishes you to know, and then of course you CAN know. And there seem to be no limits. (So yes, conceivably you can get winning lottery numbers from the future, but ONLY if the future needs you to have that information to create itself.)

And if you can understand all of that you gotta be a singular type of human being as it is very confusing. But one thing is clear, if I'm talking about it, and I'm right, then the only reason I know to even tell it is because the future needs me to know and needs me to tell it for that future to exist.

It seems to me that such mind-numbing information would have a predictably large impact on a human population, which is why people haven't maybe really known it before; therefore, it seems likely some epic event is about to occur, which is allowing the information to be known.

For various reasons the idea is floating around that the "end of the world" is in 2012. My memory is that Sir Isaac Newton actually calculated the correct year and he got 2010, but I've seen no mention of that in the record, so I'm not sure why I have that number.

### Friday, November 13, 2009

## Factoring's back door

Oddly enough a simple approach to factoring comes from not attacking the problem directly, but by instead taking a back door approach, where you start with two important relations:

z^2 = y^2 + T

and

x^2 = y^2 mod p

where T is the target composite to be factored and p is an odd prime to be determined, where notice that for a non-trivial factorization, there are an infinity of values for x, so at first blush this approach may seem naive, until you start whittling that infinity down.

The most clever approach I've found is to use k=2ax mod p, as the two new variables allow me to find that:

x^2 = y^2 + T - (1 + a^2)k^2 mod p

and now I have a way to narrow that infinity down!!! As let T - (1+a^2)k^2 = 0 mod p, and I have a smaller set of solutions, and even can have cases where no solutions exist, as:

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

so if (1+a^2)^{-1}(T) is not a quadratic residue modulo p, there is no integer k.

Remarkably enough I can now actually solve for z and y mod p:

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

and

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

So when does all that work? Well, remember that there are an INFINITY of solutions for x at first, but I narrowed that down with:

T - (1+a^2)k^2 = 0 mod

So if THAT relation holds then presumably you should have solutions, EXCEPT notice that with the solutions for z and y, the variable k is in both, and it can occur that k is a factor in both—which is a block unless T has k as a factor!!! The situation is actually even more complicated as with z and y, you can get solutions for factors f_1 and f_2 where f_1*f_2 = T:

f_1 = ak mod p

and

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

and again k provides a possible block, as for instance with a=1, if k is less than p, and z is less than p, then those relations would require that f_1 and z have the same factor, which would force it into f_2, and into T, as:

z = (f_1 + f_2)/2

And similar problems happen with y, as y = (f_1 - f_2)/2, and I call the first situation the z constraint, while the second is the y constraint.

But if none of the 4 constraining equations, the z, y, f_1 and f_2 equations give a conflict, and T - (1+a^2)k^2 = 0 mod p, then this approach MUST give solutions.

But then, how do you use it?

Here's a quick example of a factorization using all of the machinery above:

Let T=341. Then p=17 is the greatest prime less than sqrt(341) and starting with a=1, gives

k^2 = (2)^{-1}(341) mod 17 = 9(1) mod 17 = 9 mod 17.

so it does exist and k=3 or k=14 are possible, but with the first f_1 = ak = 3 is not a factor and 14 is not either. So next check f_2, f_2 = a^{-1}(1 + a^2)k = 2(3) = 6 which is not a factor but 17-6=11, and 11 is a factor. Success!

And you have a non-trivial factorization, as 341 = 11(31).

Notice that z=(11+31)/2 = 21, so no conflict, y = (31-11)/2 = 10, so no conflict.

And the equations escape conflict with the f's by going to the negative residues, so checking p - f_1 and p - f_2 is a necessity.

So there HAD to be a solution in this case. So once k existed, without conflict from any of the 4 constraints, a factorization had to be found.

The method represents a back door into factoring which allows you to leverage an infinity to factor a composite. Your factoring engine is infinity itself. And infinity is a powerful tool!!!

You can scan through p's to find a large one, though notice you can't just pick too huge a p, as then you're likely to get a conflict with one of the 4 constraints, and you can choose the variable 'a' to try and adjust around the constraints as well, as notice if a=1, like with my example and k is even, then with:

y = (2a)^{-1}k mod p, you have y = k/2 mod p, so if k is close to p, so k/2 < p, then you conflict with f_1 = ak = k, with a=1

So when k is even you already know that a=1 will probably not work well, but then you can use a=2, or a=3, or a=6, or some other.

So what is the Big O time complexity of this approach?

It cannot be calculated, as if none of the 4 constraints conflict, and k exists it is 100% success rate, but if one of the constraints blocks it just doesn't work. AND you have to solve for k, so finding a quadratic residue is important!

But some of you may know this approach leads to a method for solving quadratic residues as well, simply by reversing it, as importantly you have:

k^2 = (1+a^2)^{-1}(T) mod p and z = (2a)^{-1}(1 + 2a^2)k mod p

but z = (f_1 + f_2)/2, so you have

f_1 + f_2 = (a)^{-1}(1 + 2a^2)k mod p

So if you have a quadratic residue q, you can just let T = (1+a^2)q mod p, and factor T, to get f_1 and f_2, which gives you k, from:

k = a(1 + 2a^2)^{-1}(f_1 + f_2) mod p

So you can use this approach in the opposite direction to solve for quadratic residues! And again, if none of the 4 constraints block it MUST work. It must work. Perfection.

How do you know? Because given an integer y, x^2 = y^2 mod p has an INFINITY of solutions. I narrowed that infinity down with the critical relation:

T - (1+a^2)k^2 = 0 mod p

Trying to factor T, you have to find a p for which k exists, but going the other way to find a quadratic residues you BUILD T, so you know k exists if q is a quadratic residue modulo p, so if none of the 4 constraints are in the way, there is no way for a subset of that infinity not to work. Infinity has no choice, it has to solve the quadratic residue for you then.

So you have mathematically knowledge of a perfect method for factoring and for finding quadratic residues which reveals itself by going at factoring indirectly.

It is a perfect method.

z^2 = y^2 + T

and

x^2 = y^2 mod p

where T is the target composite to be factored and p is an odd prime to be determined, where notice that for a non-trivial factorization, there are an infinity of values for x, so at first blush this approach may seem naive, until you start whittling that infinity down.

The most clever approach I've found is to use k=2ax mod p, as the two new variables allow me to find that:

x^2 = y^2 + T - (1 + a^2)k^2 mod p

and now I have a way to narrow that infinity down!!! As let T - (1+a^2)k^2 = 0 mod p, and I have a smaller set of solutions, and even can have cases where no solutions exist, as:

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

so if (1+a^2)^{-1}(T) is not a quadratic residue modulo p, there is no integer k.

Remarkably enough I can now actually solve for z and y mod p:

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

and

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

So when does all that work? Well, remember that there are an INFINITY of solutions for x at first, but I narrowed that down with:

T - (1+a^2)k^2 = 0 mod

So if THAT relation holds then presumably you should have solutions, EXCEPT notice that with the solutions for z and y, the variable k is in both, and it can occur that k is a factor in both—which is a block unless T has k as a factor!!! The situation is actually even more complicated as with z and y, you can get solutions for factors f_1 and f_2 where f_1*f_2 = T:

f_1 = ak mod p

and

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

and again k provides a possible block, as for instance with a=1, if k is less than p, and z is less than p, then those relations would require that f_1 and z have the same factor, which would force it into f_2, and into T, as:

z = (f_1 + f_2)/2

And similar problems happen with y, as y = (f_1 - f_2)/2, and I call the first situation the z constraint, while the second is the y constraint.

But if none of the 4 constraining equations, the z, y, f_1 and f_2 equations give a conflict, and T - (1+a^2)k^2 = 0 mod p, then this approach MUST give solutions.

But then, how do you use it?

Here's a quick example of a factorization using all of the machinery above:

Let T=341. Then p=17 is the greatest prime less than sqrt(341) and starting with a=1, gives

k^2 = (2)^{-1}(341) mod 17 = 9(1) mod 17 = 9 mod 17.

so it does exist and k=3 or k=14 are possible, but with the first f_1 = ak = 3 is not a factor and 14 is not either. So next check f_2, f_2 = a^{-1}(1 + a^2)k = 2(3) = 6 which is not a factor but 17-6=11, and 11 is a factor. Success!

And you have a non-trivial factorization, as 341 = 11(31).

Notice that z=(11+31)/2 = 21, so no conflict, y = (31-11)/2 = 10, so no conflict.

And the equations escape conflict with the f's by going to the negative residues, so checking p - f_1 and p - f_2 is a necessity.

So there HAD to be a solution in this case. So once k existed, without conflict from any of the 4 constraints, a factorization had to be found.

The method represents a back door into factoring which allows you to leverage an infinity to factor a composite. Your factoring engine is infinity itself. And infinity is a powerful tool!!!

You can scan through p's to find a large one, though notice you can't just pick too huge a p, as then you're likely to get a conflict with one of the 4 constraints, and you can choose the variable 'a' to try and adjust around the constraints as well, as notice if a=1, like with my example and k is even, then with:

y = (2a)^{-1}k mod p, you have y = k/2 mod p, so if k is close to p, so k/2 < p, then you conflict with f_1 = ak = k, with a=1

So when k is even you already know that a=1 will probably not work well, but then you can use a=2, or a=3, or a=6, or some other.

So what is the Big O time complexity of this approach?

It cannot be calculated, as if none of the 4 constraints conflict, and k exists it is 100% success rate, but if one of the constraints blocks it just doesn't work. AND you have to solve for k, so finding a quadratic residue is important!

But some of you may know this approach leads to a method for solving quadratic residues as well, simply by reversing it, as importantly you have:

k^2 = (1+a^2)^{-1}(T) mod p and z = (2a)^{-1}(1 + 2a^2)k mod p

but z = (f_1 + f_2)/2, so you have

f_1 + f_2 = (a)^{-1}(1 + 2a^2)k mod p

So if you have a quadratic residue q, you can just let T = (1+a^2)q mod p, and factor T, to get f_1 and f_2, which gives you k, from:

k = a(1 + 2a^2)^{-1}(f_1 + f_2) mod p

So you can use this approach in the opposite direction to solve for quadratic residues! And again, if none of the 4 constraints block it MUST work. It must work. Perfection.

How do you know? Because given an integer y, x^2 = y^2 mod p has an INFINITY of solutions. I narrowed that infinity down with the critical relation:

T - (1+a^2)k^2 = 0 mod p

Trying to factor T, you have to find a p for which k exists, but going the other way to find a quadratic residues you BUILD T, so you know k exists if q is a quadratic residue modulo p, so if none of the 4 constraints are in the way, there is no way for a subset of that infinity not to work. Infinity has no choice, it has to solve the quadratic residue for you then.

So you have mathematically knowledge of a perfect method for factoring and for finding quadratic residues which reveals itself by going at factoring indirectly.

It is a perfect method.

### Sunday, November 08, 2009

## JSH: But it just amazes me

I figured out years ago when I FINALLY had major mathematical finds that there were these people in the mathematical field who didn't give a damn. They didn't care about the truth, so what were they doing there?

Working it out over the years I've determined that for some people the dream of royalty did not die with the transition to democratic societies around the world, so they've simply worked to set up shop where they can chase that dream.

And academia is a place that lets them play at their dreams of being royalty.

In a class society the king does not have to be the best at anything. He is simply the king.

So for these people in setting themselves up as wannabe royalty, merit does not matter.

They don't care if you're right or wrong if you're someone they've de-classed in their minds!

And you can see how far they can go with what is increasingly looking like a solution to the factoring problem.

How hard to check?

For major researchers, oh, minutes, maybe a few hours to program it and watch it go, and then there should be calls to colleagues and excited discussion, and oh yeah, notifying of security experts andintelligence services around the world.

But instead there is a dragging of the feet by people who don't want to let go.

I mentioned on sci.physics that the world may decide to 0 fund academia and I'm increasingly thinking that will happen as the modern academic world seems to attract medieval thinking, and in the medieval world it was not about truth or merit, but about class.

There has to be some way to break that out of academia so that people within academic walls do not feel free to dismiss results that they don't like.

My suggestion is 0 funding. If the money is taken away then only the best people will still remain as like me, they will find a way.

The BEST people do not need handouts or what I call white collar welfare.

So 0 funding academia is increasingly looking like the way to go, as this latest incredible example shows: you people are fighting powerful mathematics as if you can win!

But why would you even want to win?

Working it out over the years I've determined that for some people the dream of royalty did not die with the transition to democratic societies around the world, so they've simply worked to set up shop where they can chase that dream.

And academia is a place that lets them play at their dreams of being royalty.

In a class society the king does not have to be the best at anything. He is simply the king.

So for these people in setting themselves up as wannabe royalty, merit does not matter.

They don't care if you're right or wrong if you're someone they've de-classed in their minds!

And you can see how far they can go with what is increasingly looking like a solution to the factoring problem.

How hard to check?

For major researchers, oh, minutes, maybe a few hours to program it and watch it go, and then there should be calls to colleagues and excited discussion, and oh yeah, notifying of security experts andintelligence services around the world.

But instead there is a dragging of the feet by people who don't want to let go.

I mentioned on sci.physics that the world may decide to 0 fund academia and I'm increasingly thinking that will happen as the modern academic world seems to attract medieval thinking, and in the medieval world it was not about truth or merit, but about class.

There has to be some way to break that out of academia so that people within academic walls do not feel free to dismiss results that they don't like.

My suggestion is 0 funding. If the money is taken away then only the best people will still remain as like me, they will find a way.

The BEST people do not need handouts or what I call white collar welfare.

So 0 funding academia is increasingly looking like the way to go, as this latest incredible example shows: you people are fighting powerful mathematics as if you can win!

But why would you even want to win?

## JSH: But how can the Internet go down?

Some of you may wonder how such a massive and seemingly powerful system as the Internet can go down, well how can a massively complex human being go down from some relatively simple virus?

But the flu is killing people around the world as it has done and the H1N1 variant is doing even more damage.

But how is that possible?

If you have to figure that out before you can believe that the Internet that you have grown to know and love could be toast, then there is no hope, but let me also add: the defensive systems do not appear to be operating.

So it's like if your immune system is compromised like by the HIV virus. The Internet security people are behaving like immune cells which refuse to function. And part of the problem may be the massive wave of negatives about my research like by Erik Max Francis on Crank.net, where people underestimate the power of negatives.

So the situation is like an immuno-compromised patient that may be hit by a wave of viruses in a situation where its own immune cells will do nothing.

What would happen to that hypothetical patient?

It would die.

But the flu is killing people around the world as it has done and the H1N1 variant is doing even more damage.

But how is that possible?

If you have to figure that out before you can believe that the Internet that you have grown to know and love could be toast, then there is no hope, but let me also add: the defensive systems do not appear to be operating.

So it's like if your immune system is compromised like by the HIV virus. The Internet security people are behaving like immune cells which refuse to function. And part of the problem may be the massive wave of negatives about my research like by Erik Max Francis on Crank.net, where people underestimate the power of negatives.

So the situation is like an immuno-compromised patient that may be hit by a wave of viruses in a situation where its own immune cells will do nothing.

What would happen to that hypothetical patient?

It would die.

## JSH: Internet defensive response theory

I have a theory that the growing Internet is in many ways like a living organism, and wishes to survive.

But if I DO have mathematical breakthroughs where exploits because the world refuses to acknowledge the research could end the Internet as we know it, then its survival is now in doubt, while I'm fighting for that survival, but some really stupid people are blocking.

If my Internet defensive response theory is correct then the Internet will somehow, someway move to remove the blockages.

But I have no clue how, and no clue in what way, but if the Internet is growing forward as an entity, it is not a human one.

So I'm distancing myself. If some of the solutions the Internet comes up with to survive are nasty, it's not me.

I'm just an observer and I didn't think this situation was possible anyway. I figured that if I had major mathematical breakthroughs that the math people would do the right thing.

They have not.

But if I DO have mathematical breakthroughs where exploits because the world refuses to acknowledge the research could end the Internet as we know it, then its survival is now in doubt, while I'm fighting for that survival, but some really stupid people are blocking.

If my Internet defensive response theory is correct then the Internet will somehow, someway move to remove the blockages.

But I have no clue how, and no clue in what way, but if the Internet is growing forward as an entity, it is not a human one.

So I'm distancing myself. If some of the solutions the Internet comes up with to survive are nasty, it's not me.

I'm just an observer and I didn't think this situation was possible anyway. I figured that if I had major mathematical breakthroughs that the math people would do the right thing.

They have not.

### Saturday, November 07, 2009

## JSH: World should be ok, from stupidity

One of my long-time fears was that a dramatically simple solution to the factoring problem might crash the world economy, but after looking over the statistics on hits to my math blog—which has had what may be the solution for a year—it seems the world is remarkably DUMB.

So I think it's safe!

Oh yeah, after I found what I thought might be a solution to the factoring problem I went ahead and proved that P=NP by solving the Traveling Salesman Problem. Did that MONTHS ago. It's on my blog too.

The world IS stupid.

So I no longer am concerned about markets crashing as the world's too dumb for that, so it should all work out, I think.

Wow though, it is so amazing! You people going on about your business as serious as ever I'm sure. Acting as if you're doing important things and have important ideas. Working at research which has probably been made completely irrelevant.

Your lost year.

But who knows? The world maybe vastly more stupid than I even think possible and in another year I'll be making posts like this one talking about another anniversary.

So you have another year to do things that are, worthless, and irrelevant—oh, except they still are important in a stupid world that can't see solutions because you people rip on people you hate as cranks or crackpots because you're EVIL.

Factoring problem it would seem was solved about a year ago. So you wasted a year of your lives since then, it seems because your society maligns some people as "crackpot" and refuses to accept mathematics you then don't like—even if it is correct.

In a word, you are: stupid.

So I think it's safe!

Oh yeah, after I found what I thought might be a solution to the factoring problem I went ahead and proved that P=NP by solving the Traveling Salesman Problem. Did that MONTHS ago. It's on my blog too.

The world IS stupid.

So I no longer am concerned about markets crashing as the world's too dumb for that, so it should all work out, I think.

Wow though, it is so amazing! You people going on about your business as serious as ever I'm sure. Acting as if you're doing important things and have important ideas. Working at research which has probably been made completely irrelevant.

Your lost year.

But who knows? The world maybe vastly more stupid than I even think possible and in another year I'll be making posts like this one talking about another anniversary.

So you have another year to do things that are, worthless, and irrelevant—oh, except they still are important in a stupid world that can't see solutions because you people rip on people you hate as cranks or crackpots because you're EVIL.

Factoring problem it would seem was solved about a year ago. So you wasted a year of your lives since then, it seems because your society maligns some people as "crackpot" and refuses to accept mathematics you then don't like—even if it is correct.

In a word, you are: stupid.

## JSH: So Crank.net really works?

I've talked about Google searches that pull in my research highly, but I'm more interested now in the web searches that pull in a hate page against me on Crank.net, which is a website of Erik Max Francis, whom I encountered on the sci.math newsgroup years ago.

It elicited howls of glee on that newsgroup when he put the page up against me, and years later I'm beginning to see that the posters of that group were probably right and this one person has an amazingly powerful negative impact on you.

Today I have various ideas which I think are important across a lot of areas and I'm here to tell you: no progress.

While I'd been optimistic about hit data from around the world, better analytics software showing me what users are actually doing on my math blog and how they got there is discouraging. And more importantly, oh yeah, looks like last year I DID maybe, um, solve the factoring problem.

Actually I have 3 probably good basic research results which when properly developed could allow VERY rapid of factoring of VERY large numbers in polynomial time, which is probably one of the biggest advances in number theory, oh, ever.

And I can't get anywhere with it!!!

If that turns out to be true that I DID get these finds over a year ago and people finally realize what's going on then I'm going to be presenting the case that Erik Max Francis may by himself be a lot of the reason for why the world did not pick up the research—or even notice it existed!!!

I call it a social transistor effect. Like with transistors you can control a LOT of current with a small one. Negatives like what Erik Max Francis put up, can have an awesome power in our connected world.

He may be able to completely shut people down even if they are right!!!

Way wacky, way fascinating. If that is true then the "herd" mentality of human beings can reach impressive limits which can only be seen in this case because of absolute mathematical proof.

There is wisdom in crowds then. Crowds can be super-stupid. MOST of the people following along in the wrong direction.

3 ways that can potentially solve one of the big problems in number theory—factoring.

All mostly ignored. We're at one year plus now.

And I am fascinated by my Google Group ratings as well!!! I have a 1 star rating with over 8000 ratings given.

Fascinating.

I picked the factoring problem because it has real world impact. Of course "Uncle Al" can request that I factor a large number.

I will not. I cannot. I have basic research which indicates a way can be found. I haven't implemented anything.

Again I liken taunts for me to factor a large number to asking Albert Einstein to build an atomic bomb—and explode as well.

AFTER ALL if relativity is correct, why couldn't Einstein build an atomic bomb to prove it? Right?

Of course his ideas and that of others DID get picked up which is why people finally DID develop atomic weaponry.

Here is a fascinating case where for the moment it appears the knowledge is just sitting out there.

Despite the potential energy wrapped up in its use.

And I'm blaming a social transistor effect.

It elicited howls of glee on that newsgroup when he put the page up against me, and years later I'm beginning to see that the posters of that group were probably right and this one person has an amazingly powerful negative impact on you.

Today I have various ideas which I think are important across a lot of areas and I'm here to tell you: no progress.

While I'd been optimistic about hit data from around the world, better analytics software showing me what users are actually doing on my math blog and how they got there is discouraging. And more importantly, oh yeah, looks like last year I DID maybe, um, solve the factoring problem.

Actually I have 3 probably good basic research results which when properly developed could allow VERY rapid of factoring of VERY large numbers in polynomial time, which is probably one of the biggest advances in number theory, oh, ever.

And I can't get anywhere with it!!!

If that turns out to be true that I DID get these finds over a year ago and people finally realize what's going on then I'm going to be presenting the case that Erik Max Francis may by himself be a lot of the reason for why the world did not pick up the research—or even notice it existed!!!

I call it a social transistor effect. Like with transistors you can control a LOT of current with a small one. Negatives like what Erik Max Francis put up, can have an awesome power in our connected world.

He may be able to completely shut people down even if they are right!!!

Way wacky, way fascinating. If that is true then the "herd" mentality of human beings can reach impressive limits which can only be seen in this case because of absolute mathematical proof.

There is wisdom in crowds then. Crowds can be super-stupid. MOST of the people following along in the wrong direction.

3 ways that can potentially solve one of the big problems in number theory—factoring.

All mostly ignored. We're at one year plus now.

And I am fascinated by my Google Group ratings as well!!! I have a 1 star rating with over 8000 ratings given.

Fascinating.

I picked the factoring problem because it has real world impact. Of course "Uncle Al" can request that I factor a large number.

I will not. I cannot. I have basic research which indicates a way can be found. I haven't implemented anything.

Again I liken taunts for me to factor a large number to asking Albert Einstein to build an atomic bomb—and explode as well.

AFTER ALL if relativity is correct, why couldn't Einstein build an atomic bomb to prove it? Right?

Of course his ideas and that of others DID get picked up which is why people finally DID develop atomic weaponry.

Here is a fascinating case where for the moment it appears the knowledge is just sitting out there.

Despite the potential energy wrapped up in its use.

And I'm blaming a social transistor effect.

## Resolving quadratic residues

There is a surprisingly simple fundamental relation between factoring and quadratic residues using simple congruences.

Given a quadratic residue q modulo N, where N is a non-unit odd natural number coprime to 3, where you wish to find k, where

k^2 = q mod N

you first find T = 2q mod N, so T = 2q + jN, and j is a non-zero integer.

You next check each factorization of T to find positive f_1 and f_2 such that f_1*f_2 = T and then you get k quite simply from

k = 3^{-1}(f_1 + f_2) mod N

where for roughly 50% of cases you will get a k that will work, as a primary requirement is:

T - k^2 - f_1^2 = 0 mod N

So T - f_1^2 must be a quadratic residue modulo N, and as T = 2q mod N, you have 2q - f_1^2 must be a quadratic residue modulo N. That means, for instance, that if the first trivial factorization of T does not work, then none will, as f_1 mod N will, of course, remain the same.

So let's try it. Let N = 35, as that's simple enough, and q=29 (I'll explain how I picked it later).

Then, T = 2(29) mod 35, which is T = 23 mod 35.

The first possible T is T = 93, and it does work with f_1 = 93, and f_2 = 1 (so yes, trivial factorizations can work), as I get:

k = 3^{-1}(93 + 1) mod 35 = 12(94) mod 35 = 1128 mod 35 = 8 mod 35.

And k = 8 mod 35 is a correct answer.

And now you can see how I picked that example as knowing that 35 has 5 and 7 as factors I picked the first k coprime to 35 that would not give a perfect square:

8^2 = 29 mod 35

My first example was with k=8, using q = 29 modulo 35, as that's the first case where the quadratic residue is not a perfect square (though this method WILL solve a perfect square as well I should add) and is coprime to 35.

So now let's move k up one and do, k=9. And 81 mod 35 = 11, so q = 11 mod 35, and T = 22 mod 35, so I can try T = 57.

The trivial factorization didn't work here—which means none will—so I'll go to the non-trivial one:

f_1 = 19, and f_2 = 3, so:

k = 3^{-1}(19 + 3) mod 35 = 12(22) mod 35 = 264 mod 35 = 19 mod 35.

19^2 = 11 mod 35

so it worked! (It's so weird though watching it. Even though I know the underlying mathematics it seems like magic.)

And that is a factoring example, as I know k=9 is a solution, so I have

19^2 = 9^2 mod 35

so (19-9)(19+9) = 0 mod 35, so (10)(28) = 0 mod 35,

and you pull 5 and 7 as factors with a gcd.

THAT is how you use a method for solving quadratic residues modulo N: you find one quadratic residue and then go looking for another.

The result follows from the mathematics regarding a new way to factor I discovered that I call surrogate factoring.

To do further research into it, do a web search on: surrogate factoring

The gist of it is leveraging TWO difference of squares versus just considering one:

z^2 - y^2 = 0 mod T and x^2 - y^2 = 0 mod N

So I connect the factorization of T with the factorization of N—beautifully simple idea—which allows me to pull information about one from factoring the other, which is just really neat. I'm popularizing the result as some people have smeared me as a math crackpot, wrongfully, so I have to push results that should just excite the mathematical community on their own, which is sad.

Just some nasty people calling me names, and even beautiful mathematics has to be pushed on math people. Weird world.

Given a quadratic residue q modulo N, where N is a non-unit odd natural number coprime to 3, where you wish to find k, where

k^2 = q mod N

you first find T = 2q mod N, so T = 2q + jN, and j is a non-zero integer.

You next check each factorization of T to find positive f_1 and f_2 such that f_1*f_2 = T and then you get k quite simply from

k = 3^{-1}(f_1 + f_2) mod N

where for roughly 50% of cases you will get a k that will work, as a primary requirement is:

T - k^2 - f_1^2 = 0 mod N

So T - f_1^2 must be a quadratic residue modulo N, and as T = 2q mod N, you have 2q - f_1^2 must be a quadratic residue modulo N. That means, for instance, that if the first trivial factorization of T does not work, then none will, as f_1 mod N will, of course, remain the same.

So let's try it. Let N = 35, as that's simple enough, and q=29 (I'll explain how I picked it later).

Then, T = 2(29) mod 35, which is T = 23 mod 35.

The first possible T is T = 93, and it does work with f_1 = 93, and f_2 = 1 (so yes, trivial factorizations can work), as I get:

k = 3^{-1}(93 + 1) mod 35 = 12(94) mod 35 = 1128 mod 35 = 8 mod 35.

And k = 8 mod 35 is a correct answer.

And now you can see how I picked that example as knowing that 35 has 5 and 7 as factors I picked the first k coprime to 35 that would not give a perfect square:

8^2 = 29 mod 35

My first example was with k=8, using q = 29 modulo 35, as that's the first case where the quadratic residue is not a perfect square (though this method WILL solve a perfect square as well I should add) and is coprime to 35.

So now let's move k up one and do, k=9. And 81 mod 35 = 11, so q = 11 mod 35, and T = 22 mod 35, so I can try T = 57.

The trivial factorization didn't work here—which means none will—so I'll go to the non-trivial one:

f_1 = 19, and f_2 = 3, so:

k = 3^{-1}(19 + 3) mod 35 = 12(22) mod 35 = 264 mod 35 = 19 mod 35.

19^2 = 11 mod 35

so it worked! (It's so weird though watching it. Even though I know the underlying mathematics it seems like magic.)

And that is a factoring example, as I know k=9 is a solution, so I have

19^2 = 9^2 mod 35

so (19-9)(19+9) = 0 mod 35, so (10)(28) = 0 mod 35,

and you pull 5 and 7 as factors with a gcd.

THAT is how you use a method for solving quadratic residues modulo N: you find one quadratic residue and then go looking for another.

The result follows from the mathematics regarding a new way to factor I discovered that I call surrogate factoring.

To do further research into it, do a web search on: surrogate factoring

The gist of it is leveraging TWO difference of squares versus just considering one:

z^2 - y^2 = 0 mod T and x^2 - y^2 = 0 mod N

So I connect the factorization of T with the factorization of N—beautifully simple idea—which allows me to pull information about one from factoring the other, which is just really neat. I'm popularizing the result as some people have smeared me as a math crackpot, wrongfully, so I have to push results that should just excite the mathematical community on their own, which is sad.

Just some nasty people calling me names, and even beautiful mathematics has to be pushed on math people. Weird world.

### Friday, November 06, 2009

## JSH: Considering my math blog

Ok, so I don't have to post to Usenet to get readers. I do have a math blog and have REALLY hoped I could just post from my blog and that would handle everything. And I do get readers. I get roughly 30 to 40 unique hits a day according to several analytics software products. That's about 1000 people per month or roughly 10000 per year. And from 40 to 60 countries per month, and over 100+ countries per year.

So I figured, hey, that should mean that if I'm right someone would notice. But time just keeps passing, so I keep trying to delve more deeply into the data to see what's happening.

One thing with a massive impact DOES seem to be one page which is a Crank.net webpage by Erik Max Francis who I have argued with on the sci.math newsgroup and searches on my name related to math pull up his page #1. Also there is the way visitors are using my site in many cases which is as a reference.

The two points may seem contradictory: a hate page insulting me is pulled up highly with my

Usenet attacks have been personal and effective in painting me personally as a crackpot, which is easily verified by web searches. Just search on: james harris

I just did that and the Erik Max Francis page blasting me as a crank comes up #18.

But if you wish to solve quadratic residues who do you think comes up #2? Not me exactly, MY PAGE comes up!

The math is pulling up highly and usage that I'm seeing on the site comes up in a pattern consistent with that assessment: people from all over the world are coming to my site as a math reference, presumably getting the info they need, and then going on their way!

What I do with Usenet posts like this one, or the other threads on this group or threads on other newsgroups, is look to see how I can shift the patterns by doing what you could call experiments. Then I adjust strategy across my websites.

Presumably if enough time passes—think YEARS—the dominance of my ideas will pull me along, crush negatives like remove the influence of Erik Max Francis or other negative posters, and end this bizarre situation.

But history shows it can actually take DECADES, so I'm brainstorming out faster solutions, like promoting various research to see what impact it has.

Here I focus on solving quadratic residues. Prior to this I focused on my prime counting function on other newsgroups. At other times I've talked about the core error I found in number theory, where a paper seems to have triggered the math community into destroying one of their journals.

And I have other mathematical finds that I can pull in at will as well.

So you are part of one experimental foray to see how the impact of the world's first way to generally solve quadratic residues modulo N fares in a natural environment. So far nothing new!

I'm seeing the same type responses as in the past. And getting roughly the same feel as before. Which is what also occurred on other newsgroups.

My math blog has not seen a shift in usage. Country patterns remain the same. Areas of hits have barely budged though there have been really tiny mini-spikes from occasions when some posters have put up links to my blogs in response to me.

Quite simply the math blog behaves as if you don't exist, except for Erik Max Francis, where I continue to see hits coming from his Crank.net website which has a continuing influence that puzzles me.

One theory is that it's easier to control people with negatives than positives, and easier to convince with negatives than most people realize. As, he hasn't even had to update his page! But as long as it sits there, it stays highly rated, and is clearly a massive driver of my own personal negatives.

My negative numbers are HUGE. Increasingly I'm one of the few people in the world whose name is popping up on webpages around the world month after month, year after year, where most of those people have no idea who this "James Harris" guy is, but if they look, oh, he's a crackpot, but hey, this math thing works, so…

Credit that Erik Max Francis with having a site that is massively powerful, but that also means that if anyone breaks through it should be easy to prove that his site is critical in many people's assessments of value. So the negatives he puts out are VERY effective.

I call it a social transistor effect: all by himself Erik Max Francis may be able to squash people just by putting up a page on his Crank.net website, and have that squashing effect last indefinitely no matter what they do, or even how much their research is used, around the world.

He is very effective.

So I figured, hey, that should mean that if I'm right someone would notice. But time just keeps passing, so I keep trying to delve more deeply into the data to see what's happening.

One thing with a massive impact DOES seem to be one page which is a Crank.net webpage by Erik Max Francis who I have argued with on the sci.math newsgroup and searches on my name related to math pull up his page #1. Also there is the way visitors are using my site in many cases which is as a reference.

The two points may seem contradictory: a hate page insulting me is pulled up highly with my

**name**while looking at user activity on my math blog indicates they are just using it as a reference, but the explanation is bizarre and simple.Usenet attacks have been personal and effective in painting me personally as a crackpot, which is easily verified by web searches. Just search on: james harris

I just did that and the Erik Max Francis page blasting me as a crank comes up #18.

But if you wish to solve quadratic residues who do you think comes up #2? Not me exactly, MY PAGE comes up!

The math is pulling up highly and usage that I'm seeing on the site comes up in a pattern consistent with that assessment: people from all over the world are coming to my site as a math reference, presumably getting the info they need, and then going on their way!

What I do with Usenet posts like this one, or the other threads on this group or threads on other newsgroups, is look to see how I can shift the patterns by doing what you could call experiments. Then I adjust strategy across my websites.

Presumably if enough time passes—think YEARS—the dominance of my ideas will pull me along, crush negatives like remove the influence of Erik Max Francis or other negative posters, and end this bizarre situation.

But history shows it can actually take DECADES, so I'm brainstorming out faster solutions, like promoting various research to see what impact it has.

Here I focus on solving quadratic residues. Prior to this I focused on my prime counting function on other newsgroups. At other times I've talked about the core error I found in number theory, where a paper seems to have triggered the math community into destroying one of their journals.

And I have other mathematical finds that I can pull in at will as well.

So you are part of one experimental foray to see how the impact of the world's first way to generally solve quadratic residues modulo N fares in a natural environment. So far nothing new!

I'm seeing the same type responses as in the past. And getting roughly the same feel as before. Which is what also occurred on other newsgroups.

My math blog has not seen a shift in usage. Country patterns remain the same. Areas of hits have barely budged though there have been really tiny mini-spikes from occasions when some posters have put up links to my blogs in response to me.

Quite simply the math blog behaves as if you don't exist, except for Erik Max Francis, where I continue to see hits coming from his Crank.net website which has a continuing influence that puzzles me.

One theory is that it's easier to control people with negatives than positives, and easier to convince with negatives than most people realize. As, he hasn't even had to update his page! But as long as it sits there, it stays highly rated, and is clearly a massive driver of my own personal negatives.

My negative numbers are HUGE. Increasingly I'm one of the few people in the world whose name is popping up on webpages around the world month after month, year after year, where most of those people have no idea who this "James Harris" guy is, but if they look, oh, he's a crackpot, but hey, this math thing works, so…

Credit that Erik Max Francis with having a site that is massively powerful, but that also means that if anyone breaks through it should be easy to prove that his site is critical in many people's assessments of value. So the negatives he puts out are VERY effective.

I call it a social transistor effect: all by himself Erik Max Francis may be able to squash people just by putting up a page on his Crank.net website, and have that squashing effect last indefinitely no matter what they do, or even how much their research is used, around the world.

He is very effective.

### Wednesday, November 04, 2009

## JSH: This factoring thing is scary

You people need to wake up. I'm telling you the underlying math is EASY. It's a technique to solve quadratic residues modulo N.

If you think it's not possible then use all of your mathematical training to explain the second example:

So now let's do, k=9. So q = 11 mod 35, and T = 22 mod 35, so I can try T = 57.

The trivial factorization didn't work here, so I'll just jump to, f_1 = 19, and f_2 = 3, so:

k = 3^{-1}(19 + 3) mod 35 = 12(22) mod 35 = 264 mod 35 = 19 mod 35.

19^2 = 11 mod 35

so it worked! (It's so weird though watching it. Even though I know the underlying mathematics it seems like magic.)

And that is a factoring example, as I know k=9 is a solution, so I have

19^2 = 9^2 mod 35

so (19-9)(19+9) = 0 mod 35, so (10)(28) = 0 mod 35, and you pull 5 and 7 as factors with a gcd.

THAT is how you use a method for solving quadratic residues modulo N: you find one quadratic residue and then go looking for another.

Factoring problem solved.

Happy one year birthday to the solution as it's a year old about now.

If you think it's not possible then use all of your mathematical training to explain the second example:

So now let's do, k=9. So q = 11 mod 35, and T = 22 mod 35, so I can try T = 57.

The trivial factorization didn't work here, so I'll just jump to, f_1 = 19, and f_2 = 3, so:

k = 3^{-1}(19 + 3) mod 35 = 12(22) mod 35 = 264 mod 35 = 19 mod 35.

19^2 = 11 mod 35

so it worked! (It's so weird though watching it. Even though I know the underlying mathematics it seems like magic.)

And that is a factoring example, as I know k=9 is a solution, so I have

19^2 = 9^2 mod 35

so (19-9)(19+9) = 0 mod 35, so (10)(28) = 0 mod 35, and you pull 5 and 7 as factors with a gcd.

THAT is how you use a method for solving quadratic residues modulo N: you find one quadratic residue and then go looking for another.

Factoring problem solved.

Happy one year birthday to the solution as it's a year old about now.

### Sunday, November 01, 2009

## JSH: And I ALWAYS beat them

It is mathematics so I can use very simple positions which are irrefutable to win the arguments almost instantly.

What happens after you can see in the recent threads.

More interesting after that is Usenet posters will SAY I didn't beat them and maintain I'm a crackpot who is not reasonable.

And that has gone on since 2002.

I have learned a fascination with the process! People lie. And then lie about lying. And other people can come in and lie with them.

But there are areas where the lies do not work.

What happens after you can see in the recent threads.

More interesting after that is Usenet posters will SAY I didn't beat them and maintain I'm a crackpot who is not reasonable.

And that has gone on since 2002.

I have learned a fascination with the process! People lie. And then lie about lying. And other people can come in and lie with them.

But there are areas where the lies do not work.

## JSH: Not Riemann's zeta function

It was actually Euler's zeta function and if you don't understand that oddity where math people shifted to saying it was Riemann's zeta function when Euler used it before, then you will not understand a lot of the story!

There is a LOT of history here, so it's a lot easier if you understand the timeline and you actually need to start with Euler.

See: http://en.wikipedia.org/wiki/Leonhard_Euler

Euler used the zeta function to first connect the count of prime numbers to a continuous function.

See: http://primes.utm.edu/glossary/xpage/EulerZetaFunction.html

That connection is easy to see by playing with the scientific calculator that should be on your computers.

Count of primes up to 10: 2, 3, 5 and 7. 4 primes 10/ln 10 = 4.3 to one significant digit

Count of primes up to 100: 25 primes. 100/ln 100 = 21.7 to one significant digit

You may not care but for people who were interested in primes it was kind of an odd thing! And Euler FIRST made the connection with a continuous function with his zeta function.

Euler is the one who then found a way to even begin to talk mathematically about how primes actually connected to continuous functions, and was the guy who realized that the zeta function was a way to do so.

The next significant mathematician in the line to Riemann is intriguingly Chebyshev, and not Gauss.

See: http://en.wikipedia.org/wiki/Pafnuty_Chebyshev

Gauss introduced a function called Li(x), which is closer to the prime count than x/ln x, as notice say, with 1000.

The count of primes to 1000 is 168, but 1000/ln 1000 = 144.7 to one sig no rounding.

So there is this increasing gap between the count of prime numbers and x/ln x, which mathematicians sought to remove!!!

(I don't give values for Li(x) as it's not as easy to just calculate it on your home pc!)

But Chebyshev figured out a way to use Euler's zeta function to put boundaries on the continuous function which became the "prime number theorem". (Math people can get kind of weird with some of their designations.)

Riemann narrowed those boundaries and made his famous hypothesis and assuming it to be true various other mathematicians came up with arguments that should not be called proofs!

They shouldn't be called proofs unless they are established on proof but math people have this annoying habit of calling them proofs and saying they firmly believe the Riemann Hypothesis to be true, without a proof that it IS true.

It's like trying to build a castle without a foundation at all. Building it on air.

But now with ancient history done let's get to modern times.

My research starts you over at the beginning!!!

Back to Euler with the question of how to connect the prime distribution to continuous functions in the FIRST place.

I found easily enough that you have a P(x,y) function, a multi-dimensional prime counter, versus the pi(x) function where mathematicians oddly picked "pi" with the prime counting function which can lead to confusion. It's not pi. It's pi(x), the count of primes up to and including x, so pi(10) = 4. pi(3) = 2, as 2 and 3 are counted.

My P(x,y) function leads to a partial differential equation, thus easily connecting the prime distribution to a continuous function by giving a natural REASON. A simple easy to follow reason for why prime counts would be close to some continuous function.

That is not in debate.

But now it's possible to look over the entire previous history with a new tool and a new perspective as I've given what's likely to be THE explanation, so all others can be evaluated with it.

It's kind of like when Ptolemy's spheres were superseded by Kepler's ellipses.

My research greatly simplifies the area and allows you to now go back and look over what was done previously, including verifying or disproving the Riemann Hypothesis.

However, remember those "proofs" based on the Riemann Hypothesis? If the Riemann Hypothesis is shown now to be false then those arguments are clearly not proofs and a lot of mathematical edifice was shown to be, wrong.

Mathematicians have proudly proclaimed that upheavals that are a part of physics, like the shift from Ptolemy, are not in mathematics which they claim builds on proof.

But with the Riemann story you can see sly cheat throughout, like the shift from the Euler zeta function to the Riemann zeta function, and the building of "proof" on the top of a hypothesis!

Here I can give a prime counting function. Can easily step through to connecting it to a continuous function by getting to a PDE, and it's a multi-dimensional prime counter versus the single variable pi(x) that math people had before, so I can give THE answer and logically connect primes to a continuous function, for the first time in human history.

And I've been able to do that since 2002.

It's weird living history. Years from now scholars can puzzle over the bizarre recalcitrance in mathematicians in fighting what by then will be considered one of the great results in mathematics, just like we can puzzle today about why people would disagree with Galileo.

I mean it's so obvious, right? Clearly the earth revolves around the sun!

It's so obvious here, there's now for the first time a simple answer to the prime distribution!

Belief is weird. Maybe somehow, someway the math people believe they are doing the right thing by fighting the actual mathematical answer to hold on to stuff that has a lot of history, but is probably false, but it's hard to see how.

There is a LOT of history here, so it's a lot easier if you understand the timeline and you actually need to start with Euler.

See: http://en.wikipedia.org/wiki/Leonhard_Euler

Euler used the zeta function to first connect the count of prime numbers to a continuous function.

See: http://primes.utm.edu/glossary/xpage/EulerZetaFunction.html

That connection is easy to see by playing with the scientific calculator that should be on your computers.

Count of primes up to 10: 2, 3, 5 and 7. 4 primes 10/ln 10 = 4.3 to one significant digit

Count of primes up to 100: 25 primes. 100/ln 100 = 21.7 to one significant digit

You may not care but for people who were interested in primes it was kind of an odd thing! And Euler FIRST made the connection with a continuous function with his zeta function.

Euler is the one who then found a way to even begin to talk mathematically about how primes actually connected to continuous functions, and was the guy who realized that the zeta function was a way to do so.

The next significant mathematician in the line to Riemann is intriguingly Chebyshev, and not Gauss.

See: http://en.wikipedia.org/wiki/Pafnuty_Chebyshev

Gauss introduced a function called Li(x), which is closer to the prime count than x/ln x, as notice say, with 1000.

The count of primes to 1000 is 168, but 1000/ln 1000 = 144.7 to one sig no rounding.

So there is this increasing gap between the count of prime numbers and x/ln x, which mathematicians sought to remove!!!

(I don't give values for Li(x) as it's not as easy to just calculate it on your home pc!)

But Chebyshev figured out a way to use Euler's zeta function to put boundaries on the continuous function which became the "prime number theorem". (Math people can get kind of weird with some of their designations.)

Riemann narrowed those boundaries and made his famous hypothesis and assuming it to be true various other mathematicians came up with arguments that should not be called proofs!

They shouldn't be called proofs unless they are established on proof but math people have this annoying habit of calling them proofs and saying they firmly believe the Riemann Hypothesis to be true, without a proof that it IS true.

It's like trying to build a castle without a foundation at all. Building it on air.

But now with ancient history done let's get to modern times.

My research starts you over at the beginning!!!

Back to Euler with the question of how to connect the prime distribution to continuous functions in the FIRST place.

I found easily enough that you have a P(x,y) function, a multi-dimensional prime counter, versus the pi(x) function where mathematicians oddly picked "pi" with the prime counting function which can lead to confusion. It's not pi. It's pi(x), the count of primes up to and including x, so pi(10) = 4. pi(3) = 2, as 2 and 3 are counted.

My P(x,y) function leads to a partial differential equation, thus easily connecting the prime distribution to a continuous function by giving a natural REASON. A simple easy to follow reason for why prime counts would be close to some continuous function.

That is not in debate.

But now it's possible to look over the entire previous history with a new tool and a new perspective as I've given what's likely to be THE explanation, so all others can be evaluated with it.

It's kind of like when Ptolemy's spheres were superseded by Kepler's ellipses.

My research greatly simplifies the area and allows you to now go back and look over what was done previously, including verifying or disproving the Riemann Hypothesis.

However, remember those "proofs" based on the Riemann Hypothesis? If the Riemann Hypothesis is shown now to be false then those arguments are clearly not proofs and a lot of mathematical edifice was shown to be, wrong.

Mathematicians have proudly proclaimed that upheavals that are a part of physics, like the shift from Ptolemy, are not in mathematics which they claim builds on proof.

But with the Riemann story you can see sly cheat throughout, like the shift from the Euler zeta function to the Riemann zeta function, and the building of "proof" on the top of a hypothesis!

Here I can give a prime counting function. Can easily step through to connecting it to a continuous function by getting to a PDE, and it's a multi-dimensional prime counter versus the single variable pi(x) that math people had before, so I can give THE answer and logically connect primes to a continuous function, for the first time in human history.

And I've been able to do that since 2002.

It's weird living history. Years from now scholars can puzzle over the bizarre recalcitrance in mathematicians in fighting what by then will be considered one of the great results in mathematics, just like we can puzzle today about why people would disagree with Galileo.

I mean it's so obvious, right? Clearly the earth revolves around the sun!

It's so obvious here, there's now for the first time a simple answer to the prime distribution!

Belief is weird. Maybe somehow, someway the math people believe they are doing the right thing by fighting the actual mathematical answer to hold on to stuff that has a lot of history, but is probably false, but it's hard to see how.

## JSH: Psychology of perspective

It is of interest to me what we think we know, as a species, and I have a unique opportunity to watch people behave in a way that they would not, if they actually accepted my accomplishments, where for some reason mathematicians have given this window by not properly acknowledging my research!

So I exploit the window.

It IS of interest to me to see how posters react to various pieces of data and to see how newsgroups behave as so much of it is so wild and bizarre. Like it was so amazing that math people destroyed one of their own journals. So bizarre.

World history is a fascinating subject, but how much of it is true? I am now able to detect variant story arcs throughout any number of places in world history where based on the data I'm gathering, surprisingly HUGELY different things probably occurred.

What never ceases to amaze me is the confidence some of you display. This bizarre ability to BELIEVE when I can check with absolute tools and determine that you're wrong.

It's endlessly fascinating.

I think these warnings are important as I believe I'm an ethical researcher and this situation is somewhat annoying. I am destined to be one of the major historical figures, but I'm pre-that level. It is my duty to use this unfortunate situation to its limits.

And I'm learning that history is just a word. Who knows what really happened even a few decades ago. Most of you I fear, have no clue, but just know what someone told you, like, what you read in some book or something.

But people write those books, and are quite capable of shifting things to their own needs.

I study such shifting having tools far beyond what any of you seem to have, which I find puzzling. They're not complicated tools and I talk about them freely.

My own major hypothesis is that it is an evolutionary construct. I'm studying that construct as well, and how it shifts human conscious behavior and most importantly—perspective.

You "see" what your brain tells you that you see. And it happily lies to many of you in very special ways.

So I exploit the window.

It IS of interest to me to see how posters react to various pieces of data and to see how newsgroups behave as so much of it is so wild and bizarre. Like it was so amazing that math people destroyed one of their own journals. So bizarre.

World history is a fascinating subject, but how much of it is true? I am now able to detect variant story arcs throughout any number of places in world history where based on the data I'm gathering, surprisingly HUGELY different things probably occurred.

What never ceases to amaze me is the confidence some of you display. This bizarre ability to BELIEVE when I can check with absolute tools and determine that you're wrong.

It's endlessly fascinating.

I think these warnings are important as I believe I'm an ethical researcher and this situation is somewhat annoying. I am destined to be one of the major historical figures, but I'm pre-that level. It is my duty to use this unfortunate situation to its limits.

And I'm learning that history is just a word. Who knows what really happened even a few decades ago. Most of you I fear, have no clue, but just know what someone told you, like, what you read in some book or something.

But people write those books, and are quite capable of shifting things to their own needs.

I study such shifting having tools far beyond what any of you seem to have, which I find puzzling. They're not complicated tools and I talk about them freely.

My own major hypothesis is that it is an evolutionary construct. I'm studying that construct as well, and how it shifts human conscious behavior and most importantly—perspective.

You "see" what your brain tells you that you see. And it happily lies to many of you in very special ways.