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Tuesday, November 28, 2017

What is mathematical success

There is an odd thing to being puzzled at how your fellow human beings react to some mathematical object. To me these are things. A mathematical object is a bit of logic or a lot with some numbers, and usually some abstract symbols mixed in there which can DO things or express things done.

What decides how we react as human beings to a mathematical object?

That question is more interesting to me now, while have pondered it before, when was more interested in validating mathematics as have been gifted with a trying situation. So I defined mathematical proof because did not get reaction from people I thought expert who thought could check mathematical approaches that was expected.

Ok, so in general, human beings designated as experts will not consider something, does that mean is incorrect, or unimportant, or what?

You can end up kind of stuck, as how can you get them to do something when they do not so wish? And why would you try? But then how are you supposed to know with your own math research?

One thing I could eliminate on my own though was question of correctness, if I had a perfect method to check. So I discovered a functional way, and found an easy definition. So yeah, went looking because of a need and ended up MUCH better off that way as well.

mathematical proof (noun): a mathematical argument that begins with a truth and proceeds by logical steps to a conclusion which then must be true.

From my reference post for my definition of mathematical proof.

It is remarkable that NOT getting what I thought I needed, led to an advance for the human species. Where for a lot of years have wondered why that was for me to explicate, though wonder less lately.

So yeah, you have something like that and wonder around it and about it for years, but also most importantly I'd use it. It WAS functional for me. So then I could eliminate questions of correctness, and realized mathematics was embodied in logic, but logic is bigger. So logic is bigger than mathematics, and I had to handle questions of logic for a bit, but was fun. And yeah why a post on logic is on this blog, way back to blog beginning year 2005. So yes, logic is bigger than mathematics, but is also, easier in many ways.

Our reactions as human beings are so much about emotion. And through years of hammering at problems, and hammering at problems and getting nowhere, simply felt better when switched to using my imagination and pretending at times like was asking a powerful entity I called the math. And the math became my constant companion and getting answers, I wondered what value emotion in math? When asking the math a question, emotion has no value. The math does not care. Yet, if you ask in the right way, with the right question the math will just tell you, as my imagination made it feel real.

To me success with numbers IS emotional though. And the math does not care, but does not block or punish that emotion.

Like look at something like:

60*2551100302 + 2551100292 = 19924730292

For me looking at it, I have mixed emotions. Why can you square 255110029, and add result of adding one to it and squaring multiplied by 60, to have another square? I know the mathematical answer. Still feels remarkable. So I remark.

That is simply a mathematical object. It cares not.

But also feel like is mine in some sense as I am the discoverer. YOU are seeing that as a result of my thoughts and my questioning. Without which, human beings..who knows. Someone else might have discovered, eventually.

Success in mathematics to me is in contemplation, of mathematical objects previously unknown, now known. Maybe for each of us, is a question to answer, what is success?

Which is a question to try to answer for those who care, of course.


James Harris

Monday, November 27, 2017

Better with explained

With over 15 years since my find of a cool way to count prime numbers which easily answers some big questions, and surprised by a find of a brand new primary way to calculate the modular inverse, was natural to talk it out.

And yeah there is no doubt that real mathematical experts would not ignore exciting mathematical finds, which is an easy way to hide some, from people who are in awe of the people who supposedly are. But it is also how they are actually kind of honest I realized that lacking actual mathematical ability themselves, they do not find valid mathematical results to be of interest.

That was very much a shocking assessment I really only accepted this year, and reluctantly. As I pondered that thrill of a mathematical result and how much it can move a person. And then ranged to the thrill we as human beings find in other areas. Which is also partly why studied entertainment. Then came back to realize that yeah, people who experience such emotion are MOVED by the truth. The discovery itself has power. But when they are not, you can trust that emotion, and in mathematics is indicative of lack of actual ability.

Originally I leaned towards belief not being moved by certain things was indicative of rivalry. Until realized was giving too much credit. Rivalry might have been easier to handle than the dud reality of simply not being able to tell. Why my leaving some possible discoveries as bait, never worked. Eventually I'd simply collect them myself, and wondered. Now am confident as to why.

My prime counting research to my surprise didn't move things also probably because is too far away from anything practical, and things like Riemann's hypothesis are hype-zones. It has no actual value to human knowledge one way or the other, which is sad to me, to even write. And I realized over a decade ago, suspecting it is false, that even if I could prove that, there was no way that would be accepted in current situation.

Right now am sure millions of dollars for worthless research in just that area is being paid to number theorists as salary--but also guessing. Have long thought that and do believe that is true, but readily admit no longer check closely on what number theorists are doing these days.

And there has been another reason for concern for me. How many realize that the biggest concentration of number theorists on the planet is at the NSA? I sure do. If you wonder why I don't just prove certain things in certain ways, understand would rather have the knowledge spread WIDE.

Which still may not save you, if the US Government, thinking things are one thing, goes after you, guided by people who know they are frauds, protecting themselves. Another reason I am cautious about asking for help.

Will situation resolve? Yes. Eventually the preponderance of mathematical evidence will do what it always does. And the truth will win out. But does not have to happen on what those of us who know fully what is going on think of as a reasonable schedule.

The good news is mathematical industry is ok. And reality that applied mathematics actually is the mathematics is kind of not surprising. That is what is necessary for science and technology.

If that were not the case guess I would have to be more creative, to defend the discipline. But for now, reality is, esoteric problems at the fringes will not hamper human progress, I don't think.

AT least for those of you who wondered who can determine am correct, and have these great mathematical discoveries, FINALLY explained situation to you completely. Was worth the effort.

Of course have known most of these social assessment things for years, like over a decade. Some of it though often times just refused to just accept, and now decided, especially with my latest modular inverse discovery, there was no room for doubt. And realized probably should explain better full situation, for others.


James Harris

Tuesday, November 14, 2017

But does it factor?

One of my first results with what I now call the BQD Iterator was talked couple years after found it in this post with this result.

Given x2 - Dy2 = 1:


(D-1)j2 + (j+1)2 = (x+y)2, where j = ((x+Dy) -1)/D

or

(D-1)j2 + (j-1)2 = (x+y)2, where j = ((x+Dy) +1)/D

Where two cases matter for finding integers, as j will be an integer if x = 1 mod D, for the first one, or x = -1 mod D for the second.

And gave this example back then:

With D=2, and x=17, y=12, you solve as 172 - 2(12)2 = 1, and going with the minus of the plus or minus:

j = ((17+2(12)-1)/2 = 20 is a solution giving:

202 + 212 = 292

So D-1 a perfect square relates to circles but otherwise gives non-circular ellipses.

And also I noted the rational solutions for x and y:

x = (D + t2)/(D - t2) and y = 2t/(D - t2)

Because that means you can get integer solutions for an ellipse, using those solutions, because of how those squares are distributed. But like with the first one, have always wondered about:

(D-1)j2  = (x+y)2 -  (j+1)2 = (x + y + j + 1)(x + y - j - 1),  j = ((x+Dy) - 1)/D

Because you can force that to be all integers with rational solutions as all the denominators can be multiplied out. And I never checked it, to see if it ever factored D-1 non-trivially.

But have wondered, for years. Why will I not check it?

My guess? Is one of THE control equations for ALL integer factorizations. The other results from the other variant. That thing I suspect, helps control all integer factorization, across infinity. But maybe not.

May be key to integer factorization the way that x2 + y2 = 1 is key to trigonometry. Showing value of what I now call unary two conics equation. Maybe that thing controls ALL integer factorizations! To the math would be just about logic. For us? Would be so remarkable.

Maybe it just kind of scares me.

To just look at something that maybe controls so much, and wonder.


James Harris

Analyzing psychological space

Was surprised over a decade ago. Did think maybe that math grad student would be honest and help when he realized I was correct. Have had time to consider my example social problem case quite a bit through the years. And part of it is, training? Number theorists rule their roost with psychological precision. Doubt is an accident.

Over a hundred plus years to probably notice their number theory tools were not really working well, if at all. And now very practiced at deflecting attention to the reality, including bouncing off people like myself talking it.

Feel like am playing catch up, as for a LONG time felt like just needed to keep piling up discoveries, but they are more and more immune to it! Like are adapting, and learning can keep getting away with it. Possibly until my research number theorists didn't realize themselves exactly what was wrong, or that they could keep going with error even when it was outed publicly. But why would they so choose?

Need to abstract them. Ok, what is currently known?

Math students can't seem to be enticed with further discovery potential, as have tried that angle. Can't be trusted to turn on them when erroneous math is proven. And willingly apparently continue, even when must be clear that the analysis they are doing is actually worthless. Troubling.

So odd is why have been puzzled for so long. Mathematical results? Useless with them. Even if some excitement is generated for a bit? Gets dampened out, somehow. Yet, they think they're interested in mathematics? How is this cognitive dissonance maintained, except by very powerful psychological tools? Or am I assuming too much. Could be stupid simple, eh? Ponder here more?

Institutions are well trained to accept them like any other experts. And IS weird how deep that trust goes, even as web security breaches embarrass. Why not question underlying system?

US Government apparently is completely hacked and doesn't even seem to care too much. I just assume all its systems are leaking, and then world events make more sense. Shows power of military might over good sense. So figure we're safe regardless or would be more concerned.

These are psychological forces that go deep into the human animal. Behavior is very communal, and number theorists are tapping into controls that suspend human reasoning, which is really surface anyway. Human beings are NOT rational. Emotion has been noted as key.

People identifying communally must be separated from seeing as a benefit. Failure realty should be emphasized? Key? Must be some kind of reward system that creates such tight binding against mathematical proof! Absolute truth can be deflected only certain ways? Or no? Yeah by casting doubt on concept of mathematical proof, which HAVE caught them doing in the past.

Over a hundred years of history of apparent success current ones can check of people who may have owed that appearance to the error. Could be reassuring to them. They may no longer believe in truth. They may no longer believe in social systems. They may feel safe with believing that even their fake research will last indefinitely, or definitely long enough to live a seemingly successful life. Interesting.

Psychological assessment space generated. Will consider and edit over time as needed.


James Harris

Further defining social problem space

Discovering the math was apparently first problem solving exercise and after over two decades of successful discovery, am now realizing need to focus on what I call the social problem.

What have established beyond mathematical doubt is that there is a problem with use of ring of algebraic integers, where recently posted in detail talking historical reality entered field in late 1800's.

Mathematicians at that time, ironically were trying to solidify the foundations of mathematics as a discipline, and screwed up.

Problem with math error is it makes it easier to look like you prove something which is actually taught to math students. In this case erroneous ideas were not outed because also push went to supposedly pure research. With that even as a theory, would expect MORE people would crowd into the easier area, and looks like that's what happened, with number theorists dominating.

Initially did not realize math people had to be aware on some level. Now believe they were, as looking at sophisticated blocking to maintain the error, which primarily relies on silent treatment.

And can easily prove the math. Have however watched as people do not react as have expected when realize am correct, which posted earlier on, as have studied for more than a decade. As I solidify the mental adjustment. Means is not enough to just inform. Must consider human psychology more carefully.

People look to experts. Is forced on us in a complex world. In this case, with a problem so old, very much entrenched at ALL levels in mathematical community.

A practical wrinkle did enter though, as number theorists presented integer factorization as a hard problem, and web emerged using techniques like public key encryption for security. It is now doubtful that lack of ability with integer factorization was because of difficulty, but simply may have been because of lack of ability, of number theorists. And yeah, lots of web security problems consistent with that theory, but also an area where governments are SUPPOSED to check.

My latest research result, figuring out a new primary way to do modular inverse could not be ignored by legitimate researchers, so can conclude they realize they are in error, and now are simply protecting, against further mathematical discovery which then is an enemy to their success.

Social problem then is fully revealed as a cadre of people who are succeeding at being mathematicians in appearance, with it possible they have no research of importance that is valid. Whose entrenchment may help explain widespread problems with web security. And they are apparently aware that they are in error and that their research is bogus. Where they have been acting to protect their situation.

Thorny problem, indeed.


James Harris

Knowledge as entertainment or sport

There is a great thing about mathematics that you do not need to work to convince, when is correct. And have noted past success with my own research in getting published, even though I also make sure to note chief editor tried to delete out my paper AFTER publication from a now dead electronic journal, which was back in 2004.

However, it occurs to me that people seem to think AM trying to convince, when am sharing mathematical truths, which of course is a position that can collapse if they evaluate.

Like shared that story of a math grad student who I guess thought he might be able to convince me I was wrong, over a decade ago, and what happened when he instead convinced himself.

He was deflated.

That is NOT fun to witness either. And have encountered more than once. So no, there is no flush of satisfaction as if won an argument. Math just is. For me there has usually been a bit of sadness for that person. The truth should not hurt, but is choice of that person, not me.

To me the rest of the story with mathematics in the modern age is NOT with whether or not you can prove, but how others react, and I think math for those who don't avoid it, who think they value it, is like entertainment or sport.

So yeah, can easily convince mathematicians--better yet, my math easily convinces mathematicians, as one would expect. There is no mathematical basis for denial of it. But it is disappointing for them emotionally.

Is maybe like watching a favorite team lose a sporting event, when someone finally realizes am correct. They deflate and slink off. Is that cruel? They disappear. That person so jumping up and down and excited like a fan cheering on the home team--the mathematical establishment, deflates and metaphorically, just goes home.

People who do not care if the math they do is correct as long as they FEEL good doing it, or talking about it. They represent a problem I find difficult to solve. Correct mathematical arguments? Just make them feel worse. The more proof given? Worse they feel. Then they just walk away from it, to keep doing what they like.

So yes! Have experienced winning the arguments for those who wonder. So no, you do not have anything to offer by agreeing with what I know to be true. And I've watched what people do next.

Have watched, as they, slink away from the truth.

And I have pondered the behavior, for years.

It is a difference of our times I think. Possibly a product of a culture where entertainment and sports are HUGE in lives of many, including me, where others maybe think they chose knowledge, but reality is they get pronouncements from people informing something is important. Same people also often saying is TOO DIFFICULT for most to understand. So yeah, for them? Is not about knowledge, but about trust and emotion. Their cheering of mathematics is no different than others cheering their favorite sports team or celebrities.

And when math bursts that bubble? They shrink away like a morose crowd upset with a loss by the home team.

Which I realized years ago. And I call it, the social problem. That in the 21st century people of the math community do NOT care if math is correct. They care how it makes them feel. Applied math is safe from this effect. In pure math it dominates. Notice where MOST of today's mathematicians are.

What can you do though, when the truth is not appealing?

Working on solving the social problem, like with this post. Solution is to get people to value the knowledge, not their emotion about the knowledge. But how? Is such a weird problem and remarkably difficult to solve. So no, is not needed for me to work to convince. Can simply share, and is EASY for me to get agreement on my math research.

Is math after all.

Math works that way.


James Harris

Monday, November 13, 2017

Some math and attention reality

Mathematics has a huge value to our species for what we can do with mathematical things and has developed to a level of abstraction where much of it can be distant for many.

That is true in lots of areas of human interest and recently noted to myself how little I know about paint, though see all over the place, and have painted things. But how much do I really need to know to live my life?

Reliance on expertise is a necessity in our world where so much was developed over so much time and mathematics is one of the oldest areas of human interest and effort. Yet can still surprise us with powerful tools. Like one of my personal favorites is my own discovery.

Given: u2 + Dv2 = F

then it must also be true that

(u-Dv)2 + D(u+v)2 = F(D+1)

-------------------------------------------------------------
Reference blog post: Binary Quadratic Diophantine Iterator

That is not complicated really. And yeah, if you THINK the only thing you need to do is discover something interesting, then you do NOT need to be an expert in that area.

One of my favorite examples made with it, so yeah have on repeat as I think tells SO much:

(462 + 482 + 722)(1722 + 258+ 430+ 6022 + 17622) = 


            615+ 30752 + 141452 + 159902  + 1884972   =  774*210

Do you know how was done? On this blog can see easily built--and explained.

Yeah we DO need people who can develop a deep knowledge most do not need to have, like about paint like do I need to ponder its chemical structure? No. However if I noticed something cool about paint, might I expect say a chemist who specialized in paints to find curious, if true? I think so.

Am NOT a mathematician. Am someone who has found some cool, relatively simple things, including ones that challenge what we think we know, about numbers. And went to the people who are establishment, repeatedly.

Math in its usefulness is how we have so much. As human beings learned more to do with what they had, have discovered more. To me that is SO cool.

Have watched as people from all over the world come to this blog, to look at things, presumably useful. I know I like playing with numbers.

Then I think a spectator mentality takes over. Especially when you've been trained to accept pronouncements of supposedly important math which very few people are expected to barely understand. Which is ok, is like with my paint example. We just need to know that expertise is real, you know?

While with me and my math discovery, is much emphasis on simple understanding and being able to DO something with the math, which is also am sure a great draw.

The math is the indicator. The attention flows naturally--to what works well.


James Harris

Saturday, November 11, 2017

With questions on expertise

DO find it hard to value what I do, though greatly enjoy sharing my mathematical adventures, where was a HUGE wrinkle to shift from thinking would be looking for validation from mathematicians, to questioning their expertise and distancing from them.

Now make sure to note am NOT a mathematician. My training though only undergraduate was in physics. And do note am a mathematical discoverer.

Rise of the web has helped me so much, both in doing basic research, evaluating and getting that research out there, while leaving vastly fewer clues for others I guess, about how great that reach is.

(I guess because I don't know. Puzzle over web analytics and other clues, much.)

Which means for me is newer generations who escape television or those who managed to unlearn like I did and relearn web ways who have a greater grasp of the possibility. Which is so weird.

Am global with ideas.

Read my dramatic explanation for how number theory got messed up:
Explaining a historical miss

Can assure you that story is not something that does not travel. But what people do next?

There things get complicated, which I like to call, the social problem.

Has been a very difficult adjustment. From looking to simply find something new and important, to also handling other things, where would not be needed if experts in the field could simply be trusted to help with great finds.

And am doing better at valuing what I do and what I've accomplished. But it did take thinking about it that way that I should. Is an odd thing as human beings, how do we value our own efforts?

Reality is, think for most is about how others value, or how we think they do.


James Harris

Friday, November 10, 2017

Where numbers rule

One of my favorite expressions from my research is simply a sum of seven squares to get a square:

349672 + 7522 + 1128+ 1880+ 26322 + 41362 + 48882 = 357212

Where am using my BQD Iterator, and have put up repeatedly and like to use to remind that numbers DO rule. And with me? Number theory should show number theory results, with integers.

Am NOT a mathematician. Am a mathematical discoverer.

For a bit thought maybe mathematicians couldn't do such sums, as was just doing web searches but finally with one, as I do keep searching, found a paper which stated was summing seven squares to a square.

Another example I like where have not seen anything like it elsewhere, which does not mean is not out there:

(462 + 482 + 722)(1722 + 258+ 430+ 6022 + 17622) = 

            615+ 30752 + 141452 + 159902  + 1884972   =  774*210

Which is still using my BQD Iterator as that thing is really useful, and really cool.

For me number theory should have such things. I like looking at them, and understanding the relationships between integers that make such possible, and easy to generate.

Numbers DO rule. And should rule, number theory.


James Harris

Thursday, November 09, 2017

Social problem is tough as ever

Six months now with my modular inverse. A surprising fundamental discovery adding only the third basic way known to humanity to calculate one. And fifteen years plus now with my prime counting find, which just gives the answer for why prime counts connect to continuous functions. Oh, and found a coverage problem over 13 years ago. Plus my various other major finds.

Yet the blog pulls global interest from people which doesn't seem to matter much.

Feel like am missing something.

But what?

Long believed the discoveries would do the work.

Now guess need to work more on figuring out my fellow human beings.

Well, at least I've discovered enough. Have said that before then this modular inverse thing popped up.

Well glad I have it regardless. Is really cool.


James Harris

Common mistake with my modular inverse method

There is one case which can potentially trip people up with my modular inverse method, which is when you have shared factors which divide across the modulus. Will show a case copying and expanding upon and modifying a bit from my full iterative example previously given:

Need modular inverse of 34 modulo 295. r' = 34, N' = 295.

Will let m' = 23, where again is arbitrary choice but do like picking primes, and y'0 = 1 as usual, which is also arbitrary.

F'0 = 34(34+2(23)) = 34(80) = 65 mod 295

2(23)(65)d' = [65(n-1) - 1](34 + 2(23)) = [65n' - 66](80) = 185n' - 265 mod 295

40d' = 185n' - 265 mod 295

Dividing off shared: 8d' = 37n' - 53 mod 59, and notice that 37 - 53 = -16, so n' = 1 mod 59 works.

Gives d' = -2 mod 59

If you do NOT divide off across modulus will often get wrong answer. Which happened to me until figured it out, but that is part of the discovery process. Maybe is worth explaining why.

The modular approach is like 40d' = 185n' - 265 + 295j, where j is some unknown integer we don't need to know. But if you divide to get: 8d' = 37n - 53 + 295(j/5), you are assuming j is divisible by 5, which is an error. We don't know what it is and don't want to know.

So correctly is: 8d' = 37n - 53 + 59j, where we assume nothing.

But yuck, now have solution mod 59, and checking took -2 + 2(59) = 116 to work for d'.

So r'-1 = -2(23)(116) = 269 mod 295. And checking: 34(269)  = 1

Where needed answer modulo 295, and yeah 116 = -2 mod 59.

Is a situation where looks like just have to check various values.

So yeah, when divide off the modulus? Can get a range, and have to try more than one value.

So answer was d' = -2 mod 59, but value needed modulo 295, is d' = 116 mod 295.

That seems to really trip some people up.

------------------------------------------------
Here is the system:

r-1 = (n-1)(r + 2my0) - 2md mod N


Where y0 is chosen as is m, with m not equal to r, and n and d are to be determined. They are found from:

2mdF0 = [F0(n-1) - 1](r + 2my0) mod N

and

F0 = r(r+2my0) mod N

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

Copied from my post: Modular Inverse Innovation

It is also derived there.


James Harris

Wednesday, November 08, 2017

Much work to a simple explanation

Figuring out a problem that entered into number theory in the late 1800's was a daunting task. While clues came early, like with a dead math journal, was just too astonishing by much, where over a decade to ponder bears fruit with my latest:

Easily explaining a historical miss

And it is unfortunate became so much a single person's diligence, as still thankful for those two anonymous peer reviewers who approved my paper over a decade ago, and the original decision to publish, despite what happened later. Where also there is that math grad student example where at least he did confirm my research for himself, and told me that but then, in my opinion, bowed to the enormity of the social pressure.

It is stunning that a relatively easily explained miss in trying to establish foundations of mathematics as a solid discipline can be so easily explained and yet so hard to handle, where that is work in progress.

My continuing discovery has helped, both by drawing interest and helping my confidence in pursuing the story. Like how easily could explain prime counts to continuous functions, like with:

Differential equation and prime counting

Again, powerful pure research, where problem with acceptance I believe? Sadly best explanation is people fearful of disruption of current funding for number theory research which then is bogus or unnecessary. That we in our times, can know why the count of prime numbers links to continuous functions like x/ln x is a blessing. That any would fight such knowledge, is a curse and consequence of error.

But yeah, huge thing that helped me move forward more confidently was finding a THIRD fundamental way existed to calculate a modular inverse. Where show, give an example and derive at:

Modular inverse innovation

The evidence is overwhelming. The social consequences? HUGE. There is no scenario where proper mathematicians ignore ANY of the above. And no one who loves knowledge should either.

A story long in the telling, with over a century of history, with a historical miss. Now known. What YOU do next though? Will help with where things go from here.


James Harris

Easily explaining a historical miss

They were trying to solidify the foundations of mathematics as a discipline, in the late 1800's. Gauss was already gone. One of the most influential mathematicians of all-time, he had died in 1855. Yet one of the areas they felt had to be addressed as they tried to handle all numbers were what we call gaussian integers in his honor. Numbers like 1-3i where note (1-3i)(1+3i) = 10, behave a LOT like integers, so were considered integer-like while not being rational.

Those mathematicians working to turn mathematics into a formal discipline seized upon roots of polynomials. Considering roots of monic polynomials, meaning lead coefficient was 1 or -1, with integer coefficients seemed to work! These roots encompassed gaussian integers and more, and they tried various tests where things seemed ok. They decided had the ring on which number theory could rest, which they called ring of algebraic integers. Found a link to Wikipedia on algebraic integer where oddly enough they don't talk the history. Algebraic numbers are a ratio of algebraic integers.

What we know now that those mathematicians apparently did not, is that a problem can be shown with a generalized quadratic factorization.

In the complex plane:

P(x) = (g1(x) + 1)(g2(x) + 2)

where P(x) is a primitive quadratic with integer coefficients, g1(0) = g2(0) = 0, but g1(x) does not equal 0 for all x.

But to show the problem you have to force symmetry, but also do something unintuitive and ADD an extra variable. That is actually counter to how math students are taught to go, as training is to remove extraneous variables, not introduce them!

Introduce k, where k is a nonzero, and new functions f1(x), and f2(x), where:

g1(x) = f1(x)/k and g2(x) = f2(x) + k-2

Multiply both sides by k, and substitute for the g's, which gives me the now symmetrical form:

k*P(x) =  (f1(x) + k)(f2(x) + k)

The purpose of forcing symmetry is to allow for solutions where also need one more function, then can solve for the f's using the quadratic formula.

And introduce H(x), where: f1(x) + f2(x) = H(x)

Where easy algebra allows for me to solve for the f's using quadratic formula:

f1(x) = (H(x) +/- sqrt[(H(x) + 2k)2 - 4k*P(x)])/2

And H(x) is a handle for every possible factorization with the g's, and has a key constraint: H(0) = -k + 2.

Copying from post: Simple Generalized Quadratic Factorization

If we let our extraneous factor k equal 1 or -1, no problems, and you get solutions that are roots of monic polynomials with integer coefficients when use integers for x and have P(x) and H(x) with integer coefficients. But let k equal some other integer, and one of the g's must be out of the ring as:

g1(x) = f1(x)/k

Which can show with an example. 

Let P(x) = 175x2 - 15x + 2, H(x) = 5(7x - 1), and k = 7.

Plug everything in, and you can solve for the f's, and with x=1, get an interesting conclusion, as then you find that one of the solutions for 3-sqrt(-26) has 7 as a factor.

Have ran into problems where people learning a convention with square roots can see that as a single number. But note that 1+sqrt(4) = 3 or -1. So is improper to say it has 3 as a factor, while one of its solutions does.

That we cannot see the factor of 7, with 3-sqrt(-26), does not change what is mathematically proven with our generalized quadratic factorization. However, it is a standard result in number theory that cannot have a non-unit integer as factor in the ring of algebraic integers.

It is hard to imagine those mathematicians over a hundred years ago coming across such an argument to find there was a problem with their attempts at formalizing sturdy foundations for mathematics as a discipline.

Today we know far more, especially about impact of over a hundred years with this result, as number theory became the biggest draw for people entering the field, where I think is because was easier. Is a quirk of mathematical fallacy. Results in number theory went further and further away from integers, where is now dominated by complex analysis, ever more abstruse, and much of it? May simply be wrong, but how would you know?

As a discoverer came across this problem with another argument where got a paper published in a dramatic story, as I used the problem to show one result by the rules, which could also be shown wrong by those rules. That is, I proved a result with the ring of algebraic integers declared, used valid expressions and operations in that ring, yet got a conclusion not true in that ring.

What happened then was telling. 

Have talked it in detail in this post: Publishing a contradiction

A math journal possibly being brave published my paper, then a chief editor tried to run from it, and later the entire journal shutdown.

Given that was over 13 years ago, best guess as to what was the purpose? Is what was obtained: millions of dollars in continuing funding primarily from governments for number theory research which may simply be bogus.

Such a simple thing in trying to establish solidity to mathematics in simplicity of purpose. Such a difficult thing to pull off in actuality. Cannot fault their motivations. Their lack of ability was unfortunate. One wonders if Gauss had still been alive would he have quietly figured out there was a problem? Don't want to take hero-worship too far, but his brilliance was demonstrated and his gaussian integers do not have a problem.

Who knows.

What we do know is that in running from the error in our times, which thankfully have learned to demonstrate more easily now with my generalized quadratic factorization, is that fraud took over number theory research. Motive? Money, status and false recognition.

There is no other conclusion available. The math is simple enough.

Scientific progress and technology were protected by relying on applied mathematics. Which is great!

But for the error to be fixed requires acceptance of pure math research.

And the bravery to accept mathematical truth, in order to once again have number theory focused on the pure pursuit of knowledge.


James Harris

Tuesday, November 07, 2017

Answers can be satisfying too

Like to emphasize the importance of questioning, which to me is more fun, but there is also a lot of satisfaction with answers, which to me usually with the best ones, raise more questions.

Where for instance did have a social need myself with my latest big discovery, as kind of wondered in the past what it would be like to have something of its nature, and how might that move things?

Now nearly six months with it, am finally calming down from the basic excitement, and can look at behavior which is more telling from the people who should be cheering it, and very loudly.

A major discoverer can find, but again can see what happens if others choose to be silent about that find.

With this result is completely clear.

Which lets me dig into motivations without doubt. And in this case is simple fraud am sure, primarily about money. The result shows am a person who deserves attention with research that does, and also have a result over a decade old showing a problem where is highly likely number theorists are currently getting paid, for things of no value to the world.

It's like free money. Especially since the error I found means that you can probably appear to prove just about anything in number theory, so is not like you even need to pretend much work at it. Math errors are just that way. Wow.

Am glad too have shied away from money things myself, as does help to keep perspective, and keep me from pressure to NOT talk these things out! Here is no big deal. Look at data from my blogs with curiosity but not monetary motivation.

The global thing though! So yeah, the ideas will still draw the attention. Is just social structures around other things like recognition in a variety of ways depend on these other social structures.

So I have the global reach--and thank you readers from around the planet--but nothing else.

Which shows how much things you may think are about one thing, are actually about a system. And yeah, I do look at celebrities differently now. In many cases when you see them, you're looking at people with vastly less reach than I have. Definitely less permanence to their levels of attention in comparison to me. There is no proper comparison though. Discovery makes the difference. Is so cool.

But weird. Is interesting to process. And like to talk it out too. Where will admit am not talking everything so much. However, yes, some answers are nice to put down in writing.

Billions of people on our planet and a few will notice something, but why?

Considering things I've noticed think is a process not easily explained. Discovery is the reward though, so from perspective of a discoverer, lost nothing myself.

Gained so many satisfying answers, and along with them, of course, came more questions.


James Harris

Money and social problem

From my perspective is obvious am looking at a situation where mathematical progress is running into resistance from people engaging in now a rather obvious fraud, which is very lucrative for them.

Reasonable doubt is further removed from others by important discoveries, where my latest has been key to my deciding to put more pressure and to again explain fully what is going on.

The modular inverse is part of fundamental mathematics and is easily explained. For instance 2 is the modular inverse of 3 mod 5, and vice versa as 2(3) = 1 mod 5. There were two primary ways to calculate a modular inverse beyond such an easy example, and I found a third.

So why would academics see a monetary benefit from ignoring mathematical discovery?

Answer is, public funding flows to number theorists primarily for salary, since they do so much work with their minds, presumably. My research shuts down further research in some areas while opening doors in new ones, which is typical with major discovery. Some questions get answered, while new ones arrive.

By not acknowledging those discoveries though, number theorists can continue being funded for research that is not needed, and also as found a coverage problem, can receive funding for research that is not valid. Presumably that is quite lucrative for them.

So yeah, for the people at the top, could also be about prizes and accolades but for most number theorists? Talking about a paycheck.

Other academics who become aware of the problem, may not realize how stark the explanation is, as simply fraudulent activity primarily for the purpose of obtaining public funding against the public good.

So am explaining it for you.

These assessments help me as well. Is an interesting position am in, as the discoverer. Like to think is unique but others have faced resistance to truth. And also have with modern academic system.

The scale though as these discoveries are at such a high level. These people are probably trapped now in their fraud. Is hard to see how they can expect to casually walk away from consequences at this point.

Well, explains why it doesn't matter what I discover. For these people? There is no discovery great enough, as they are simply then shown to be more greatly at fault. They are enemies of the system that funds them.

Interesting.


James Harris

My reference social problem example

Year ago, but definitely after I was aware there was a coverage problem impacting number theory, was posting on Usenet math groups through Google Groups and on one a person identifying himself as a math grad student at a prestigious university offered to help. He said if I explained to him in detail via email, and he agreed that it would be a plus for me in getting things moving. I verified he was a student at the university.

And I confirmed the offer of help with him, and he emphasized and I agreed, as wasn't that big of a deal for me, and hey, maybe could move things. Back then might have already moved to a quadratic example following same path as my published paper, which the chief editor tried to delete out AFTER publication, before the math journal keeled over and died. So am explaining things carefully to this math grad student and giving him things to work through for himself.

Eventually his email replies took longer and longer, and one had things about wandering around at 4 am think it was staring up at the stars, until he'd finally emailed had stepped through entire argument, which of course is correct. And then was the begging off, where I knew what to expect after the weirdness about wandering around late at night.

That coverage error entered into the math field in the late 1800's and allows people claiming to be mathematicians to make arguments that LOOK correct because of the error. When recognized, it removes the research of top number theorists, where noted yesterday Andrew Wiles as one who loses his claims to fame. For those who need reference as to why error with math can be so potent or a refresher, here is an article on mathematical fallacies. The coverage error should be part of that article by now. The people profiting from it though am sure is why is such a slog. It has been worth hundreds of millions US to a community, which is annual.

Understand why that math grad student, so confident I guess that I was wrong, would crumble when realized I was right? He verified the truth himself. So had nothing to do with math. Had everything to do with humans as social creatures. Hence the social problem.

My research eventually focused also on: x2 - Dy2 = F

A fundamental equation which was known but clearly NOT researched enough, as my discovery of a third fundamental way to calculate the modular inverse flows from THAT equation. Looked back through my posts and looks like first posted a modular solution September 2012, with this post, which means over 5 years ago.

2x = r + Fr-1 mod N and  2my = r - Fr-1 mod N

So yeah, had almost five years of having noticed that modular inverse before figured out a way to solve for it using these equations back in May.

So a math grad student realizes there is a correct argument which undermines the underpinnings of modern number theory, and decides he'd rather stick with the system. By now he could be a full professor, maybe even married with kids, as is over a decade ago.

Did he make the right choice?

Of course, I say, no. Turning to deliberate error could give him social approval, and maybe a career, but is the wrong choice, in my opinion. And to some extent one might argue, his probably social assessment of the odds have not been shown to be wrong, so far.

That is my reference example. Shows you a case where I know of a mathematician who carefully stepped through the argument showing a coverage problem, doing work himself, went through apparent emotional distress, then concluded was better off accepting error than help me fight error in the system.

Note also, he helped then condemn later students to being trained in error, with less opportunity to know was being done to them.


James Harris

Monday, November 06, 2017

Benefits of patience though

So yeah, wasn't really affected by lack of proper acknowledgement of my research by math community as reflected in discoveries because discovery was the point. Which is yeah a benefit of NOT depending on other people to do the right thing. Never got too distracted with that social.

Where time my first major discovery now in 1996, where used what I now realize was a modular approach to packing of spheres. Which made the problem SO easy have struggled with accepting solution! Which isn't terrible. Guess first talk that discovery on this blog here. And there am more hesitant I think but at least talk approach. Wow, that was back May 19, 2008. So that post was 12 years after the paper that amazingly was rejected by a math journal as TOO SIMPLE.

Next big thing which was HUGE was when finally got frustrated enough to invent or discover as debate with myself, tautological spaces. Where started with:

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

Deliberately looking to introduce a new variable, which I put as 'v' for victory. Took a few weeks to figure out that would work, and that happened in December 1999. But was YEARS before had what thought was something, in 2002. (Yeah was still after that Fermat's thing which am NO LONGER considering important.) When promptly tasked myself with finding something else and figured out my prime counting function. My favorite thing to show from it? Is the sieve form:

P(x,n) = [x] - 1 - sum for j=1 to n of {P([x/p_j],j-1) - (j-1)}

where if n is greater than the count of primes up to and including sqrt(x) then n is reset to that count.

My way to count primes. Have had now for over 15 years and yeah, DID push that HARD, contacting a LOT of people in math community including leading researchers in area. That could have messed me up. Did for a bit. Got very upset. But then I realized there was something bigger.

And has been over 13 years since cleverly published a math paper which correctly gets to a conclusion under established math ideas which can also be shown to be incorrect by them, which is the coverage problem. Knew would have an uphill slog when began to see implications of THAT one, and yeah, when that math journal keeled over and died. Talked why publication matters recently so linking there.

At this point wasn't really any doubt something was wrong. Had THREE major results which had all been rejected in some way by mathematical community, where with last got publication acceptance, then weird. And it gave the full answer as to why with all: was dealing with people who lacked what I had, real achievement.

But I kept going. Eventually got this blog in 2005 and eventually started talking my results primarily here, as they continued to pile up. Until we get to now and my latest major result.

There was another way to do a modular inverse. And I found it. Last way goes back to Euler and 1763 with his totient theorem, found out researching recently.

So the human species has a new third primary way to calculate the modular inverse as of less than six months ago.

Yeah has been worth it. And better for humanity as well that I just kept going. My method for the modular inverse:

 r-1 = (n-1)(r + 2my0) - 2md mod N


Where y0 is chosen as is m, with m not equal to r, and n and d are to be determined. They are found from:

2mdF0 = [F0(n-1) - 1](r + 2my0) mod N

and

F0 = r(r+2my0) mod N

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

So yeah, have the discoveries. Where in contrast if had let other objectives or emotion rule could STILL be stopped, focusing exclusively and bitter over the prime counting function or even more so over the coverage problem as so dramatically proven, where there is no doubt. Even got publication! Even if things got weird after.

To me shows value with making sure that the point is discovery. Would rather have more of the truth anyway.

Oh yeah, and there were some REALLY cool other things jumped over maybe really should at least mention. Like improved upon Gauss with reducing binary quadratic Diophantine equations. And decided that an idea about prime residues? Was an axiom decided to call prime residue axiom. And could go on.

The discoveries piled up enough have to think for a bit to keep up with them all. Ok yeah, don't actually try to do that any more, as would keep missing things. Is too much work and why do it?

Discovery was the point. And I kept discovering.

Have fun debates with myself at times about whether something is or is not a major discovery. When yeah, it probably is. But have enough can try to use a fine criteria. Is a fun thing for me, will admit.

Benefit of patience with all that other is more truths found, for me, and for humanity.

Who knows when will all be properly acknowledged. But as years go by am definitely more calm about it all. Focused less on math discovery these days. Still curious though, with that urge to discover new truths.


James Harris

Privilege of discovery

Do like to emphasize my discoveries now, and also make the point that like past discoverers I do have a lot of ability to put messages that will be in the history books.

And right now am looking at people who appear to believe more in social things without appreciating that down the line in history books they are simply villains with maybe a sentence or two covering their behavior.

They will be stripped of any honorifics. And the record will reflect they betrayed their world.

It is a fact that there is a problem with use of ring of algebraic integers, which have demonstrated repeatedly. Is a fact that I have major discoveries that do things like explain the prime count to continuous function connection and more recently give a new way to do a modular inverse, as well as more.

Belief that you can simply hold silent against such things is a repudiation of the pursuit of truth itself.

Enemies of the human species I designate you. And for some of you the consequences of someone at my level so doing are exactly what you think they are.

What I don't understand is why you would give up your lives in this way. It still puzzles me.

Meaning your lives considered to be decent citizens. Don't get too excited there. Am giving myself room to be prepared for people who will probably act very pitiful when pushed out of positions and consequences catch up to them.

Getting myself used to the idea, and for others.


James Harris

Why an uphill battle

For years thought I only needed major discoveries. But have put them out, and watched as world will apparently react to them, and then do nothing else, as presumably looking to establishment figures within mathematical community.

So am talking more about why those in number theory at least might see major discoveries as a threat, which is a wacky story.

My latest major mathematical discovery highlights just how much you have to do to compete with me, and shows how much work is needed to suppress.

A new fundamental way to do the modular inverse?

Apparently roared around the globe like other things. Then you get that quiet, which creates an uphill battle for the truth.

Well am letting you know now. That quiet could be criminal.

For some of you? May mean the end of your academic careers.

The quiet you may think is just usual academic behavior could be important in enabling a continuing fraud which is a threat to the national security of your respective states.

If asked, I will recommend maximum penalties in prosecutions--to send a message. And to protect the mathematical discipline from such behavior, in the future.

When valid math is your enemy

Was surprised myself to find there is another way to figure out a modular inverse previously unknown. Which is the kind of discovery should be cheered! And represents a surprising advance since last known advance was with Euler's totient in 1763. Which I found by web research. Am NOT a mathematician. But am a mathematical discover and over a decade ago found something else surprising.

That find over a decade ago was of a coverage problem with ring of algebraic integers, which underpin modern number theory, as for instance algebraic numbers were defined as a ratio of algebraic integers. And is like if you only consider evens, and 2 and 4 and 8 are ok, but you get to 6 and claim that 2 and 6 do not share factors, you are wrong. They do not in evens but is not a proper ring.

I demonstrated with a paper. The journal which first bravely published the paper, later went through gyrations where chief editor tried to delete out of electronic publication. Then a bit later the journal keeled over and died, suspending publication.

Why might this thing be so huge? Because could bring question on ANY paper that has ring of algebraic integers declared as the ring or is dependent on that ring. And weirdly enough impacts Galois Theory where THAT is WAY hard to explain as out of my zone of expertise.

First off have demonstrated it more easily with quadratic. Where a key equation is:

g1(x) = f1(x)/k

Where that k messes so many things up. If k = 1 or -1, then you have number theory as was known. But if you choose to let be some other integers, then you know something is wrong.

The full factorization is in the complex plane:

P(x) = (g1(x) + 1)(g2(x) + 2)

where P(x) is a primitive quadratic with integer coefficients, g1(0) = g2(0) = 0, but g1(x) does not equal 0 for all x.

The other substitution is: g2(x) = f2(x) + k-2

Multiply both sides by k, and substitute for the g's, which gives me the now symmetrical form:

k*P(x) =  (f1(x) + k)(f2(x) + k)

The purpose is forcing symmetry. So the f's can be forced to be algebraic integers. But here is what is difficult for the emotions, they STILL can be so forced, when k is say, 2 or 3, and then you are forced that one of the g's cannot be when P(x) is irreducible.

That full post is here.

So yeah is easy to demonstrate the coverage problem. The k does divide off, which you can see with an easy polynomial solution, like let P(x) = x2 + 3x + 2, with k = 1, and g's equal to x.

So easy math but like what major mathematical argument falls?

Well the work of Andrew Wiles goes into the garbage bin for one that is dramatic enough I think.

And people get SO territorial. And the emotion is what's driving what I call the social problem. People think certain things can NOT be considered. As if human need or social status matter to the math. Reality is, this result cleans house across the top of the modern math field, and he's just a good example.

Near as I can tell, if just the simple thing in this post is acknowledged, he may have nothing in his career of note.

Yet now he's a celebrated number theorist. World famous. Encyclopedia articles in his praise. Prize money won.

If you were him, what would you do?

Valid math is the enemy of these people. Which is why suppression has to be so stark, and am noting with my latest discovery. They have to see real discoverers as enemies.

For a bit I called them pseudo-mathematicians. And say they engage in mathematical fiction.

They have I think willfully embraced error since I put the truth out there over a decade ago. That is SUCH a dramatic story too, with the dead math journal. The error though? Is how they made their careers.

They'd probably be unknown to you without it.

Have felt more sympathy for them in the past, but hey, they're teaching NEW people to go down wrong paths, with celebrated approaches! When by now may know very well, do not work!!!

You can stay emotional and support them if you wish, but their math does not actually work. You can learn from them if you wish, but using their techniques your math will not actually work.

And my discoveries are not going anywhere. Eventually the war of suppression will fail.

So dramatic--war of suppression. Primarily seems to just be--silent treatment. Relying on quiet to send the message that a major discovery is not really one. Is very effective too.

Truth does win eventually though. Why? Because it says the same. Same today, same tomorrow.

Their work is wrong today, is wrong tomorrow. And in time the truth just grinds down the techniques that resist it because is relentless. Never lets up.

And the enemies of mathematics who managed to get in charge, will be fully revealed for who and what they are.

So, if you wish to continue with them, then you accept their fate, whatever the world decides.


James Harris

Sunday, November 05, 2017

How much worth it? And more on social problem.

Maybe does bother me some people seem to think have been grasping for recognition of ME, when to me is about the math and what's right. And reality is, as have noted before, situation without formal and appropriate recognition for my math results, didn't mean a pause for me, at all.

And my latest result is still overwhelming. Who knew? There was another way to do a modular inverse. And I found it. Last way goes back to Euler and 1763 with his totient theorem, found out researching this morning.

So the human species has a new third primary way to calculate the modular inverse as of less than six months ago.

Yeah has been worth it. How many humans get to say something of that nature? Is rare air. My equations:

 r-1 = (n-1)(r + 2my0) - 2md mod N


Where y0 is chosen as is m, with m not equal to r, and n and d are to be determined. They are found from:

2mdF0 = [F0(n-1) - 1](r + 2my0) mod N

and

F0 = r(r+2my0) mod N

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

Just stare at them, is what happens. My mind just wondering. But is like with another result where like to stare:

P(x,n) = [x] - 1 - sum for j=1 to n of {P([x/p_j],j-1) - (j-1)}

where if n is greater than the count of primes up to and including sqrt(x) then n is reset to that count.

My way to count primes. And answers there.

That feeling is one hard to describe, and I do think it helped me that there were people who probably kept thinking I wanted something else. Maybe is all they have. Social approval.

Looking for the cheers, or for fellow humans who will tell you how brilliant you are, or award you in some way.

While I wanted the math. That's what I found. And think about my results and ponder at times.

Discovery DOES change you. Was one way before, am another way now.

Was definitely worth all the work, and the years of, pondering.

What I call the social problem in my mind is a demonstration of human beings who lost belief in truth, and turned to social things. So is a challenge, to the math.

Which to me is so weird! So yeah, you convince some humans for a bit, but why is that worth it? I guess that part still troubles me. I think I understand, but then I still wonder.

Mathematicians who have quit caring to be correct, how to handle such a thing? They have embraced deliberate error.

Yeah, may as well put it that way. Past giving them benefit of the doubt.

Am the discoverer. Really that's what I do. These other things are just, tedious in one way. Unless I think am discovering a way to solve the social problem!

Ok, I am the discoverer. Just one more problem to solve. People who demonstrably show a disdain for mathematics, who have taken over the discipline. Great discovery? They just try to ignore.

Nothing seems to move them. And have discovered much! Must ponder more carefully I guess.

These are social creatures. Key to their behavior? Is emotion.


James Harris

How can you know? Social problem.

Reality can be so uncaring can seem cruel. And have noted a bizarre situation with number theory where is SO easy to know am correct, but how do you handle the emotions?

For instance, there is one key equation in an important way I solve for a generalized quadratic factorization:

g1(x) = f1(x)/k

Where that k messes so many things up. If k = 1 or -1, then you have number theory as was known. But if you choose to let be some other integers, then you know something is wrong.

The full factorization is in the complex plane:

P(x) = (g1(x) + 1)(g2(x) + 2)

where P(x) is a primitive quadratic with integer coefficients, g1(0) = g2(0) = 0, but g1(x) does not equal 0 for all x.

The other substitution is: g2(x) = f2(x) + k-2

Multiply both sides by k, and substitute for the g's, which gives me the now symmetrical form:

k*P(x) =  (f1(x) + k)(f2(x) + k)

The purpose is forcing symmetry. So the f's can be forced to be algebraic integers. But here is what is difficult for the emotions, they STILL can be so forced, when k is say, 2 or 3, and then you are forced that one of the g's cannot be when P(x) is irreducible.

That full post is here.

Easy algebra but HUGE consequences that rock number theory for over a hundred years all the way back to the late 1800's. And why would you bother to believe? If the truth hurts?

Hence what I call, the social problem. Is where emotion has taken over reason in math community, and yes, I DO understand. Is so hard. And the math does not care.


James Harris

Saturday, November 04, 2017

Suggestions for press inquiry

Decided to do a post helpful for members of the press. There are actually several avenues for verifying are some big stories around my research, where maybe can guide to questions that can break some of the easier ones to verify. Questions should go to a number theorist. The higher up the chain the better.

1. My paper published where chief editor literally tried to delete out of publication.

Is easy to verify had a paper published, and as have noted EMIS maintains original. Is just a matter of making inquiry to a number theorist about the matter. Asking a couple of pertinent questions:

a) Is the paper correct? Answer is, the paper uses correct methods but gets to a conclusion that other number theory invalidates, revealing a problem.

b) Note that I claim the above, and simply ask, is that true?

If number theorist has looked over paper, then you either have that person, or if lies, you have on record, if claims otherwise, like simply states I'm a crackpot or crank or something derogatory.

2. My find of a clever variation on older ways to count prime numbers.

There is no doubt I have a way to count prime numbers, but its importance is the matter to verify.

a) Are there any known ways to count prime numbers that lead directly to a partial differential equation? Answer is, mine is only one. Press person might get probed on expertise with such a question, how do you know about such things?

b) Is an explanation for continuous functions connected to prime numbers of continuing interest? Answer should be yes. Here is harder as press person would need some SERIOUS expertise or do some major research.

The questions oddly enough stand on their own. And could probably really mess with the mind of a number theorist, especially if DID know of my story. If needed to show an example, then can use this link.

3. My recent find of a new primary way to figure out a modular inverse.

My latest result is very compelling as is less pure math than above, as finding a modular inverse is part of modern public key encryption techniques, which I found out by researching, after my find made May 9th of this year.

a) Is a third primary way to do a modular inverse of interest? Something not derived or related to extended Euclidean algorithm or Euler's theorem? Answer is, yes.

b) If I showed you one, would you go on the record about it?

Where have posted this link a LOT, as talking it up.

Turns out wouldn't take many questions. Each of these things are HUGE, and a slightly curious press person could just ask person you know with an interest in math if could handle them.

How hard can they be?

Getting a bit more creative as realize, yeah, plenty of things could happen if certain people were asked just a few pertinent questions. Maybe if I help the press out with a bit of guidance...well it could happen. May as well try.


James Harris

Shorter version of full modular inverse example

Humanity now has a third fundamental way to calculate a modular inverse. Here will step through using my method for calculating the modular inverse showing less than in a previous post where put in a LOT of detail. The calculation is determining the modular inverse of 101 modulo 1517, and the answer is 751. System being used is at end of post.

So: N = 1517, r = 101. Further I choose m = 47, and y0 = 1

Then: F0 = 1491 mod 1517

And find:

295d = 499n + 162 mod 1517


Will go with smaller modulus, so want 295 to divide across.

So: 499n + 162 = 0 mod 295, and find: 34n = -27 mod 295

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

So can recurse! Need modular inverse of 34 modulo 295. r' = 34, N' = 295.

Decided to choose m' = 23, and y'0 = 1

F'0  = 65 mod 295

And find:

40d' = 185n' - 265 mod 295

Dividing off shared: 8d' = 37n' - 53 mod 59, and notice that 37 - 53 = -16, so n' = 1 mod 59 works. And also then have n' = 1 mod 295 as was solving to divide 40 across.

But I have d' = -2 mod 59

And checking took -2 + 2(59) = 116 to work for d'.

So r'-1 = -2(23)(116) = 269 mod 295. And checking: 34(269)  = 1

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

So now with: 34n = -27 mod 295

Now can get: n = -27(269) = -278 = 112 mod 295

Which is our solution for an equation modulo 1517, so have n = 112 mod 1517

Found a value to allow a coefficient to divide across, so is valid modulo our N.

And from before:

295d = 499n + 162 mod 1517

And now

295d = 499(112) + 162 = 56050 mod 1517, so d = 190 mod 1517

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

r-1 = (112-1)(101 + 2(47)) - 2(47)(190) =  111(195) - 1173 = 751 mod 1517

And 101(751) = 1 mod 1517 as required.

Here is the system:

r-1 = (n-1)(r + 2my0) - 2md mod N


Where y0 is chosen as is m, with m not equal to r, and n and d are to be determined. They are found from:

2mdF0 = [F0(n-1) - 1](r + 2my0) mod N

and

F0 = r(r+2my0) mod N

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

Copied from my post: Modular Inverse Innovation

It is also derived there.

Friday, November 03, 2017

Full iterative modular inverse example

Figured out my own way to calculate the modular inverse which is then the third fundamental way now known. Here will give a full iterative example showing how can tackle any modular inverse calculation.

Post walks through finding the modular inverse of 101 modulo 1517, which is 751, with my discovery. Talk a lot at beginning as talk through setting things up.

To get something that will need to iterate probably need to go larger than have done in prior examples, so will use  N= 37(41) = 1517. Like using composites as presumably should be a little harder but doesn't really matter, I don't think, with my method. Let's use r = 101. And will use m = 47. There are several variables where you get to choose. With m, I like to choose such that its square is greater than the modulus. May not be necessary but with early research will just go with what I like.

Have given the system LOTS and is in this post but at end, as I want to just dive into the method, up here. Will let y0 = 1 as usual, just is easy. Further research may indicate should give it some other value.

Then: F0 = 101(101+2(47)) = 101(195) = 19695 = 1491 mod 1517

Tempted to use the smaller negative, but will continue with it. Next equation:

2(47)(1491)d = [1491(n-1) - 1](101 + 2(47)) = 998n - 193 mod 1517

590d = 998n - 1193 mod 1517

Here both coefficients are even, so will use 1517-1193 = 324.

590d = 998n + 324 mod 1517, and can now divide off shared factors.

295d = 499n + 162 mod 1517

Will go with smaller modulus, so want 295 to divide across.

So: 499n + 162 = 0 mod 295

Which is: 204n + 162 = 0 mod 295, so: 204n = -162 mod 295

And can divide off some factors: 102n = -81 mod 295, so: 34n = -27 mod 295

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Whew! So can recurse! Need modular inverse of 34 modulo 295. r' = 34, N' = 295.

Should note how much smaller the modulus is with next iteration. There will be a smaller modulus to recurse with next guaranteed as of course each coefficient will be smaller than N.

Will let m' = 23, where again is arbitrary choice but do like picking primes, and y'0 = 1 as usual, which is also arbitrary. May turn out should pick some kind of way and have some other value for optimization. Am just demonstrating here though, so is ok.

F'0 = 34(34+2(23)) = 34(80) = 65 mod 295

2(23)(65)d' = [65(n-1) - 1](34 + 2(23)) = [65n' - 66](80) = 185n' - 265 mod 295

40d' = 185n' - 265 mod 295

Dividing off shared: 8d' = 37n' - 53 mod 59, and notice that 37 - 53 = -16, so n' = 1 mod 59 works.

Gives d' = -2 mod 59

But yuck, have solutions mod 59, and checking took -2 + 2(59) = 116 to work for d'.

While as a solution to divide off coefficient still also have n' = 1 mod 295. Is maybe kind of notable how that works? Am looking for a solution modulo 295, so the 40 will divide off. And n'=1 will work, so while found modulo 59, is also value modulo 295.

So r'-1 = -2(23)(116) = 269 mod 295. And checking: 34(269)  = 1

So yeah, when divide off the modulus? Can get a range, and have to try more than one value.

So answer was d' = -2 mod 59, but also d' = 116 mod 295. Ok, on to next thing then.

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So now with: 34n = -27 mod 295

Now can get: n = -27(269) = -278 = 112 mod 295.

Which is our solution for an equation modulo 1517, so have n = 112 mod 1517

So yeah, again how that works is found a value to allow a coefficient to divide across, so is valid modulo our N.

And from before:

295d = 499n + 162 mod 1517

And now

295d = 499(112) + 162 = 56050 mod 1517, so d = 190 mod 1517

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r-1 = (112-1)(101 + 2(47)) - 2(47)(190) =  111(195) - 1173 = 751 mod 1517

And 101(751) = 1 mod 1517 as required.

And some thoughts. Actually is better when nothing divides off the modulus. Doesn't seem to even want to let you cheat some kind of way. The math behaves as if, yeah there were shared factors to divide off. Then you MAY have to increment like I did, so maybe is good that popped up in this example.

Whew! Definitely something to let a computer do. But will admit for me is interesting to work through in detail.

So is demonstrated how you can iterate with the system, and then walk back to the final answer.


------------------------------------------------
Here is the system:

r-1 = (n-1)(r + 2my0) - 2md mod N


Where y0 is chosen as is m, with m not equal to r, and n and d are to be determined. They are found from:

2mdF0 = [F0(n-1) - 1](r + 2my0) mod N

and

F0 = r(r+2my0) mod N

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

Copied from my post: Modular Inverse Innovation

It is also derived there.


Why discovery rules

Found myself comforted by over 15 years with my prime counting function where just noted that milestone a few months ago like with this post where talked. And with my latest major result less than 6 months old, where found humanity's third fundamental way to calculate a modular inverse, am a lot better off with years of experience.

And yeah, discovery rules. It is really impossible to completely contain major discoveries. They just find ways to move. As the years go by admit am not focused on math discovery so much any more. Spend time pondering humanity, and the motivations that can drive human beings to do things which can be negative, neutral or positive.

Also work hard to keep myself grounded.

Number authority helps me greatly in many ways, as I contemplate things. Like one of my favorite things to just stare at and consider, I found it:


(462 + 482 + 722)(1722 + 258+ 430+ 6022 + 17622) = 

            615+ 30752 + 141452 + 159902  + 1884972   =  774*210


Copied from my post with some number examples where I collected some of my favorites.

Is like a technology though. If I just saw that without knowing how was built, would it impress me?

Yeah, I think it would. But knowing how it was built, using my discoveries, the emotions are different am sure. But how do I know?

Others might experience things one way, where for me? Oh that's just something of mine. Still cool though.

Discovery rules in many ways. One of the big ones? Is because discovery keeps things interesting.


James Harris

Thursday, November 02, 2017

How modular inverse reveals social problem

Back May 9th posted my method for calculating the modular inverse. A surprising find, represents a fundamental result in modular arithmetic, and is of course important in modular algebra. If asked if such a thing existed, am confident a pat answer would have been that if any other possible were available, would have been found long ago.

There were only two basic ways known to the world before that date to calculate the modular inverse besides brute force, where one is with use of extended Euclidean algorithm, and the other is done with Euler's theorem, where used a Wikipedia article as a reference. And you have plenty of ways around both of those where people have tried to speed things up, with good reason.

My approach will note here looks better fit for the computer age. Allows for optimal multi-threading out of the gate as just the first thing I noticed quickly without much research done on optimization.

Other basic methods literally have over a hundred years of people considering them, though am sure much more effort recently. Found out as was researching that a popular public key encryption approach involves calculating a modular inverse. My ideas might speed things up, dramatically. But don't know. Just reaching there.

Regardless is fundamental human knowledge which presumably would be welcomed.

I thought it might, but have enough history with certain behavior to be prepared if it did not. As the months went by realized I needed to be more direct about the situation, as there was no room for doubt.

Notice this result leaves no room to hide for people actually fascinated with numbers. By itself without my other evidence it is proof there is something wrong with the discipline of number theory, when not embraced with joy.

Is satisfying to be able to emphasize that there is no room for doubt. And while I believe the result is innocuous which frees me to push harder, without careful consideration from people who care for knowledge world governments cannot be so certain. The risk is not only unacceptable, it is against human interest. Fundamental results can quite simply, open doors.

How could things be this off where a major discoverer has to emphasize that people trying to ignore a major mathematical discovery, easily determined to be one in a fundamental area related to global security, could be hoping on an approach that has saved them before--silence?

Because human beings can do such things.

The math does not care.

We should though.


James Harris