## Saturday, December 30, 2017

### Modular inverse and pondering obvious

We understandably have few opportunities as a species to ponder obvious things which somehow escaped us, until they did not, as how do we know what we do not know? Infinite knowledge exists. Humanity will only learn, so much.

Emotionally much shifted for me in 2017 which was in many ways my most satisfying discovery year ever, as found my own way to calculate a modular inverse!!! But was such a clean and simple result as well, like a simple gift from the math, to our species. But how could something so easy be missed for so long? Here will talk a simpler start to the path to finding it, which I did not use.

Like consider: x2 - Dy2 = F

Is a fundamental equation. Is known you can reduce other binary quadratic Diophantine equations to it, and I DID have my own most general method for so doing. But still didn't occur to me just thinking about it to consider it modularly:

x2 - Dy2 = F mod N

And yeah I don't bother with congruence symbol as to me is tedious unnecessary extra. And now why not realize that the D with Diophantine equations MAY be a quadratic residue modulo N? Let's assume it is, and let the residue that is squared be m, so have:

D = m2 mod N

So: (x-my)(x+my) = x2 - Dy2 = F mod N, and you can consider some residue r, and notice something useful.

If: x-my = r mod N, then: x+my = Fr-1 mod N, or vice versa as can shift r.

So the modular inverse pops up! Also you can solve for x and y, to solve modulary, which is where I focused for years. But why not naturally ask, can I solve for the modular inverse from there? Answer is, yes. Adventurous readers can test math ability by trying to solve for the modular inverse from THIS point, if can. If you struggle, can just check my derivation.

Where we got there, quick and easy, doing things that apparently human beings do not tend to do!!! But why not? I didn't even take that quick and easy path years ago. Figured it out after.

If still haven't seen my method in action, you might want to check out a post with a full iterative example where I find the modular inverse of 101 modulo 1517, which is 751. 101(751) = 1 mod 1517.

Found myself curious enough tried to work back, and came to conclusion had figured out solution for x and y modularly by September 28, 2012, and traced back best I could with this post on my math group.

Yet wasn't until May of THIS year that I wondered if could do something to find the modular inverse with that, and posted one day with some musings:

Chasing modular inverse

Oh, wow, so that's dated May 5th, which is the date of discovery. Then days later on May 9th posted something cleaner, and have used that date as forgot:

Modular inverse innovation

Ooh, good thing making this post! So was May 5th when had discovery. But May 9th post I just think presents better.

So why new?

Because human beings think certain ways which can miss things. But how do we know what we do not know? We get to know what we mysteriously didn't know, when our species finally learns it. And discovery is not something we fully understand. But should be something we respect.

And make no mistake, a third primary way to calculate the modular inverse is a really big deal.

For me was a puzzle why was left for me to find, but at least removed doubt from people without a clue, who maybe relied on faith in human beings.

So is a perfect litmus test. Like reality finally felt sorry for some of you.

If you understand mathematics in actuality, you know to go with the math, always.

If instead, you simply trust humans? What's wrong with you?

For those who think they love mathematics, who cannot be convinced by math, you need to go into politics or something else as mathematics, then is NOT your area.

For me was just really cool. Was nice to have such a simple and clean result, and figured with an applied and pure math result maybe other things might change. With a history of major discoveries, will admit you do get a different attitude. To me now, major discoveries are very emotional yes. And extremely satisfying, but also, so very curious.

James Harris

## Friday, December 29, 2017

### But does destiny rule?

Feel like lucked out, as now with better perspective is clear to me how my discovery followed a familiar trajectory. Now do accept that it started in 1996 when I figured out a modular approach to packing of spheres. And found a proof so stunningly simple would puzzle me for over two decades as to how new. If had been recognized then, would have stepped onto the global stage like so many, early and young, who discovered.

Was spared though, and went on to build the framework for more automated modular discovery with:

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

For me has always been a story I thought I understood, as if was in control of it all. Now as think back at how steady the work has been, feels more like a compulsion. Is a similar trajectory seen in mathematical discovery. Major discoveries beginning in twenties, and usually peaking after a few decades.

Would struggle with some things with far more peace than my predecessors. No rude questions. No curious humanity wondering how.

The web would be more constant, and part of a process. At times was like was a friend that could understand me, along with the math, which I began to imagine as an entity. My two best friends, though the web was so much still a mystery, and I like to note that the math does not care.

The web would enable me to test global in my own way and in my own time. As the web helped me process and accept. The web has been my friend, and still not sure why, or exactly how. So many things have been like had the web watching over me, trying to protect as best could. Still there was a mental price to pay, in learning to face fears in time, but was so much in private. Until could possibly handle the full enormity of being a global figure. Am I there yet? Am not sure.

I knew I would be the only major mathematical mind of the early 21st century. There is no one else.

No one else is even close to me. No one else on planet Earth appears to be even trying.

I have no peer in mathematics in our times. There is a loneliness there. I do not wish it.

Where is the humility I fought so hard to regain? Why does it keep leaving me? What does it take to stay on solid ground? Still there is a truth there. Maybe should just face it, until can yank it back down to earth.

Been trying to cut off discovery since 2010? Like a rocket engine that still has fuel. Kept telling myself, I have enough. There is no competition. There is no reason to continue.

There was no way to stop, until the rocket engine blessedly ran out of fuel.

Studying celebrity, I look for those destroyed by it, then analyze with what I call a psychological dissection. Piecing through each case, understanding the forces that tear a human apart, from the attention.

Attention, in and of itself, can kill a human being. Nothing else needed. We're not built for it at levels possible in our modern world.

And I was spared. Why? In retrospect, realize it aided discovery simply enough, I guess. These math results would find their way into our world. I was simply their portal. Simply used, brilliantly. Which pleases me. Is what I wanted, to discover.

Still I do wonder. How much do we really control in life? Or how often do we actually simply find ourselves, especially as older adults, coming up with reasons why.

James Harris

## Thursday, December 28, 2017

### Basic overview of social problem

Being right is more fun. Correct is a better rule. So yeah, very much weird, even if easy to explain how in the late 1800's, ironically, some mathematicians screwed up when trying to firm foundations of modern mathematics as a discipline in the late 1800's.

Where I can prove that happened with a simple quadratic factorization, thankfully. Still is so hard to process.

Mathematicians back then failed to realize that they were not including necessary numbers when focused on ring of algebraic integers as roots of monic polynomials with integer coefficients, like:

x2 + 7x + 3 = 0

Monic just means 1 or -1.

As comforting as the simplicity may have been to them in focusing on such expressions,

Was too easy by much. Is even easier to show how they failed, where actually found something simpler, backing up a published proof, where love THAT story as chief editor tried to yank after publication and a little later journal keeled over and died. That's so wacky.

So there is no reasonable doubt and there is no doubt under established rules. These people try to flip their own rules though, when it pleases them, like no one would notice?

What is clear to me is that a political explanation for that wacky is easiest.

Have explained recently how modern number theory could have LOTS of bogus things, while working mathematics keeps working with post: When math is correct

And yeah, with number theorists sidelined from anything important to our world OUTSIDE of encryption, is not pressure from what I've seen for problem to be corrected. Clearly, world does not feel a need for them to be correct! Which creates a fascinating situation have pondered much.

Most people just do not care. Like to say the math does not care, but most of the planet doesn't either. They have no clue what number theorists are doing, and couldn't care less.

So yeah, they DID succeed at making themselves irrelevant, which makes for a much harder social problem to solve.

As we're talking about math error, number theorists can appear to prove just about anything they choose. So I look with a certain cynicism at what they put forward. Means they can also pick winners and losers, which am sure math students learn quickly. Is not what you know with this error that matters.

Human civilization is rather new. Human beings have a thin veneer of rationality and again, is simply ironic really that mathematics became so afflicted. Is very ironic, given that mathematics has the possibility of absolute proof.

Which I have used. And you can check how that has gone, so far.

James Harris

## Wednesday, December 27, 2017

### Simply definitive factorization

Much of what I have is challenging, in many ways. And even for me that can be daunting which is why like to have those results where can look at with nothing controversial.

One of my most important results follows from a generalized quadratic factorization but while there are important posts which cover everything, one of my recent favorites just focuses on the basic facts. For me can be comforting and will copy from it and link to it.

So yeah seen in LOTS of posts are some important things.

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 simplest example is: P(x) = x2 + 3x + 2

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)

Reference post: Simple generalized quadratic factorization

That post has nothing controversial. Is just a matter-of-fact stepping through a way to solve for the g's by solving for the f's using a rather simple technique. And I will at times just go over it.

Can feel weird how simply absolute mathematics can be. You have just these facts and logical connections to get to a conclusion, which is truth. And it just is. Like 2+2=4, like to say to myself.

Look at things now, over and over again, and just wonder. Is weird for it to feel weird. But sometimes, yeah, say to myself, I found that. And eventually, move away, think about other things.

The results are there for me whenever I need them. Whenever I want to wonder.

Which can also be shared for those who want that as well. Is just knowledge.

James Harris

## Monday, December 18, 2017

### Two squares and powers of 50

12 + 72 = 50

482 + 142 = 502

1462 + 3222 = 503

21082 + 13442 = 504

115162 + 134122 = 505

823682 + 940242 = 506
____________________

Result follows from a simple rule.

If:  u2 + 49v2 = 50a

Then: (u - 49v)2 + 49(u + v)2 = 50a+1

Which is just using my BQD Iterator. And u or v can be positive or negative, so I selected to eliminate trivial solutions. Let's divide off shared factors where can, and highlight a pattern:

12 + 72 = 2(52)

242 + 72 = 54

732 + 1612 = 2(56)

5272 + 3362 = 58

28792 + 33532 = 2(510)

102962 + 117532 = 512

So yeah, found myself curious again, where have already shown with powers of 10 and powers of 5 from those, and realized hey, can do 50 as well. And I do find it interesting looking back at overlap between lists, without seeing any different values.

The more general reality is that there always exists nonzero x and y, such that for an integer n equal to 0 or higher, and an integer m equal to 3 or higher:

x2 + (m-1)y2 = mn+1

Where yes, can extrapolate from rule shown above, as actually I used the more general form:

If:  u2 + (m-1)v2 = ma

Then: (u - (m-1)v)2 + (m-1)(u + v)2 = ma+1

My BQD Iterator is tool being used to get those values. And on this subject can see more with this post from 2014.

And am endlessly fascinated by it, even as I use it in what I see as marketing ways. So here use it to do sums of two squares to powers but have also used it to do arbitrary sums of squares to a square, as it shows that squares and exponents have this deep relationship.

These relationships control integers, whether you know of them, or believe in them, or not.

The math does not care. That infinite coldness of absolute knowledge has bothered me at times, but now I simply appreciate it, and all it can do.

The math has infinite knowledge. And the math can do so much where I like what has been shown to me. But I DO appreciate the knowledge.

The math knows.

James Harris

## Wednesday, December 13, 2017

### Simply lucky is best am sure

ENDLESSLY fascinates me that in the late 1800's mathematicians got lost trying to finally solidify mathematics into an established discipline. Which is also remarkable to learn as before was more of a hodgepodge of tools. And in our times I got to figure out what went wrong with an argument so simple, a quadratic will do.

And talk much as IS important, and they seized upon roots of monic polynomials with integer coefficients as being a good idea for members of a ring. Where monic just means 1 or -1, so for instance something like: x2 - 4x + 3 = 0

To show a flaw with their reasoning I need a generalized quadratic factorization.

In the complex plane consider:

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.

Simplest example is: P(x) = (x+1)(x+2)

Where I throw a wrinkle into the mix by doing something no respectable mathematician would do--I add an extra factor I call k.

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)

That extra factor is JUST to force one of the f's to have k as a factor, while I can force the f's to be roots of monic polynomials with integer coefficients. And if all seems very distant, consider means can show that 3-sqrt(-26) has 7 as a factor.

Have talked in detail with this post:

Easily explaining historical miss

Have given a more dry and purely mathematical explanation with:

Non-polynomial factorization short argument

Which is useful for well-trained math people, and it IS all very easy though. Was simply lucky that something this HUGE could be handled with a quadratic and a trivial leap that I force an extra factor which is something math people are trained NOT to do, and in so doing can blow up a hundred years of mathematical thinking.

So they did NOT solidify the mathematical discipline, but instead weakened it, but our species was ok, because most math we use was developed long ago, as I recently explained.

There I note that certain folks are simply not needed for valid mathematics. And yeah, when realized that years ago knew had a bit of a slog to getting these things properly recognized.

Of course have plenty of the demonstrated ability NOW for being the kind of person to figure such a thing out. Actually first came across problem with an entirely different approach using tools I developed, as created my own math discipline. Those same tools also let me do lots of remarkable things. And I found my own way to count prime numbers with a clever tweak on old ideas. Get tired quickly trotting out a list of things, so will stop there. Wow, yeah I have a lot can trot out there for being the person would discover such a thing, I guess.

But also has been lots of luck am sure in it all, though also, fundamental results often DO come from simple, is a good thing. Simple is so cool.

But we humans are clever by much I guess, while also can fall into odd traps, like NOT adding extra factors! While I'm NOT a mathematician, I proudly note, so to me? Why not? So I did.

When will situation resolve? Did you read that part about certain folks NOT being needed?

I'm just the discoverer. I do not make the knowledge. I simply found it, and often wonder.

When humanity feels like it, I guess. Meanwhile, I enjoy the knowledge.

There is such a thrill with discovery. Do appreciate knowledge gained.

Mathematics is endlessly fascinating to me.

James Harris

## Monday, December 11, 2017

### Working discovery reality

There are those things which can simply challenge points of view, like one of my favorite discoveries as connects to so much across number theory. Apparently it just connects integers to each other across infinities and with years to ponder, still find myself wondering. Is much about an iteration.

Given:

u2 + Dv2 = F

then it must also be true that

(u-Dv)2 + D(u+v)2 = F(D+1)

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

My reference source says I found it back in 2008, but just started calling it a Binary Quadratic Diophantine iterator in 2014, where like to call BQD Iterator for short. And it has a unique pedigree, as results from using my method to generally reduce binary quadratic Diophantine equations on:

x2 - Dy2 = F

You switch signs by shifting the sign of D. Where I found out that is the most important expression for integers in number theory, partly because I could generally reduce to it! Where of course can generally reduce:

c1x2 + c2xy + c3y2 = c4 + c5x + c6y

Where am going over old territory quickly as have brought up all before. So yeah, you have this simple relation at the heart of number theory, and can do SO much focused on one expression, like recently found a new primary way to calculate the modular inverse using:

x2 - Dy2 = F mod N

And key here is noticing how discovery propagates. So I finally used tautological spaces against a general binary quadratic Diophantine equation, and found a better method for reducing! Then that method of reducing at some point I used on a fundamental equation and got my BQD Iterator. And then I began to play.

One of my favorite demonstration results like to use lots now to just show:

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

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

Found with it.

Number theory is supposed to have numbers.

The BQD Iterator behaves like the core equation of number theory, clearly commanding much. Still I find I ponder.

Such power in a result draws GLOBAL interest and not surprisingly. And as the years go by will admit have thought more about a responsibility. Should I have it? I say, yes. Discovery doesn't make me but I can feel better that way. The math may not care, but I should. You end up feeling a responsibility for the knowledge which doesn't need me beyond discovering it, but feel like can help others in appreciating properly.

If you do not think a planet of humans notices useful working mathematics then I think that you do not believe in math. Web enables. Knowledge moves very well in our times.

Believing in math is an interesting thing. Either you do, or you do not. Actions tell.

There is a sharp divide there that intrigues me. If you believe it IS the math which is important then focus is there. If you believe actually other things are more important, others can tell.

For me contemplating a global attention reality has been daunting for some time. I wonder how I am to know how, as grew up just outside of Tifton, Georgia, USA. Growing up in a small town rural area of South Georgia has given me much, and I ponder how much works for me now.

It amazes me global attention. Like to talk about the math as like a friend as that is comforting for me, and my imagination makes it feel real. The math is the initial why of the attention and the math stays with me, no matter what.

To my mind when you discover? You present. And let others do THEIR jobs.

Today am a person known around the globe who while struggling with the attention has kept my appreciation for what drives it.

Is interesting that belief in mathematics though, even if does not matter to use. People can rely on math tools without understanding them, or face consequences without believing in them, but some also believe in the beauty of truth. And if you are such a person, is a good thing am sure.

If you believe in math, then you know what discovery can do. Not just for what we know now. But for all our species may ever know.

As someone who is thankful for the knowledge, am working harder to focus on being a responsible bearer of the truth. Besides, it also is SO much fun!

Discovery? Works.

I discovered the BQD Iterator, and am thankful for all it can bring, even though will never know it all. Can just be grateful for more opportunity opened up to our world.

James Harris

## Saturday, December 09, 2017

### When math is correct

Our math has developed with our species and greatly enabled much. Without math humanity could not exist as it does today. And thankfully much of what is useful? Was developed long ago.

Was very hard for me in many ways when began absorbing that the mathematical field had gone astray in the late 1800's as seemed so impossible. And am thankful for over a decade now to consider, which allows me to easily explain the mathematics part at least.

However, I think it is natural to wonder how that could be possible in terms of the human side of a very advanced world we think, where mathematics is so important to just about everything technologically related. Where I found a problem arrived in the late 1800's, when MUCH of the math that is used in science and technology came long before.

Like modern calculus was developed by Sir Isaac Newton who died in 1727, and Gottfried Leibniz who died in 1716 just did a web search. And what they were doing is WAY beyond what most people learn today, if they even get to calculus. Where like if you THINK you are mathematically advanced, can you look at a cable hanging between two electrical poles and calculate the mathematical curve you see there?

Pondering was very humbling for me. And consider, you can hold a rope loosely between your two hands, yet calculating that curve so many know? Requires advanced mathematics, which pushed my limits to understand, as a teenager.

Turns out that curve is a catenary. Figured out back in, yup, the 1600's thanks to the Bernoulli brothers. Where here's a link to article about Jacob Bernoulli. Where came across the problem as a teenager and was humbled when I struggled with learning a solution to the problem with variational calculus, trying to work through to understand. Which only made sense to me many years later. Felt weird to struggle again with math.

We're lucky that so much in math got well developed long ago.

When we look at mathematics from the late 1800's turns out gets specialized in very specific ways while yes, there has been much relevant to technology but that is applied math which has no problems of which am aware. But you find that MUCH of the prestige in modern math field is attributed to pure mathematics and MOST of it goes to number theorists who primarily claim their research is useless in the real world.

Is very weird situation folks, as when you find out they are full of error, you start noticing things. Like who was one of the most acclaimed recent mathematicians who tragically died in a car accident with his wife? John Forbes Nash Jr.

An applied mathematician who gained quite a bit of celebrity including a movie made about him, check his mathematics only awards. Yes, he got the Nobel in Economics but famously there is no Nobel prize for mathematics things directly. (Some think maybe Nobel had something against mathematicians, I think. Where did I get that from? Regardless, he didn't set up any awards for them.)

Used to rip on mathematicians about not giving him ANY awards and then they gave him one.

Related to my criticisms? I'd think not. But yeah applied mathematicians are immune from the problem I found, as it cannot help them. It actually means number theorists can gain success with results which appear to be true, which actually are not. Such a thing could only be useful in exploiting your fellow humans, if you knew of it, with certain kinds of things. Applied mathematicians do not have option of using it, even if they wished.

Studying behavior towards applied mathematicians from others in mathematics is useful.

It does not take much to get a grasp of the actual situation, once you get suspicious.

There is a muddled picture I think for most people as to them is just mathematicians and what is with this pure versus applied and useful versus not or whatever? And there is how people can get away with mathematics which is NOT correct! Which I studied first over a decade ago.

So how do YOU know?

Well I get suspicious as soon as a mathematical result is emphasized as being worthless for anything practical, does not lead to any follow up results, and is incomprehensible supposedly except to a few leading mathematicians in the world. They've switched up the value of human knowledge.

Today many otherwise sensible people take it for granted that knowledge can be abstruse, useless and only understood by a select few, which those who know their human history know in the past was attributed to priesthoods. The Catholic Church at one point I think emphasized to people that reading the Bible could mess up their minds and they should only rely on Catholic priests for guidance. One of the things Martin Luther brought back I think, as going by memory, is the simple notion of reading the Bible again. Which was part of the Protestant revolution where was recently celebrated over a half a millennium since his famous Ninety-five theses.

But are distant from math now, now aren't we? If I got anything wrong above maybe will catch it with editing later. Main point though is the religious aspect to emphasizing trust from the people.

Reality is math developed for its usefulness. And continues to so develop so humanity is doing just fine. Esoteric reality of number theory means most people don't know and don't care about it, even though there has been a practical aspect with computer security.

Which is a very different subject and I tend to shy away from THAT subject lately.

So yeah, turns out is easy to explain how there can be this bizarre error that entered into mathematics in the late 1800's which did not derail science and technology, but did shift mathematical community, to focus on people who behave like a modern priesthood. With true believers trained to simply accept their proclamations, they can gain adulation, reward and continuing employment, when people who understand how things went wrong in late 1800's can reasonably suspect they have few if any correct mathematical results. Is so wild. But they are simply not needed for valid mathematics.

Good news is mathematical industry is doing just fine, regardless. Our species continues to progress and no reason to expect that will change from this particular problem I found.

When math is correct? It works for us. And vast amounts of useful mathematics is doing so much for our world, no worries.

James Harris

## Thursday, December 07, 2017

### Explanation and unary two conics equation

Much to my surprise came across a simple explanation for size of integer solutions to:

x2 - Dy2 = 1

I like to call it the unary two conics equation, as depending on sign of D, you can have ellipses or hyperbolas. And traditionally people in mathematics like to attribute it to a guy they also say should not get credit, which is SO wacky to me. Why not just correct the error? I like name I give it better. And traditionally its integer solutions are of interest as solving it rationally was known to Fermat, who liked to play intellectual games looking for integer solutions.

And integer solutions can seem kind of random, if you don't know the why, like consider:

98012 - 29(1820)2 = 1 with D = 29

But next one:

112 - 30(2)2 = 1 with D  = 30

Giving smallest positive non-trivial integer solutions.

And turns out that key to size is D-1, and not D unless is prime. Which is the kind of mathematical wrinkle that can confuse humans I guess? Which is why wasn't just noticed centuries ago? But even if they noticed, how would they explain? I can explain why.

Also people have to be curious. I've tried to research this subject and apparently number theorists gave up on finding an answer long ago so claimed there was NOT one. So no way they'd ever find one, eh? So primitive to me, how some people choose to think a certain thing, in order to not learn, satisfied with a belief. So was declared no answer existed. Interesting.

Where lots of the easier solutions are going to be when D-1 is prime. And hard starts if has small prime factors, and harder the more of them you have. Like if D-1 has 4 as a factor and is not square? You start getting the bigger ones like example I showed, where also is key that D is prime.

There are also other things make for easiest answers, like trivially found solutions when D-1, D+1, D-2, or D+2 is a square, as well as when D is divisible by 4, for when D+4 or D-4 is a square, where kind of didn't even wish to give all that but figured why not for completeness. Oh yeah, so smaller integer solutions for those more boring cases as well.

The general result around factors of D-1 is vastly more interesting to me.

With explanation in hand just jumps out at you if you scan through tables of integer solutions. For me? Was so wacky wild years ago. But of course for me now is old news. And also, I discovered so my emotional relationship with the information is different.

The result then is that D-1 is the control in general for size of solutions, and its prime factors as to how small and how many are what matters. Turns out the math has to work harder for integer solutions if D-1 has lots of small prime factors, especially if D is prime. And hardest if D is prime, D-1 has lots of small prime factors and 4 is a factor, except for the exceptions with various squares, like if D-1 is square. Which then was the previously unknown explanation. Easy answer which is backed by mathematical proof!

Is slightly convoluted though when all the details are laid out. So yeah, D-1 is most important, but D impacts also if prime! And yet there are all these exceptions when you get easy, like D plus or minus 1 or 2, and yeah, I guess it could be hard to tell by looking over solutions trying to guess. And results could look random to a human without all the rules?

Where copied from this post used as reference. Have another post soon thereafter talking two conics equation size which is way detailed.

Is very interesting to me these relationships between integers which control things. There is that mathematical precision, of course, to the machinery of integers in relation to each other.

James Harris

### When people fear simplicity

How do you deflect human beings all over planet Earth from proper appreciation of important mathematical discovery and why would anyone bother? SUCH huge questions where feel forced to ask and find answers.

After years of focusing on mathematical discovery, for which am very grateful, have begun studying more the lack of appropriate acceptance of remarkable discoveries, which I call the social problem. And in theory mathematics should not have such a problem, as people can prove things with rigor unlike in most areas of human intellectual pursuit.

However, have noticed that today's humans who are into mathematics are programmed to believe that often a supposedly brilliant mathematical argument is too difficult for them to understand. And they are trained to think that some mathematical things are only available to a small group of experts to fully grasp.

Have experienced some of that training myself, and noted when talking about paint in a prior post that there is legitimacy to specialization arguments, as there's too much for any one human to know enough in ever subject. And we DO trust experts to gain specialized knowledge well in excess of what most need to know.

But I have focused on simple approaches.

And what have I noticed more recently? Training to push people to suspect simple approaches in mathematics! They're being programmed to dismiss such!!! Presumably then, they are convinced that there must be something complicated and convoluted as to why a seemingly simple explanation must be wrong or unimportant? I guess so.

Is so harsh though as if you think your hard task is to find some great math? When is more interesting after, as you find out your fellow humans have been programmed to dismiss such, and as years go by, puzzle much. The math of course does not change.

So have shifted to functionality yet again, like with my new modular inverse method. But even then there is that fascinating pause where I wait to see what happens. Like have experience with my method for reducing binary quadratic Diophantine equations. And over more than a decade with past things, can then be that puzzle to tease out. Is a fascinating thing, this social problem with so much specialized programming of humans. Which is very effective.

And very clever I notice. So modern people in the math community who see themselves am sure as intelligent operators take it for granted that simplicity in a mathematical argument is suspicious, and accept that they should not understand very complex mathematical arguments supposedly important. And even when given absolute proof, can be expected to question even THAT as supposedly only expert mathematicians have authority with them.

So the establishment mathematicians can tell them anything without concern of being questioned.

So obvious when laid out, as why bother programming humans in such a way, if your math is correct?

Am all for faith in demonstrated expertise, but the questioning mind should also check routinely, as human beings are so adept at disappointing each other.

But of course I have much faith in demonstrated expertise, as I can demonstrate with numbers and math so effectively.

It actually takes understanding human beings at a deep level to be so effective at manipulating them against truth. Why bother if math is your focus? Such techniques are also used by governments am sure. Are also used by religions, which I know as grew up indoctrinated in a fundamentalist Christian one adept in such techniques, and escaped it. Is also why take a longview as these things are very difficult to crack. The people at their mercy? Enjoy it deeply. It feels right to them in a way few things in life do. Is like a drug. Why would mathematicians have such techniques as well? Such psychological manipulation sophistication takes quite a bit of effort to master.

Of course I have given the answer there too. How number theory went off the rails as I like to say, in the late 1800's with a subtle error one might call diabolical. And people had to be controlled by practitioners to maintain. So a system developed to train them over more than a hundred years.

People so thoroughly trained that they get hostile to valuable new mathematical knowledge. You present and they treat you like you crashed a party to which you were not invited when you thought discovery was the ticket to entry.

With math that is actually correct I cherish simplicity. Emphasize mathematical proof, and like demonstration with actual numbers. For me such things are the comfort, not faith in some humans.

Am a discoverer though. Makes a big difference I think. I trust the math.

James Harris

## Wednesday, December 06, 2017

### My modular inverse method

Discovered there was a third primary way to figure out the modular inverse, which is NOT related to work by Euler or in any way to Euclidean algorithm.

Like to talk as a system, where that system is:

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

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

When using have focused on two coefficients as key in an equation with two unknowns so you have a degree of freedom.

Those key coefficients are given by: 2mF0 mod N and F0(r+2my0) mod N

Like with my detailed modular inverse example in this post, where calculate the modular inverse of 101 modulo 1517.

So r = 101, N = 1517. Then I get to make arbitrary choices for m and y0 where maybe future research will give more guidelines there. And choose m = 47, and y0 = 1 as is easy.

Plugging those in and you can get detailed calculation at referred post, reach:

295d = 499n + 162 mod 1517

Here you can see those key coefficients as one is 295 and the other is 499.

So with d and n, the two unknowns, I can just pick for one so that I can solve for the other.

What I do is choose n such that 295 divides across as I want to pick a smaller value, as yup, need another modular inverse. And show how to recurse with my detailed example. Ultimately finding that d = 190 mod 1517 and n = 112 mod 1517. And you end up with correct answer that modular inverse of 101 modulo 1517 is 751.

There is one area where people can mess up, which is if you make a mistake with modular arithmetic as to how to divide across factors shared with the modulus.

So yeah, is a full method, with which you can use in general to solve for the modular inverse, though practical details of a full algorithm are not things I focus on. But immediately realized, as someone with coding experience that you can multi-thread effectively.

With other techniques, when I look at them, is like, what's the point? But with my system you have choice, which can shrink those key coefficients, and also you can loop down paths which split by choice. And if have multiple such paths, presumably some will be shorter and a smart algorithm could maybe figure out better ways? Or you can just run several at a time and use one which reaches answer first, of course.

Is advancement of human knowledge at foundation level of number theory which is at the earliest stages. Is weird too. No such fundamental result was supposed to be still available to be found. Presumably ALL had been found before. Yet here it is, in our 21st century, less than a year old the discovery.

That is so wild.

James Harris

## 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