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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.

Mathematicians instead weakened considerably.

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