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Friday, April 27, 2018

Considering modular impact

Once focus is on modular methods it does become clear that you get a HUGE difference in terms of mathematics. And does seem pertinent that Gauss had started much. Also Euler had done some things where usually am thinking more about Gauss myself. And then we get this remarkable pause, until the 21st century.

And I found out there was another primary way to figure out a modular inverse, where Euler had one, and another goes back to ideas shared by Euclid, where tried to explain those simply. And have talked my modular inverse discovery much! As was a telling result for me, nearly one year ago back May 5th, and shifted my thinking on other things as well.

Now can show that x2 - Dy2 = F mod N, is the key equation controlling behavior of integers along with my BQD Iterator. But modular goes much deeper as my first major result back in 1996 relied on for packing of spheres. And even what I say is an axiom reveals lots.

Like p1 mod p2 where just consider primes modulo each other makes no sense to claim they have a preference. And THAT by itself resolves the Twin Primes Conjecture and refutes Goldbach's, if accepted. Where did an overview recently and also one can web search: prime residue axiom

When consider fights that have emerged in this area, they aren't even interesting to me.

Human beings can be weird in predictable ways with thousands of years of known history. There is nothing to learn there.

Yet is worth noting that there are recent mathematicians who tried to erase Euler from his zeta function and hand to Riemann, who am sure would have been horrified at the attempt. While I noted a simple explanation for that question he was asking--which I found back in 2002.

Modular methods lead to a true modular algebra, which can manipulate algebra itself, and in so doing give VAST analytical ability to anyone who learns. The math does better algebraic manipulation than any human can manage. Who knew? That is actually kind of freaky. And I found out when improved on Gauss with reducing binary quadratic Diophantine equations, which I did years ago.

Have also talked some of my history with modular with a Google Group posting on my MyMathGroup, where checked and found out yeah, can link to it.

Good news is that the math apparently has been picked up rapidly. Explaining how THAT is clear to me feels kind of complicated and also kind of awkward. I think has to do with emergent reality which is in process.

That web enabled reality have discussed before. Is kind of interesting I guess. So social problem is not that big of a deal really in one way. However is still troubling in many ways.

How does it impact me? Um, I don't talk that either, any more. Have discussed in the past.

So modular was the next big thing in mathematics, and waited for quite a bit of time, until the 21st century to reveal just how big. Makes sense though.

Human thinking keeps advancing in certain ways, for some reason there does seem to be a logic to timing often, as modular?

Well modular is perfect for an age with computers. Even my story actually involves web much.

Interesting.


James Harris

Monday, April 23, 2018

Knowledge reality in age of the web

Knowledge does move fast in our times. And readily admit rely on that much with my own research. So yes, can now say definitively that I had one of the greatest mathematical discoveries, when figured out there was a simple explanation for why count of primes connects with continuous functions like x/ ln x and that ROARED, through math circles.

Then other things didn't happen and our world learned some things, while most humans did not.

My assumption was to assume I was missing something, which is pursuing simplest explanation which actually worked out great.

Then in 2004, when demonstrated with a published paper that with existing rules a person could create an apparent contradiction from a coverage problem apparently that roared too, through math circles. That coverage problem DOES allow mathematicians to fake mathematical discovery should note. And world learned more, but most humans did not.

And I assumed maybe I was missing something, or who knows, maybe was even WRONG. Which was the smart thing to do, so I pondered: How do you know mathematical truth?

And I came up with a functional definition for mathematical proof.

Is is SO cool. And is it unintuitive that I assume that I'm wrong and get MORE? Is one of the best things ever. In math it is better to work with the possibility you are wrong than to ever rest on certainty you are right. As math can be SO subtle.

If you ARE right, try as you will, your math will handle every attack. Math does not care.

So yeah, web spread the information widely and I learned could simply use blogs, as less and less relied on other means like Usenet. Though DID go on Usenet through Google Groups deliberately in the past to maximize wide distribution and lessen censorship ability.

When one looks at how others reacted to my discoveries is not telling about me, but is about them.

And there is some naive I think. Mathematicians who do not realize they were accurately judged from then on by their behavior, even if also judging their peers.

World apparently judged as well but assessing is harder. Nations can be so cagey. There is a use for academics even they may not fully understand. Do they really need to be correct?

But we DO understand emotion with truth. Is so telling really. Which confused me greatly for years as I got this JOY from answers. I really did want to know. Took quite some time for me to more fully comprehend how could there be these people who did not.

Where lately I noted I increasingly was worried less about correctness when proven as solidified tools for determining, than with trying to handle my global attention reality.

But so much faith maybe from some in a world they never understood.

Like you think recognition is from awards? That's funny. Or maybe you think the press coverage is what makes things happen?

There was a time that I did, will admit, which is embarrassing to me now. But is better to be honest I think. But how was I supposed to know better?

Took experience for me to learn.

And yeah, if you think brilliant mathematics and great discovery is primarily about winning awards or getting covered by press people then you have no clue why and how math discovery works.

And should those of us who do know inform such people who do not? In the past I thought yes, with persistence. Now am like just share well as one can--and trust, as now am more interested in talking to others who know.

We know.


James Harris

Saturday, April 21, 2018

Better with simplicity and integer rules

Great advances in mathematics can lead to stunning simplifications. Which definitely makes easier to check. But can also fascinate as wonder about how our species learns.

Like with integers have quite a few results that show that key to their behavior is a single equation: x2 - Dy2 = F

I like to call it the two conics equation. As dependent on D, either gives hyperbolas or ellipses.

But even more important than the explicit form is the modular: x2 - Dy2 = F mod N

My most recent result MAY show how given one quadratic residue m modulo a composite, you can easily find another, by leveraging the power of the modular inverse. But haven't tested it, as is my hottest result yet, in many ways. But also is just a few days old where can take me YEARS to thoroughly test.

While will be a year come May 5th first posted then on May 9th more objectively talked a path from that modular form, to a new primary way to solve for the modular inverse. Feel like have talked it better with post:

My modular inverse method

That was made in December 2017. Does take time to process these things, where for me lots is emotional, which exasperates me somewhat. So I like to remind: math and emotion? Do not mix well.

Much of the power with integers apparently comes from an iterative form that I decided to call a binary quadratic Diophantine iterator, or BQD Iterator for short. And here is a key post:

BQD Iterator is a very surprising tool

The math apparently, as I still hedge, uses just a few simple tools to control all behavior of integers.

Such stunning efficiency with a few simple mathematical tools with power over all of number theory? Is the kind of logical thing which I find appealing.


James Harris

Friday, April 20, 2018

Power to publicizing knowledge?

For years have talked quite a few major mathematical discoveries. My original focus not surprisingly was in sharing with established academics where things did not go as expected. Of course figured primarily just needed to be correct!

And then ran into so much wacky when talked to the people I trusted to value mathematical truth. I was so very disappointed by mathematicians was able to reach somehow, like by email. And that disappointing behavior from them happened over and over and over again.

Maybe have been others all over the world similarly, like me, disappointed by the reality versus the expectations for people who supposedly should LOVE math and cherish mathematical truths!

The web allows rapid and easy distribution of information.

Have watched now for over a decade as has been clear the knowledge has spread widely. Which is also cool. And I do not think directly validates but stands to reason that important math would get used around the planet.

Like recently discovered a third primary way to solve for the modular inverse. Historical record shows that Euler gets one, and another goes back to ideas shared by Euclid. And is almost a year since I discovered.

My one published result, demonstrated how mathematicians can fake math in number theory. And that was over 14 years ago! Why wouldn't they talk it, except to keep being able to use it? Or to hide needed corrections if known? Do they really think few people know? Maybe.

Could go on and on with HUGE results. Am looking at behavior though not of people who believe they're known to be running from the truth. I think these are people who learned rules before the web, and are comforted by their ignorance of web reality.

I HAVE tried to share still with establishment people but they have apparently decided can just ignore me, which to me is naively believing others do not know. But what if you let them know you do?

How easy? Just tell a mathematician a somewhat cryptic phrase? But then again, maybe not.

If you DO know, just tell a mathematician: We know.

Am thinking about other recent movements on the web, and thinking of something that should be safe. I think. Who knows.

But my suspicion is that an establishment which believes that established media still rules does not realize the reality. Web rules.

We DO know.

Monday, April 16, 2018

Finding quadratic residues with modular inverse

Am curious about numbers often and find myself trying things. And realized had not yet used one of my innovative ideas in a way that looks possible. So figured may as well try a few things. And doing so public helps my process.

Have focused on x2 - Dy2 = F, as a fundamental equation which controls much of the behavior of integers.

Importantly, realized could solve modularly, as for some integer N, will be true that D is a quadratic residue, such that D = m2 mod N, for some m.

Which means have a useful factorization available:

x2 - Dy2 = x2 - m2y2 = (x-my)(x+my) = F mod N

Where then, for some residue r, x + my = r mod N, and x - my = Fr-1 mod N

And used this direction to find my own way to get the modular inverse, but am thinking could be used to find m, as well. Though why bother? Well, am curious.

Can solve for x, as: 2x = r + Fr-1 mod N, and 2my =  r - Fr-1 mod N

If I simply pick r, then get its modular inverse, and set F, then I have set x.

And with this path can simply set y = 1 mod N, and F = 1 mod N, and then find D, from D = x2 - 1 mod N.

For example, let N = 119, where I like to use composites for these types of experimental musings, as primes can be too particular. This approach does not care though if prime or not, so using a composite covers more territory.

And let's let r = 2 mod 119, for easy. As then also r-1 = 60 mod 119.

So: 2x = 2 + 60 = 62 mod 119, and x = 31 mod 119.

And 2m = 2 - 60 = -58 mod 119, and m = -29 = 90 mod 119

And D = 312 - 1 mod 119 = 8 mod 119

And yeah 902 = 8 mod 119, as required.

Well still ended up squaring x, and definitely does not seem practical to me. Will do one more easy.

So will go up one, and let r = 3 mod 119, again for easy. As then also r-1 = 40 mod 119.

So: 2x = 3 + 40 = 43 mod 119, and x = 43(60) = 81 mod 119.

And 2m = 3 - 40 = -37 mod 119, and m = 60(-37) = 41 mod 119

And D = 812 - 1 mod 119 = 15 mod 119

So of course: 412 = 15 mod 119

And my curiosity is satisfied, for now. Though does not look like did much more than look at examples of the modular factorization. Obviously you can simply square numbers and get residue modulo N, directly without taking this slightly convoluted path.

Of course I set F = 1 mod N, just playing around. But can set to minimize x, or who knows. Just playing around just a bit, so will not speculate much.

Oh, but you could use the BQD Iterator, to get another set of solutions for x, with the same D, if the same r holds. That of course shifts y and F. And then you can solve for m.

Wow that might work. Don't even have to do another square either, as have D. And could be a way to loop through all possible m's for a given D. BUT it could just give the same m, as before. And maybe r does shift when you iterate. Will not try it myself though. Just note it here, for now.

Of course would bring BQD Iterator up, as use it so much. But think is the key controller across integers. That thing is SO wild. Is like this super key to just about anything with integers. Where just have to use your imagination. And that I guess I have.

These kinds of things can be stubs, where later may notice something else which follows. So is worth it to have these initial musings public, as may lead nowhere interesting, but then again, could lead to something else really cool. And how would I know at this point?

Just musing on some math for me is so much of the fun.


James Harris

Tuesday, April 03, 2018

Knowledge recognition reality

Much has shifted in the search for knowledge because of the ease with which a person can share knowledge gained. Like on this blog have plenty of mathematical results. And can from objective sources get a feel for the global interest.

For this blog according to Google Analytics, which is a source have used for somewhere around a decade, there were visits from 66 countries so far this year. And that is from 746 cities, with people using 55 languages. Where just read from what the application told me as went to it for this post.

And thanks for the interest! Should acknowledge my appreciation to those visiting here from all over the world. My focus is on the primary reason being--useful information.

Of course there is also question of official recognition of some kind which with me has only happened once with a paper which was published, and then things got wild as a chief editor tried to just delete out after. But have told that story SO MUCH, as yeah worked at processing.

So how can there be such a disparity? Well the answer is that math actually moves best on USE and that was more difficult to track--in the past. Which meant establishment sources were the best bet for trying to guess at what mathematics was actually doing out there. But that was so much about a gathering of people most passionate--in the past. In our times? Is debatable what establishment represents.

But also in our times, if you have math ideas? And you put them on the web? Then conceivably you can get objective data from others who are trustworthy sources to get a feel for how your mathematics is moving around the planet. That does free a researcher from the establishment.

It has occurred to me the establishment may not like that reality.

Would rather have objective evidence than to me, empty accolades, as how do they know really? And who are they, often? Reality of attention to your mathematics from those who toss out awards or whatever are rivals, with grudging approval dragged out of people who should wonder why they didn't figure out instead. They can be forced to know YOU because of what you DID figure out that they did not. If they aren't rivals then maybe then they think are better than you, but with what mathematics?

Objective reality does not care about social things--at ALL. Either the math works, or it does not.

Either the math is an improvement or it is not.

Wonder if is still taught that reality. Maybe is too harsh to admit to students such honest rivalry or the brutal reality check that no matter what you do, if your math is not good enough it is like you did nothing at all?

The solidity potential of mathematics can make competition stark. Few winners with absolute truths, but so many trying.

In our times you can have some mathematical discoverer cheered by community, which is actually tiny if you consider a planet of billions, whose work just does not move outside some small circles. But his buddies, as is often a male, like him. How far will that make math move? If not actually useful? Not far at all.

Is just easier to check in our times.

In the past might have been enough for much more in mathematics--being well liked.

You cannot compete with someone like me on the accolades of people like you. Their praise is as worthless to me as yours, if your math discovery does not justify interest.

If you need praise but somehow convince yourself you are into discovery, then maybe you really should go into some other area, like politics? Or entertainment? Because the math does not care.

Does push mathematics as a discipline BACK to hard, objectivity without delusions of importance of celebrity.

Mathematics is about logic, consistency, and absolute truth at best.

Humans can make into something else for a time, and call flawed ideas mathematics, but what works keeps pulling back the body of accepted mathematical ideas to better. Math is just SO useful.

More and more I think people are learning to trust objective data and my story helps! We need facts.

Web helps there as well. Readily admit started checking more to be aware of what happens--after the news fades. Can be a pressure to just chase after the latest without watching people over time. Web is really great there, as long after news organizations have moved on, you can consider what a person is doing, or not. If they do anything on the web. But also you can usually especially with academics at least keep up with where are working, or not, if is at a university.

Reality is, the rush of attention can fade so rapidly. While for me? Have been pondering global attention levels for over a decade now. Is just SO steady which puzzled me for years. Realized had been trained wrong. Human interest builds over time with math. People keep building more things in actuality. So yeah, is the actual way.

The math becomes more in demand as is proven in the minds of more as they use, and the need continues to grow, as humanity continues to progress.

And you can have someone like me, who is global on a scale few humans can actually comprehend, which I say as have worked for YEARS at trying.

But yeah, I invented a new math discipline, where the math does algebra demonstrably better than any human. Discovered for the world a new primary way to calculate the modular inverse. And even figured out the 'why' of the connection between counts of prime numbers and continuous functions like x/ln x. Among other big things actually but just grabbed for a few quick cool ones.

Try to compare with someone else's discovery in the 21st century. I know I don't even bother.

So yeah, the Google Analytics data is probably an underestimate, which is ok. I like to use as a baseline.

Math moves when useful. Human interest is so much about what works. And with the web, in our times, you can watch human interest pull that needed knowledge.

Is a great thing really. Your discovery? Can MOVE, and YOU can watch.


James Harris

Monday, April 02, 2018

Where identity is key

One of those things did years ago, had me wondering why I bothered. But again shows how tautological spaces which are what I call complex identities can do MUCH when it comes to algebra.

Copying from my post September, 28, 2011:

...result that in general x2 - y2 = (x+y)(x-y), which can be constructed from within the tautological space as an intrinsic property:

x+y+vz = x+y+vz, x= -y - vz + (x+y+vz), squaring both sides, gives:

x2 = y2 + 2yvz + v2 z2 - 2(y+vz)(x+y+vz) + (x+y+vz)2

x2 - y2 = 2yvz + v2 z2 - 2(y+vz)(x+y+vz) + (x+y+vz)2, and setting v = 0, I have:

x2 - y2 = - 2y(x+y) + (x+y)2, which is: x2 - y2 = (x+y)(-2y + x +y),

so: x2 - y2 = (x+y)(x-y).

Source: Under the hood

Was like the coolest thing to me when realized was possible. Have pondered and wouldn't be surprised if you can build every part of algebra using tautological spaces, and in so doing relate everything back to identity.

So, given: x+y+vz = x+y+vz, you have x+y, and x-y, and a standard result. Follows logically from the tautological space itself. Oh yeah, also need some basic operations of addition, multiplication and need the distributive property. And the integers, should not forget. But realized also just need -1, 0, and 1, actually as of course from there can get the rest.

Reducing mathematics to identity? Resolves questions of truth.

That basis in truth then conceivably is the basis for mathematics itself: identity.

If so, then you just need identity and mathematical operations, and can build ALL of mathematics.

I like that idea. May as well make a post that goes ahead and states something else have been pondering for years now.


James Harris