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Saturday, October 20, 2018

Meta and my modular inverse method

Realized that self-reference is key to my way to calculate the modular inverse. And turns out meta just means self-reference, which is easily checked on web with search: define meta

My approach relies on focusing on: x2 - Dy2 = F mod N

When D is a quadratic residue modulo N, you can easily solve for x and y modularly, with D a quadratic residue of N, so there exists m, such that m2 = D mod N.

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

So x - my is a residue I like to call r.

And with: r = x - my mod N, then Fr-1 = x + my mod N, and the modular inverse makes its appearance. That is what got me thinking these expressions might be useful to find it.

Can use x = r+my mod N now, with x2 - m2y2 = F mod N, to get:

(r+my)2 - m2y2 = F mod N

Multiplying out: r2 + 2rmy + m2y2 - m2y2 = F mod N

So have simply: r2 + 2rmy  = F mod N, so also: F = r(r + 2my) mod N

Now will have F self-referencing itself with two control variables n and d, where F0 is initial F value, and y0 is initial y:

F = r(r + 2m(y0 + d)) mod N
nF0 = nr(r + 2my0) mod N

So can subtract to get: nF0 - F = r(n(r + 2my0) - r -2m(y0 + d)) mod N

Which is: nF0 - F = r[(n-1)(r + 2my0) - 2md] mod N

And now can use that nF0 - F = 1 mod N will exist.

And using that: 1 = r[(n-1)(r + 2my0) - 2md] mod N, and can get to my modular inverse with:

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

Notice then: nF0 = F+1 mod N does put some pressure on the math. For instance if factors are shared between F0 and N, then those would have to divide off. As yeah, F is being found so that these things are true.

And can again go to F = r(r+2my) mod N, where multiply times r-1 mod N, to get:

Fr-1 = r+2my mod N

Which is true in general, but now with my solution for r-1 mod N want to use with F0 to get:

F0[(n-1)(r + 2my0) - 2md] = r+2my0 mod N

Where can simplify to:

2mdF0 = [F0(n-1) - 1](r+2my0) mod N

Which is a lot more detail to add to what I derive here.

As a simple example consider: N = 101, r = 35

I pick m = 11, and y0 = 2 mod 101.

Then F0 = 35(35 + 2(11)(2)) mod 101 =  35(79) mod 101 = 38 mod 101

And now can relate n and d to each other:

2(11)(38)d = [(38(n-1) - 1](35 + 2(11)(2)) mod 101

so: 28d = [38n - 38 - 1]*79 mod 101 = 73n - 51 mod 101

So: 28d = 73n - 51 mod 101

Where a neat trick is that 28d = 73n - 51 will work! And n and d are otherwise free to be whatever so can solve for that, where notice can find with:

0 = 17n - 23 mod 28, so: 17n = 23 mod 28

So can recurse, looking for modular inverse of 17 modulo 28. Point of showing is to notice how you get to recursion, and to emphasize that with next iteration the modulus must be smaller, as now down from 101 to 28. Oh yeah, also then can cut at least in half, guaranteed--with each recursion.

That is because with modular can use positive or negative. Like as continue with the example notice how can go negative to further shrink things a bit with:

-11n = -5 mod 28, so: 11n = 5 mod 28

Yeah can see that n = 3 works there, which gives d = 6.

Plugging into our expression for modular inverse then:

r-1 = (3-1)(35 + 2(11)(2)) - 2(11)(6) mod 101 = 57 - 31 mod 101 = 26 mod 101

Also, notice that stepping y forward 8, so d = 8, gives:

F = 35(35 + 2(11)(2 + 8)) mod 101 = 35(35 + 22(10)) mod 101 = 35(53) mod 101 = 37 mod 101

Where picked those values because then F0 - F = 1 mod 101, so n = 1 mod 101.

And yup, r-1 = -2(11)(8) mod 101 = -75 mod 101 = 26 mod 101

So yeah, more than one n and d can work for the final answer.

Of course will give same answer, and indeed, 26(35) = 1 mod 101.

Decided should show that path that leads to recursion to highlight the full method to show why is complete.

Notice YOU get to pick m and y0, where notice m has to be coprime to r. This post was more me emphasizing the meta aspect, as is intriguing to me that this approach works by having the equation for F reference itself. Is SO cool.

That is so weirdly easy, huh? These are the kinds of things really make me wonder.

Our species could have gone through its entire existence and never found it.

That really rattles me more than I like to admit. Especially as contemplate more than a year after finding, in a world where some appear to think they can ignore major mathematical discovery.

With some people there is just no respect for knowledge.

To me? They're funny. I get to have the last word. Being the discoverer DOES have its perks.

My words will echo through the rest of human history. That rattles me even more. So I must state to help me accept. It is also a responsibility. Yeah I'm going to go back now to NOT thinking about that too much.

There are two primary ways previously known to calculate the modular inverse, where have talked that in a prior post.

But now humanity knows a third.


James Harris

Saturday, September 22, 2018

Checking me reality

Most of my mathematical research has huge implications which makes checking me on it easy. Which may seem counter-intuitive, so thought would talk a bit about how that works.

Better math should have clear indicators.

1. There should be things never before doable from it which we can now do--as a species.

Mathematical advance MUST mean humanity can now do more. That means you can check me on things human beings just could not do before something of mine came along.

2. In general, people tend to use things that are useful and to move to things more useful than prior.

So yeah is just common sense that if there ARE better mathematical tools then people will go for them. That is not hard to find if is happening.

3. Human emotion is telling by behavior.

Which is something I quietly check routinely. And it is just human emotional reality--why put your efforts in an area where you know you will NOT get a benefit?

With plenty of things now doable that were not, am hesitant in laying out things in THIS post I think are cool. Is not really about my opinion. Human interest of others is more interesting I think with regard to certain things. The math does not care, but people may.

Checking how people react? Why not? Some people may over-estimate their ability to hide their emotions. To me is kind of funny.

Where started checking in that way years ago. Web makes it easy.

So yeah, I have discovered things that allow the human species to do things it could not do before. Have looked at indicators showing global interest and also global use, because there are things that can now be done easily which were either difficult, or impossible before.

And have noted human emotional reality by doing some web searches on certain areas to try and get a handle on current research in number theory.

Is fun. Passes the time. Though admit for me also is kind of weird. People can be so wacky.

Math does not care. Mathematics tells the truth regardless of social consequence.


James Harris

Saturday, September 15, 2018

To what audience then?

Have wanted to move from posts which to me were trying to push certain people to acknowledge what am certain are important mathematical truths. But it can be hard not to be emotional, which part of me thinks is a path to being more convincing. However philosophically prefer to stick to the value of simply expositing the truth.

By the rules established my most compelling results for others should be around a published result. Where feel important to emphasize how wacky that got back in 2004. And recently emphasized was demonstrating how a correct argument under all the established rules of mathematical rigor could nonetheless lead to a wrong conclusion!

That result is as valid today as was back then, of course. And is simply a demonstration of how declaring the ring to be the ring of algebraic integers can lead to a result outside that ring despite starting with expressions valid in the ring and only using ring operations.

More recently defending that result found myself fascinated with a general factorization where have repeated over and over but also I like the math.

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

Can solve with some simple things: g1(x) = f1(x)/k and g2(x) = f2(x) + k-2

Multiply both sides by k, and substitute for the g's--to force symmetry:

k*P(x) =  (f1(x) + k)(f2(x) + k)

Forcing symmetry lets me solve after introducing a handle function: f1(x) + f2(x) = H(x)

Then have a defining quadratic:

f12(x) - H(x)f1(x) - kH(x) - k2 + k*P(x) = 0

Can now solve using quadratic formula but main thing is that with integer k, have algebraic integer solutions because is a monic quadratic with integer coefficients then.

And with k = 1 or -1, you are forced to have algebraic integer solutions for the f's and the g's. Ok well that's cool.

But move to other integers like k = 2, and g1(x) = f1(x)/k blocks one of the g's from being an algebraic integer if P(x) is irreducible as then have: g1(x) = f1(x)/2

(Oh, if get confused can use simplest P(x) which IS reducible to get a handle on what is happening.)

That is so devastating to so much of prior thinking in mathematics.

But with such elementary methods, backing up a published proof.

So why does nothing happen now to officially address and fix the problem? How can a mathematical community continue in error against such simple proof?

Will leave as rhetorical. At least is a pure math result, so had no impact on applied mathematics. Which means doesn't matter for our science and technology at THIS point at least. There may be applied mathematics down the line, yet to be discovered though from the corrected mathematics.

The wild thing then is that we now are looking at one of the g's which is still integer-like but cannot be an algebraic integer, so what is it?

Years ago I pondered. And pondered, and came up with something. That something is talked about in the first post for this blog.

These new to us numbers have intriguing properties. Or I say new to us, as in new to humanity in general, while now have been contemplating them for...how long? Guess really since 2003 before my paper.

Were the reason really for that paper. New numbers previously not catalogued. And a human species still not quite fully coming to grips with them.

But the math does not care. To the extent these numbers rule? They do not care what we think.

Will still work at shifting the tone as ponder what audience finds the math to be of interest, and push myself to give up on others who do not.

You cannot force true curiosity and why bother?

Mathematics can demand the most the human mind can bring to barely understand. Without interest?

There is no way the math will make sense at the outer limits of human knowledge without working hard for that understanding.


James Harris

Wednesday, September 05, 2018

Have been surprised so some math reality

Should admit if also for myself as DO read through my posts wondering at times, years later, have been surprised at how things have gone with some recent results.

Like last year was back almost to holding my breath with the modular inverse find. Luckily am smart enough to realize...ok, was VERY surprised. So to recap though have talked relentlessly back May of last year discovered there was a new way, MY way, to calculate the modular inverse. Explain modular inverse in overview here.

Yet here we are.

Is clear to me that mathematicians are going to play the ignore game--again.

Which is very weird. Also FINALLY trotted out a general method for reducing certain types of three variable quadratic equations. And figured hey with a UNIQUE result showing something number theorists couldn't do, maybe something would happen. Yet here we are.

Guess mathematicians are going to play the ignore game there too!

So what happens? People use the results. I can watch evidence on web search but also there is other evidence. There are things you can do with my math research that humanity as a species struggled with, without it.

Like in this case strongly suspect my modular inverse method can be made VERY much faster than anything prior. And being able to generally reduce even a certain type of three variable quadratics for the FIRST time in human history?

Well, we live in a three dimensional spatial reality, you know?

Oh well. Our species can be so wacky. What can I do about reaction of OTHER people to major mathematical finds? Not much. But have explained that plenty of times.

My primary job is just to make the information available, for the good of humanity. It's a really cool job too. However so far? Doesn't pay very well, monetarily.

You cannot make any particular humans beings be interested in math, in reality. Fantasy though?

Well lots of people are all over that. Real math can be VERY hard, to discover.

So I get to be the major discoverer who gets to deal with a variation on past themes.

Ok. There was a slot available. To me? Is actually maybe more interesting this way.

Am working on not being surprised is all. Now have covered enough ways to be a major math discoverer guess that's it.

Well eventually will get over my surprise. Actually doesn't change things much for me--as long as I'm not surprised!!! You'd think I'd know by now though. Like have over 16 years with my explanation for the prime distribution thingy. Still part of me was clinging to something which I find hard to let go.

Was a view of the human species that I held but must relinquish. We're just more emotional as a species than moved by fact.

You have to appeal to people's FEELINGS whether you like it or not. Oh.

Ok. Yeah needed to put that down. So can remind myself.

Maybe will get around to that--eventually. Is not like it's hard to do. Still part of me is like, really?

Are you sure?

Just bothers me on principle I think. More and more am into logic.

Part of me thinks I simply need to expand the problem space. But what's my motivation? Eh?

Knew that was coming.

Maybe my motivation is the same as always: I just want to know--more.


James Harris

Saturday, August 25, 2018

Simply weird math more effective

Have been gratified that one of my telling recent results has gained a bit of popularity on the blog.

Reducing:

c1x2 + c2xy + c3y2 = c4z2 + c5zx + c6zy

where the c's are constants.  And have talked how to reduce that in general to a form:

[A(x+y) - Bz]2 - Am  = (B2  - AC)z2

And that is simple enough. (That m is actually a simple function of x,y and z. Why don't I give its explicit value?) Is VERY easy. So yeah is possible in general to reduce a three variable quadratic. And near as I can tell that is new!!! But to me is just one more example of better mathematics where I have piles.

And have emphasized how makes easier with this post:

Quadratics easier with more degrees of freedom

Which also links to where those A, B, C variables are explained. And am wondering if there are other folks wonder as I do, how can stunning advance not get more visible attention like from established mathematical people?

Am curious. Lots of times through the years for over a decade have felt confident some answer or advanced mathematics would move things in social ways. And still looking for that to happen.

Have plenty of math which to ME is simply weird. It is also demonstrably more effective than prior math and apparently is dominating the world with use, but where is the chatter? Where is the celebrity? There is much mystery in that area.

Consistently apparently people in established mathematics have chosen to keep crap. Like my most stunning result still was showing how to write a perfect argument under established rules which was wrong under those same rules. Contradiction!!! Got that published over a decade ago, and things got messed up.

(Yeah you need to be seriously clever to figure out how to do it, and realize a bit of splash would be to get a publication demonstrating. I may be the only person in human history with THAT achievement on this scale.)

People apparently use my results, which is good. But then don't bother to do anything about the other, which I call the social problem. Was a reason distanced myself from mathematicians. I am NOT a mathematician. Am a mathematical discoverer. To me numbers are interesting and I'd prefer to know how they work!

Like what my published paper actually did was show that declaring in the ring of algebraic integers can lead to contradiction, as I could use mathematics correct in that ring which would lead to a result outside the ring.

Talked it all in what I think is one of the most important posts on this blog:

Easily explaining a historical miss

Where actually thought that might do it! Yet here we are. If you are a math person and you declare things in the ring of algebraic integers you are engaging in an error which was proven to be one, over a decade ago. (Does anybody though? Will admit, have not checked so don't know.)

My favorite expression to ponder in that area is beautifully simple.

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

Solving for g's in general is easily done with some substitutions, where one will seem superfluous, but is important. To solve with my approach we will need a new variable k, and two new functions.

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...rest of it is at the post:

Simple Generalized Quadratic Factorization

Like to copy from my own posts. So copied a bit there. So yeah elementary methods blow up prior established number theory which people kept on doing even though the things that do not work? Do not work. Where there is plenty of evidence is bogus math--if you look for it. I quit looking years ago as found was depressing.

While I piled on results like even found there was another basic way to find the modular inverse, which then of course is my modular inverse method and went on and on about it for a bit.

Situation of course is a HUGE advantage to me which I finally started admitting, and there is no motivation for me to stop mathematicians from doing fake research except being a good citizen. And have tried some things, but admit will not fully address again until 2028, if necessary.

Why is a huge benefit to me? Removes competition. Makes me bigger in human history. And other things as well where I don't like to explain. Things like not having to behave certain ways or lecture or even bother with academics really.

So why would people prefer fake math? Because it can be easier. Notice even with my simpler results the discovery was actually, well took centuries. With the fake math people can pile on fake results. There's an infinity of them. More than enough to support current number theorists--worldwide.

Yeah, my results probably remind how hard real math can be. Its discovery can elude endlessly. (Yeah I know that rhymes.)

If it weren't for my curiosity would gleefully leave the situation alone, maybe. Possibly am too cynical about myself. But it is relevant to me that at this rate I'm it for the early 21st century and without any real competition from scientists either, would probably be it completely. Like in the future there will be no reason to name anyone else as significant in science or mathematics during this period. Which I think is cool.

Is also kind of sad. But feel like is my duty to seize the opportunity. Is a very rare one. And actually should not exist at this time. So yeah my competitive nature relishes the advantages that puzzle me still. But who knows, maybe someone else will step up. Time will tell.

If weren't so curious, could just let it go and appreciate the gift that appears to be mine.

Ok yeah, am competitive. Why wouldn't I be? But oh yeah, much mystery here! People are behaving in ways that blow up a lot of fictional scenarios. Well yeah fiction around discovery is usually completely out of whack. So that is NOT a surprise. Guess could ramble on some more, but why bother?

Would just go in circles. Am curious. How is it possible? Yet regardless the attention that tests me. And the certainty of a future that...but how can I really know?


James Harris

Friday, August 10, 2018

Some things mysterious to me

Have been blessed with mathematical discovery can call my own, and able to share on the web have received a HUGE amount of validation in ways, without the formal things thought would happen. Which is ok with me. I do like the quiet. And feel like have been helped by being able to consider fields I opened up in my own time.

Yet the details about behavior from others is where find myself wondering at times, and is SO mysterious to me. Like consider:

u2 + Dv2 = F

then it must also be true that

(u-Dv)2 + D(u+v)2 = F(D+1)

May not look like much but have used it for so many things! Like consider this numerical example:

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

_____________________________

Where summed 7 squares to a square just for fun. Turns out figured out how to sum as many squares as you want to a square, though the results do tend to get kind of huge. For me is just playing around.

Also did this one:  (189750626/109552575)2 approximately equals 3.0000000000000000833

Copied and added bold to highlight here, from post: So much from one thing

Where noticed a unique case could exploit.

Now what happens if I try to talk my results with the math community directly? Well consider my encounter with mathoverflow where tried to answer a question. So I gave a method for in general finding solutions for something really simple for my research to handle.

Given is the Diophantine equation:

(x2 + ay2)(u2 + bv2) = p2 + cq2

You can find an infinity of solutions as long as c = ab + a + b, using BQD Iterators. Where if you follow that link it links to the post on mathoverflow and you can read over what they KEPT, while my answer was apparently deleted.

They'd rather not know how to do it, apparently. But that is mysterious!!!

What kind of sane humans would prefer to NOT know how to do some math? I find that behavior mysterious. And have noticed it appears to be a weird thing with certain people in the math community. Part of me wonders, if I could interview these people, what would they say? How would they explain?

Oh yeah, one of my favorite things is to get to number authority! Is fun to watch the numbers do as the math says they will.

Like with this example copied from here:

(922 + 962 + 1442)(70482 + 6882 + 1032+ 1720+ 24082) = 

15079762 + 1279202  + 1131602  + 246002 + 49202 

Where yeah, to me? Is just for fun, and it feels cool. Finally created an entire post of number examples, which includes this one:

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

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

Just like to stare at it for some reason.

People have to WANT answers. We can be very emotional creatures.

If are puzzled as to what I think is the significance of my examples try to find others that are similar--from anyone else, from anywhere in the world.

Far as I know they demonstrate unique mathematical efficacy. But am NOT a mathematician so yeah, if you know of something please share!

Can just comment if find something. I DO search for things but yeah may miss much that way.

And like with that mathoverflow thing where a perfectly good answer was apparently simply deleted? I think reality with people claiming to be interested in math is they may actually be trying to get something else. But what?

Pretend may be more fun to them as real math can be very difficult, especially to discover.

But for people who want answers? Here they find them, which is why this blog trends highly in web search am sure!

For the people who are serious about their math?

I have some math that can help.

And when you actually need the best method for real work because you are a serious person? Then thankfully there are answers where I can get indication that you exist!

But with certain other people though who claim interest in math who dismiss or ignore perfect answers? They are so much a mystery to me.

Does the math care? Nope. And I think most of us should not either. People can play pretend.

I would rather know perfect math and have the BEST knowledge available.


James Harris

Thursday, August 09, 2018

Some quick web search reality

Like to talk about web authority as kind of this new thing, when I say that authority is when one entity has information needed by another. Like you seek appropriate medical authority when concerned about your health.

So will give some of the types of web searches that I do when trying to check on audience reaction to some math result. Will do these searches now as well to talk them.

Web search: three variable quadratic reduction

Got my blog at #2 with a search in Google. However, importantly most of the other search results seemed unrelated though maybe the #1 talked the subject but will not click on it to be sure.

While in Bing got #1 and a link to my Google Groups where have posted a PDF. Curious, eh?

Web search: reduce binary quadratic diophantine

Still in Bing and do not get any of my math in the top 10, which...ok went further and didn't find anything of mine in top 50 search results. And got #1 in Google with that search. So can witness a case where Bing didn't link to it highly but Google did.

Your results may differ. However remarkably just about anyone in the world with web access who does these searches? Is likely to get similar results. Is weird, eh? (If not feel free to comment. Ok I rarely get comments. But reader will know if tries.)

Ok, how about more new? Web search: modular inverse discovery

In Google have #1 and #2 with that search.  And in Bing have #1 and #2.

So yeah the bigger the result the more dominance have often found, early. Like not even bothering with searches on my prime counting. Its dominance has shifted over the years and last I checked was on the wane. So there apparently is a certain amount of celebrity involved.

Hot items were my focus here with good reason! Wanted results that impressed, But in general the idea here is to get objective information from web authority.

And those are the web search engines I check routinely. And I try to do action searches which show a NEED which is to be fulfilled, like wanting to reduce three variable quadratics.

If wonder how web search works suggest do your research.

Web authority is gaining more and more influence around our planet, as when people need information more and more they turn to it. I know I do.


James Harris

Wednesday, August 08, 2018

With what impact? Should I do more?

For me the validation for my strategy was in research where am STILL giddy over my find just last year of a way to calculate the modular inverse can call my own.

Should show some math! Yeah just want to put it again--my modular inverse method:

 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

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

For over a decade have felt my primary job was simple--discover.

And part of meaningful discovery is correctness, of course. Then I just felt needed to share, and others would do the rest. Where remarkably enough finally recognized others have done much.

Results shared here have consistently trended highly in web search, as does this blog. Where noticed that DID depend on my efforts to maintain a certain tone, as well as explain, well.

So yeah have watched in the past where the blog would slip in search engines and would return when I made adjustments. Which is all about people who find things here--useful.

Have noted that in the past I sought publication. Where after one publication with a surprising and disappointing story, found that door firmly closed. Did not matter how correct my results, as editors made sure I knew, my research was NOT welcome.

Have made it very clear am NOT a mathematician. And readily admit have been harshly critical of the established mathematical community, with good reason. That will not change until that community behaves appropriately.

To me the results are what matter. And feel good to have them, and am confident that mathematical truth will keep winning out over time. But should I do more? There is no point in me trying to get published again, as THAT resistance has been clear to me. And am not interested in doing all kinds of things which I think ignore the mathematical reality, as if am begging for approval or trying to convince!

Mathematical proof is all that matters.

If there are people out there with suggestions though, am curious. Have LOTS of social media, as well as there is ability to comment here, or post on my math group. So yeah, people can easily suggest things--if so wish.

Here primarily will continue to try to focus on talking some math which I think is of interest, with a focus on an audience that appreciates it. Is just more fun that way, and also I think better respects the reality of demonstrated support, which web search reflects. And thanks to you, if one of those who has shared my math, and to any and all of such folks!

Occurs to me can be a lot of bravery there.


James Harris

Monday, August 06, 2018

When innovation simply surprises

Belief can have an impact on discovery where have a great example I think from my recent realization about the three variable general reduced form for the quadratic case.

So the general equation is:

c1x2 + c2xy + c3y2 = c4z2 + c5zx + c6zy

where the c's are constants.

And noted recently could show where was just as easy to find Diophantine solutions for the three variable as for the two variable case! Is kind of wild really. Where have had the primary argument since 2008 so recently just recognized it. Where I guess that's new. The general reduced form for the three variable case is just: u2 - Dv2 = Fw2

Where yeah, humanity has had the ability in principle to so reduce since November 2008, since I'm a human and I'd figured out how, but then wandered off to other things before fully realized.

The simple innovation to reducing a seeming VASTLY complex expression? I subtract it from a complex identity I call a tautological space, and that space does complex algebraic manipulations--for me.

Of course interested folks can go out and check if there is anything else out there to generally reduce three variable quadratic equations. Tried briefly myself and found nothing which does not mean is not out there. But if not, consider just how much it can matter to ignore discovery, especially if you are someone who needs, yup, to generally reduce a three variable quadratic! That is a pure and applied mathematical discovery.

Another favorite of mine where innovation simply surprised is around 16 years old, as thought about counting primes. And yup, noted as did others centuries ago that with a prime p, you can count composites by dividing by it and subtracting one. Like up to 10 there are 5 evens: 10, 8, 6, 4, and 2 itself. And up to 10 there are 3 numbers with 3 as a factor: 9, 6 and 3

But was like, why bother counting that even 6 again? And figured out the math will let you automatically count at each prime excluding counts from all lesser primes, which gives an innovative form for the composite count.

Going to copy much from a prior post, and add some highlighting with bold, but main thing is, notice the simple mathematics! But that math does what I just noted above--counts composites at each prime without bothering with counts from smaller primes.

(Should note below that [] is the floor() function. For example [10/3] = 3. So is just dividing without bothering with the remainder if any.)

The main workhorse is the dS function though which is where I had to do the most work in figuring out its form:

dS(x,pj) = [x/pj] - 1 - (j-1) - S(x/pj, pj-1)

where S(x/pj, pj-1) is the count of composites that multiply times pj to give a product less than or equal to x, where notice that pj must be less than or equal to sqrt(x) or the composite count given by [x/pj] - 1 will not be correct.

So the dS function is the count of composites for a particular prime excluding composites that are products of lesser primes. So it gets a count of integers with a prime as a factor, subtracts 1 for the prime itself, and then subtracts the count of primes less than it. And finally it subtracts the count of composites multiplied times that prime.

Reference post: Composite counting functions and prime counter

Where not only do I get a simpler mathematics for counting primes, weirdly enough, but it also leads to calculus and an explanation of why you connect to continuous functions.

Mathematics is logical. We humans can BELIEVE something is hard, but does not make it hard because to the math? Nothing is hard.

To the math all mathematics just is. There is no such thing as hard or easy from the perspective of, the math.

One advantage I think I have is a healthy disrespect for our abilities as human beings.

We cannot compete or compare with the math. We can only learn from the math.

And for us humans belief can matter much! Like I wandered off from a three variable reduction for years thinking wrongly that x, y and z were harder than just x and y. But had so much fun like with my BQD Iterator that I don't regret it.

While with my prime counting function lucked out that I didn't even bother with what was known before struggling at figuring out my own way, and found that my common sense and the math agreed!

The math DID have a way to do something that made sense! And it dawns on me that maybe there are humans who do not respect infinite intelligence.

We can figure out some things, but the math knows ALL. If something makes sense to us? If it actually does make more sense then there may very well be a mathematical way.

But you have check to know. The math does not care what we know.

Glad I checked. And humanity gets more tools as a result. Is win-win.

So why does one human out of BILLIONS find such things? Who knows really but I think is because I believed simple solutions might be there, so I went looking for them. Is interesting how much belief can impact our lives and determine what we know and even what we can discover.

Better thing I think with math is--try something. Especially as can do so quietly, if only just curious.

And who knows? Maybe you will find something regardless worth sharing. But even the exercise can be good for the mind.


James Harris

Monday, July 30, 2018

Importance of certifying authority

Have various claims like I claim to have a degree in physics from Vanderbilt University. And also claim to be a US military veteran as served some years working primarily in medical centers. Regardless of what I claim though, much depends on what others certify, where in those cases is straightforward.

So Vanderbilt University is the certifying authority as to my physics degree, and the US Government is the certifying authority about my status as a military veteran. But what about with math?

Let us say for an analogy that I show up at some agency of some kind and wish to get benefits as a veteran, and that agency checks and because of a mistake the US Government does NOT certify my status as a veteran.

Then of course I would not get those benefits at that time.

And what could I do then? Could protest. Might claim is a mistake! But would have to work to get the proper facts certified to have anything happen.

Which of course is a scenario that has not happened. Just found relevant to consider what I might try if it did.

In actuality, when needed both the US Government and Vanderbilt University have certified. And thinking about this post realized I took it for granted! But now I don't. Is a deep thing really and much appreciated.

And for so many am sure relying on a certifying authority is something that happens routinely for them.

With mathematics the certifying authorities are leading mathematicians around the world. And presumably when presented with an important major mathematical discovery they would certify that discovery as one.

(Should at least mention web authority, but also to note to my knowledge is not officially recognized as a certifying authority. However is extremely important in my story! So deserves this aside.)

Notice the mathematical discovery should drive the certification as the knowledge is what is important.

If leading mathematicians do NOT certify a result as a major discovery then reasonable people can presume that it is not.

So when you work through how things are supposed to work, you can realize my job does end with a mathematical discovery once shared--if I've made one.

One reason to make this post is to clarify my own position that refusal to acknowledge my major results is a social problem which I will not address directly until 2028. That has to do with audience for a particular post. Because was concerned was making posts which were trying to be convincing to certifying authorities.

Instead now my posts are meant for people who can check my math for themselves who also might wonder how discovery can not be properly recognized. And reality is: I wonder as well.

But main point is, is not about me there.

Also helps you understand limitations on what I can do with such behavior.

So will not address people who SHOULD behave certain ways with mathematical discovery until 2028, with posts here. Which is just about the intended audience for my posts.

The behavior has gone on for more than a decade so yeah, I should think also about others who may wonder about it. However posts that talk certain things should NOT be making a case or trying to be convincing to math people who I think are behaving badly. That is what is not to be done further for now until 2028, if necessary then.

Have put LOTS of my explanation out already though, where tried to put all under label: social problem

So if curious for what have already written on subject clicking on that label below is a way to get lots.

Which lets me talk things as settled when I know they are. And discuss my mathematical discovery in a more fulfilling way.

Is more fun for me that way, can be helpful for others, while also lets me talk about whatever, as long as am NOT trying to make a case. Am at best sharing facts as I know them to an interested audience.


James Harris

Friday, July 20, 2018

With what ambitions?

Back summer 2002 was pivotal for me, because quite a bit had finally happened. Like had mysteriously to myself even, found myself suddenly curious about counting prime numbers and within a few weeks had my own prime counting function. But then also thought had finally figured out a way with my complex identities would call tautological spaces to prove Fermat's Last Theorem.

Luckily for me my way to count primes was bigger than I realized then, and also soon would discover there was something VASTLY bigger than Fermat's Last Theorem which would take me well over a decade to fully understand.

To me some of the most fascinating mathematics can ponder endlessly where key to understanding is so simple really. So much is easily proven with a relatively simple factorization using only elementary methods.

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. Then would use: g1(x) = f1(x)/k and g2(x) = f2(x) + k-2 and get to symmetry:

k*P(x) =  (f1(x) + k)(f2(x) + k)

But would be YEARS before had that simple route. And am SO GLAD as to me the emotional impact is the hard part, and now have had over a decade to process many things. So yeah, the math turned out to be easy.

But got a publication early, after a bit of work!

So yeah did the pitching to journals for a bit, including trying to get an editor who was at my alma mater Vanderbilt University to publish. But he would not, even after I visited to explain in person. Yeah was disappointed there. But did give me a reason to visit my old school for a bit.

Persevered until found a journal which would consider and send out for formal peer review. And of course by early 2004 was trying to get a handle on actually getting published, and THEN so much very strange occurred as the chief editor went wild and tried to delete out my paper.

But yeah, where was their ambition? Have talked one of the BIGGEST errors in the history of intellectual thought. Why weren't math people lined up to be a part of history in the making? And it gets more interesting with the story of a math graduate student have talked with:

My reference social problem example

And yeah still going back to 2004, and over 14 years later now can get perspective on what those people probably realized back then--would be a hard fight.

What is fascinating about it to me, is that in my opinion, you get a very pragmatic and objective assessment of the mathematical community that proof would NOT be enough. And even that publication in a formally peer reviewed mathematical journal would not be enough.

Of course that is naive. Mathematical proof has the characteristic that it will not change. And NOW can get some perspective of how can simply grind against social resistance, with relentless patience.

Yet will admit has fascinated me that getting a place in the history books even if took so long might be after you were gone was never enough in case after case. Where have noted my shrug, and admit made things easier for me! I would not have to share credit.

But it is SO weird. For years would rush to get to results and wonder if my habit of sharing relentlessly might lead to me being upstaged! Those fears turned out to be unwarranted. I would be allowed to leisurely make some of the biggest discoveries in the history of mathematics, without competition.

And as I piled on results learned to be less worried myself about where things would go with mathematics. You just have to believe in it, until could also move onto other things.

Am NOT a mathematician. These days find am involved in a lot more than mathematics, where plenty of other things are extremely challenging as well. Guess need to keep myself occupied? Is more fun that way.

Relaxed quite a bit more with my find of my own modular inverse method. Which is what allows me to raise that question again, of how could intelligent people within the mathematical community get things so wrong about how things would go?

Was as if, time after time, people who could check, better than me for some time actually that I was correct, behaved as if the truth would never matter. Which to me? Said they had no faith in mathematical proof against social things. They must have firmly believed that social would rule out over mathematical truth.

But how could they learn such a thing? To me seems so bizarre.

Could go on and on about what is a deeper mystery harder for me to resolve as involves other people. Have theories. But yeah for quite some time was confident certain things MUST happen once proof was released to the world. What apparently did happen was people used the results. Which is cool. But so much else didn't which is a mystery to me readily emphasize.

However, people who are REALLY into something will get excited over their favorite subject. Like imagine Major League Baseball and a relentless homerun hitter. Fans could not contain their excitement. Other ballplayers could not ignore the competition. And sports is a ready reference to me.

Clearly have faced people just not that into mathematics in reality, regardless of any professional label or personal claim.

So in short? I realized, were not really mathematicians. Well that was helpful to me as became my opinion. Simplified things for me. After all, true mathematicians would be excited about mathematical truths regardless of social consequence. And would quickly have been serious competitors with advantage of being in their own area. For me had training in physics, but even then only undergraduate.

Is a mystery to me, will admit. Without questioning folks, wonder how could get an answer.

Yet here we are in 2018, so am looking at facts over 14 years later. So yeah, even on ambition alone is remarkable how clearly there was a consensus that mathematical proof was not enough to guarantee a place in history.

So fate would dictate would be my place, alone.


James Harris

Thursday, July 19, 2018

Talking some celeb-reality

Realized years ago needed to not myself be a source of misinformation about my own story. Which is why try to be careful and objective, with a focus on facts. But had to correct how I talked anyway as realized believed things not true from growing up in the 20th century.

Most of what I thought I knew about celebrity was either a complete fiction, or was just wrong.

Now I say celebrity is JUST the conversation in some excited way around a person, thing or topic. Which necessarily dies down rather rapidly, but can be re-energized with something new.

In contrast, my ideas draw attention from usefulness. But you know it's like Euclid is one of the most known names in the world, but how much celebrity? Very little. But there is some. Reality though is most talk of Euclid is about people forced to take geometry in school.

Consider Euler in contrast, where there is not that kind of thing. And most celebrity around Euler is with math people, especially mathematicians.

Web authority has emerged as very important in our times and have talked some. What will add here as maybe useful is that this blog is one of my most linked, but my greatest objective reach is with my open source project which is one of my least. The search domination of Class Viewer has been there for over a decade and according to Google Webmaster Tools is one of my least linked things.

Will not give numbers but Class Viewer is an order of magnitude smaller in reported links than this blog. Have puzzled over that for years. Feel like am getting a better sense of why.

To me the total number reported of links is irrelevant as is link quality that matters. That is, it matters WHO links to you more than total number of links. And have watched things grow in reach as links drop, possibly as weaker links are eliminated.

My own analysis of impact of negative statements is they merely drive reach, since the truth is more important. And at my current level has dropped to almost zero impact. Which was kind of a relief. In the past I over-rated the impact of any saying negatives, but turns out people do check.

Almost never notice anything negative about me or my research any more. Am sure could go find, but why bother. As have noted, people who draw attention on a global level get negatives as well.

And guess that is more of what will talk. Have admitted my global attention level has been challenging for me. That is both from the idea of it, in terms of the numbers and with having to deal with other things related where I do not discuss in detail but are actually positive

So yeah, you get globally known then that gives some obvious things, where money is not directly connected. To make money you generally need an agent who takes a percentage which I despise as a system. And would just as soon innovate to something better.

For years funded my own research simply working at what I'd call a day job, which to me was best. But unfortunately have found my global level does actually mean businesses take on a risk with me working a regular job.

Is unfortunate to me.

Have become more chatty about certain things. For me so much is in the challenge and the result. Like say I rework the entertainment industry and eliminate the current agent system? Wouldn't that be something?

Yup. Am confident THAT is what is in process now. But is a web reality. Web is figuring it out.

Once established then I can just make money with the new system the web is still building. Which will be really exceptional, when arrives.

As their money collapses, am confident certain people will either adjust and recognize, or will simply disappear as with money people? Money really is the only leverage they have.

To me is unfortunate needs to be this way. Spent a lot more time than was necessary looking for a different solution. To me so much was about me being more reasonable than others realized.

My own guess is that soon enough, we will have global changes that will eliminate how agents have operated in plenty of places from the entertainment industry to the press as well, and others where that unfortunate system of a percentage of the work of someone else has held sway. We will all be much better off, am sure.

Celebrity as it existed in the 20th century is quite simply--going away.

The web is bringing in more new and better, where web authority will rule much.


James Harris

Wednesday, July 18, 2018

Some things settled in my mind

As much as have talked being unsettled by many things including my mathematical discovery, it occurs to me could be helpful to others to talk some things that are SETTLED in my mind.

One thing that is officially settled in actuality though have noticed people think there is room for debate is my one publication in a mathematical journal, where yeah, always feel a need to also inform things got wacky after. So yeah AFTER publication of my paper the chief editor went against protocol and tried to just delete out of the electronic journal, which left a gap where he lied claiming my paper was withdrawn. I say now was a lie as implies was something official when was not.

The journal managed one other edition before it shutdown. Its hosting university scrubbed all mention of it (last time I checked), but my paper lives on, along with all the other papers published in the over ten years the journal was active. Saved though by Europeans.

And was surprised to notice a post thanking certain people trended higher here. Link to it so is clear and for future reference as currently is #1 on the blog. Which to me? Came out of the blue.

However yeah reality is, you cannot have major mathematical discoveries without people noticing, I guess. And they drive a LOT of things which others can notice. Like that is how this blog by its name is high in search results. And is a large part of my global presence which I have noted is on par with a university. That is a pure merit reality enabled by the web, and I have NO control over such things.

Oh yeah so thank you to people all over the world who I admit are still mostly a mystery to me. But you are also heroes in my story. Thanks to you could get other evidence besides just my ability to prove that my ideas were important. One does ponder what is most important, and yeah, mathematical proof is, but how do you know for sure? One's mind can play tricks on you.

But being global is such a massive thing has taken me years to process and am still working on. Which is something only others out there around planet Earth could give. Is never something I could take.

Can just present my ideas like my math but cannot make people do all the other on such a scale.

What is settled then is a formally peer reviewed and published result, which was to demonstrate a tragic flaw in established number theory that lets someone produce a fake paper that appears to be correct with full rigor. Where paper was dated in 2003, while the journal had dragged a bit in publishing in early 2004, then all that wacky happened.

Is a shock. Was not surprised when there were mathematicians who clearly simply ran away from the result. And some may wonder: with what impact?

Reality actually is, biggest impact in my opinion was in me not having to share credit for things. (So selfish, eh? Yup. But figure should be honest there.) Otherwise there probably was a lot of political impact, and probably biggest impacts were social and on the entertainment industry. US economy probably lost a few billion dollars or so, and mathematical discipline will shrug as have explained.

As a major mathematical discoverer, will admit, to me? I like that not sharing credit thing for the most part. And also was so HUGE to me to get my own result like my way to calculate a modular inverse. That puts me in very rare air. Can legitimately be in a club shared with Euler and Euclid on that result alone. Which JUST GOT last year. For all I know? If ANYTHING prior had been different either I might not have gotten and humanity might not have known, or someone else would have that honor. So am like, is all good. Am ok with how things have gone because have gained so much.

To the mathematical field will all be a blip in history soon enough. Mathematics is just so HUGE really. Have a greater sense of the immense scale now than had before.

So yeah for me is all cool really. Some may notice places on the web where there are disparaging things said about me or my results and is, a shrug. For any global person can probably find that to be the case. Is just how humanity behaves, or some humans behave badly should say.

For folks who can check the math, and who understand web authority which is a relatively new concept, then there are obvious things. Like no one gets status at my levels for nothing, or for wrong things either. Just look around, and try to find someone if you think otherwise. I can shrug at major celebrities. And think to myself, wow, so that's one way to get attention, for a little bit. So cute. (Yuck. I actually DO think that too.)

Who failed most massively in this saga though? Some might think I'd say academics. But I say governments. Which is so much political. So to me, there is much settled. Where the math is the easy part.


James Harris

Tuesday, July 10, 2018

Scale of mathematics impresses me

There are many comforts for someone like me, especially as appreciate just how important mathematics is to the human species. It does help someone in my peculiar situation, relax. As intense as some things have felt, in the big scheme of things are barely a blip in mathematical history.

Eventually the corrections to mathematical disciplines where needed will be in place. Mathematicians will confidently move forward with greater discovery, and so much now that seems so big, will simply...be dismissed.

People who were thought to be something they were not will for the most part be written out of human history, and so much will be a few sentences in the record.

It does bother me just a bit will admit. You think about massive amounts of human effort over a period that seems long to us, but is less than two hundred years. So much will be reset.

Web is already starting in other areas too. History books of the future will look so different in so many areas, as the truth becomes more dominant.

Web enables so much. Makes it so much harder to hide certain things.

To have my place in that picture of the grand scale of mathematics and human discovery in general is daunting. Years ago began working with that in mind, not concerned so much any more about the moment, but looking more often to the future.

Mathematics just does not care. It really is that simple. Still I felt so guilty with the truth. Still find it hard to comprehend at times. And is not my fault. Mathematical truth is just what is correct.

Mathematics is bigger than us all. Greater than us all.

And some of us get a little bit to write in that story. Is ok. It has to be.

I tell myself that it must be ok, and then I feel better. I DO care. But what good is that really?

Does my giving more time, work better? (And did I really? What did I control?) Or was it simply my worst conceit?

History will judge best am sure. Let future humans figure it out. I did what I felt was best.

So maybe I gave time, and in so doing changed the course of human history. But with what result? Time will tell.

Reality though was handling as much as I could as fast as I could. And still could use more time myself! Maybe in reality was more selfish than I want to admit.

And humanity is ok anyway. Of course our species is. Is greater than any one of us, as well. Or even all of us, currently alive.

Humanity is greater than us all.

Still we are needed. If for nothing else, for future humans to be born.

Maybe this post is to release the guilt then. And to admit, there was mercy shown. But was from a human, me. But then again part of me just doesn't think was mercy really. Was just me running away from the pain of others, as best I could. And round and round I go.

Is hard really, balancing things in my mind. I decided that the information is available. Those who were still misled had opportunity. The best mathematical minds, I expected to find the truth.

Web enables. Becomes a test then, I'd tell myself. The best will pass.

But yeah, so often just did not want to deal with the pain of others. Born into a world which would teach them an untruth, and now is a lie. And what chance did they have?

And what about me? Why do I get to be the person? And why so alone with so much? And I DO care, am sure.

The math does not care. The math is just, perfect.

Oh well, this post is moody much. But maybe it is time for me to address things I'd rather not. Did I play God with the human species?

Am confident I did not. But yeah, I made certain decisions and more than I prefer to admit, people live in a world that is a result of some of them. But THAT is about the power of mathematics, and was also a surprise to me.

How much would be my decision, alone.


James Harris

Friday, June 29, 2018

Relief on academia

Am relieved to be back to feeling that academia on the whole works ok. And feel like gained quite a bit from my education including my four years of college for my degree in physics. But also am looking at a current situation which highlights I think where academia can fail, but shows how the web is helping much.

The situation in mathematics with number theory is in MY OPINION, a total fail, where a demonstrated flaw with number theory has not been properly addressed now for over 15 years. And 14 years since a published paper with a wacky story that I think is telling. But mathematicians were weird about things anyway. ONE of their big claims was that their field was immune from upheavals, like had happened, ironically in fields like physics.

They were wrong.

However, it is of interest to muse a bit as am thankful to be back to being confident that MOST of academia prizes best knowledge. Which is probably mostly correct. So is interesting to consider how one of the most logical could go so wrong.

And yeah, mathematics is potentially dominated by logic and what can be proven. But what if people knowingly claim something shown to be false is actually still true? Then what? Oh well academics are not SUPPOSED to do that right? But what protections if they do anyway?

And there is the problem. Possibly we're lucky then that a problem can be shown where absolute proof lets us know, and then we can puzzle over why humans would behave that way.

We are communal creatures. Human beings can easily believe things NOT true for the sake of community. While knowledge can be very disruptive.

So with my situation, have demonstrated valuable information which is needed, and web let's me know. Web lets others know as well. But is like with my improvement on binary quadratic Diophantine reduction. People can just use the best and STILL support the current math community.

So you get an academic version and what people who want to actually best solve a problem use.

Which is the great news actually. As attention is indefinitely and relentlessly gathered to the correct result that will dominate over time. Web just lets discoverers watch, which is different from the past.

Means you can relax about many things. And is very wild actually though I try to talk casually. But more and more appreciate how things draw attention. And yeah, reality is MOST people are not listening direct to academics in any area. If academics try to force that one, notice what happens. Reality is at best you might get a snooze-fest. People just do not want to be forced to hear them. Though occasionally some popularizer may gain a bit of celebrity, but in math? Not any that I've noticed.

And even among academics in a particular field, most are unlikely to pay attention much to any particular academic at any given time.

In our times, appears that pursuing the traditional paths is often a way to be lost in the herd. You will never be noticed much. (But then again, why would you want to be?) Whereas here, look at what I have.

For someone like me there is a 24 hours a day global attention reality enabled by the web. Am POTENTIALLY known in over 100 countries. I find that troubling often. However that is checkable from objective and third party sources.

Reality is, am personally, bigger than most math departments can ever imagine being. And can conceivably rank with the biggest and most prestigious on the planet. But of course, I like to joke, they'd be terrified to invite me to talk my math. I'd crush them on every level. Which is sad to me, but true. They cannot compete with me. I wish they could. I wish they would try to be better.

Sounds so arrogant, but is the truth that explains so much behavior. May as well state it. Is the socially disruptive reality. Have thought about for over a decade now. Still find it hard to process. But is much easier now than before! So there is progress. I simply have some of the best math ideas out there right now. Still just feels so, off. And is easy for me to watch my ego just go wild.

It IS important for me to try to stay grounded. And I make myself rely much on data from others. Where there has been nothing like that ability to check in human history before.

And there is just no way to compete against that attention reality and there should not be. Is a pure merit reality.

Academia has long been built to give advantage to establishment people which I think will go away. We DO need institutions dedicated to bodies of knowledge and a best process for bringing new people into those organizations and taking care of them.

But from my own experience I know you do not need to reward people to convince them to discover.

Worse you can end up with those who pursue the rewards and rely on fraud, and work to hide that reality. So the reward can be counter-productive. For me is comforting to just be outside such things. And has been important for instance that I am NOT a mathematician.

Academia should focus on best knowledge without inappropriate credence given to assumed authority bestowed upon humans, as we know better now.

The knowledge is what will last. And how do we know really?

Truth demonstrates.


James Harris

Monday, June 25, 2018

Some of my explanation

Mathematics is one of the most prestigious areas of human intellectual activity. And it makes sense to step carefully when making challenging assertions, where will admit have bounced around as to how I talk certain things. Here will give my own explanations for what I think has happened around my own mathematical discovery.

Like to me one of the coolest things ever is:

Where pj is the jth prime:

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

That summation will count primes if you make sure n equals the count of primes up to sqrt(x), but no higher. And an example of what it gives is P(100,4) = 25.

Which I copied just now from one of the posts where I talk my prime counting discovery.

That result leads to a partial differential equation and is part of the explanation of the link between the count of prime numbers and continuous functions. Which makes it one of the greatest discoveries in the history of mathematics, in my opinion. So now I hedge.

The lack of immediate pickup--and appropriate cheering--from people who claim to prize mathematical knowledge is easily explained, by a result I found later. I figured out could make an argument that looked like a correct mathematical proof by the accepted rules, which could also be shown to be flawed--by those accepted rules. And the paper is dated 2003, but actually was published in 2004, where things got wacky after. Give a good recent overview I think with post:


So what gives then? Well I figured out there were some numbers...well. I covered a lot more recently with a basic generalized 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.

Taken from this post, which I look at LOTS. Figured out how to solve for the g's to show how you can get all algebraic integers, and then keep going, and realize there are other numbers previously not catalogued. Where is like if you stumble across people who only know evens, and then claim that 2 and 6 are coprime, and you end up arguing with them about the existence of 3.

Is boring. Today I face problems I consider to be about people for whom the truth is irrelevant, like they have a math religion now. And mathematicians are the priesthood, and they just believe what they are told, I guess. As these things are old news.

And have done things I think are cool in lots of mathematical areas. Like even found something that apparently puzzled Ramanujan, but am not sure, of course.  And Euler did work in the same area too! Which I thought was great. I DO wonder what they'd make of my BQD Iterator and the simple explanation.

Of course also managed more recently to be able to group myself with Euler with my own modular inverse discovery, which is about as cool as it can get in mathematics.

So yeah have a little bit of lots, and just gave some highlights which occurred to me now as both definitive and easy for me to trot out, without having to work too hard.

So what gives with other people talking? I really do not know. Actually am not sure what people are talking or not. This blog pulls global attention according to Google Analytics. But then am puzzled at what am not noticing happen. Which doesn't mean is not maybe happening. Feel confident though that if certain things were happening, I would notice.

Oh, some may think race is a factor. I figured out long ago why it is not. Turns out that American racism is primarily a problem of the United States as modern version was invented here. Nation ran into a contradiction with claims of believing in merit rules, and people having slaves: so invented what is commonly considered to be race. And science has debunked race as a social construct.

Mathematics is global. The United States is influential but not that influential. American racism in my assessment has had no impact on my story with my mathematical discovery. Turns out quite simply that humans challenged with the truth can resist it. Like check out this story for sad and bizarre perspective, on my blog Beyond Mundane:


So yeah, simpler explanation is that I have big enough math results that challenge enough people they just have far as I can tell decided to simply not acknowledge them. And just keep doing what they were doing! Which does not mean understanding numbers as well as they could.

There is a certain amount of naive too, I think. Modern web is SO new. Where will admit am very lucky that web is here. But yeah, certain people who do not fully understand how the web works, may be unable to accept how information travels.

People DO know, but then what? Gets very complicated there and often involves perceived self-interest but also pragmatism, especially with confronting establishment things. Besides a lot of people are VERY cynical about much. Mathematicians doing fake research? To them may not at all be a surprise.

I WAS surprised for years. Now I'm not. Is harder for me though.

You have to believe in so much to learn enough to discover as I have. And then to have that discovery shatter so much of my faith in so many institutions? Was VERY difficult for me. But now am more at peace with much which is really about time.

Of course helps to have kept getting monster discoveries routinely too! Guess that's natural.

Truth can be hard to handle for lots of reasons. And yeah there are people who prefer the lie. Or can believe with what I think is religious fervor, and for them is just a matter of faith.

But, for the truly curious that is not a path they would take. Is not satisfying. And more and more am trying to speak more to that audience of folks I figure appreciate the truth. Numbers do not care. These discoveries are way cool, and for people who want to understand mathematics, the truth matters more, am sure.

And yeah, may as well note, these answers intrigue me as well. Is just still surreal to think of as, answers I figured out. As years go by is easier to distance myself though, which is also kind of weird. I figured out, but...how really. Does it matter. Yeah to me! But then again, don't I know? I guess so.

So much that are simple answers though. Math really does not care what we think.

Let those others who are the lost keep with their mathematics as a religion. History will catch up eventually. Which is what history teaches us.

In the meantime, we can just have fun, appreciate the truth, and appreciate how hard it can be to know, and accept.

Would I speed things up if I could? Like get official recognition and changing math textbooks, and getting interviews with global press on the constant? In the past would have said yes, and now am thankful for the time. So no, and believe in a process. More and more feel like world knows better about how these things should go.

Besides I have my global status, but without so much of the fuss. More I learn, more appreciate how great so much actually is with how things have gone. Takes years to process. Is just reality.


James Harris

Monday, June 18, 2018

When truth matters more

Much shifted positively for me with my find May of last year of a new way to calculate the modular inverse. It was important for the world of course. But for me personally it was a different experience could compare with my discovery of the prime count to continuous function explanation, now over fifteen years ago.

To me there are similarities in terms of ease of checking from others, and celebrity around the subject area. Where yeah, celebrity is kind of boring, because is simply human chatter, but hey, is curious too.

Also importantly for others provided something really easy to check, and could give perspective on social aspects to the mathematical community not necessarily so easy to delineate without such a result.

People can claim all kinds of things, but how often in human history do you have a major mathematical find with which to test an influential group of them? Very rarely actually.

I like to say let's you x-ray certain social things, as a metaphor for how much can then peer into both the math community and into minds within that community. Where will admit have puzzled over thoughts revealed for over a decade now.

Mathematics can enable perfect tests, of human beings. And you can trap people into giving their actual feelings beyond any possibility of useful lies.

Is weird though, as now over a year later, yup, think same pattern is trying to play out that happened with my prime counting finds. Is actually a bit more refined as well, as if certain people believe they have learned lessons from there.

Which is a depressing subject which have decided to avoid talking much. Besides is weird: what kind of delusion would lead anyone to think can actually suppress major mathematical results? Has never succeeded in human history, with good reason. And regardless, why would any human try?

For me of course is fascinating to watch. The effort betrays a lack of basic understanding of knowledge, and why we humans appreciate, and how we use. Looks like some believe in celebrity as the basis for ALL interest, which is not even remotely close to reality. One reason I decry the celebrity model for academics. And routinely now dismiss those who clearly crave attention to themselves for their discovery.

World needs the knowledge. The knowledge is what our species will keep using for benefit of humanity.

Celebrity is actually not well understood in our times. I studied it carefully, and understand better how much is illusion.

Especially with modern celebrity which actually is: simply excited human discussion around a person, topic or event, usually. Which is why fades so quickly, always. People will only talk excitedly for so long. Disrupting celebrity is actually surprisingly easy--just shift the conversation. Eventually am confident we will crush celebrity out of intellectual areas including mathematics and the sciences.

Knowledge is best valued on use.

For those who wonder, web enables worldwide attention very rapidly, so useful knowledge sweeps the planet very quickly.

Have watched over and over again through the years and regardless find it, unsettling.

How people discuss though? Can be complicated.

The result also validated for me an approach which focuses more on discovery. Is just more fun to keep figuring things out than worry about, human chatter on the subject of what you discovered! Is more cool too.

Also shows irrelevance of official recognition in the modern age. World apparently is in a slow meltdown of prior systems and couldn't care less. And am a fan of the journal system, but relish a lack of need for journals. But am in a unique position.

As the world's latest major mathematical discoverer, of course I don't need journals.

Wonder when will have to stop saying am latest? For now there doesn't seem to be any competition there though. When there is another, I'll just note am a major discoverer, I guess. Should I even bother though?

Checking me is easy. But who has the will to let the truth matter more? That intrigues me as well.

Is a dynamic arena but have found my results with some web searches.

Like search: modular inverse innovation

Or search: modular inverse discovery

Others may work as well, of course.

Is dynamic as, is about others. I have no control over web search results, of course.

So yeah, this post was really for people who may have wondered: was there something wrong with the reaction? Answer of course is, yeah. But is lots naive really. And I wouldn't worry over it.

Of course I've lost nothing. Recognition at best comes from those who can appreciate the accomplishment. And the web helps you to know. Thank you to those who do.

Good thing though is how easily can check my math, if you wish. And then have no further doubts, if you had any.

Know it helped me. But I'm at the center of the storm is how I feel it is, and is like, out there are the hurricane force winds, and here, is calm.


James Harris

Friday, June 01, 2018

Simpler modular form for BQD Iterator

Was surprised to discover that considering x2 - Dy2 = F mod N, is also the path to a simpler form for my BQD Iterator.

Key is the factorization available from:

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

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

And I can iterate to get: x + Dy + m(x + y) = r' mod N

Then x + my + Dy + mx = x + my + m(my+x) = r + mr = r(m+1) = r' mod N

And have the result then: r' = r(m+1) mod N

Then with k iterations, have for kth iteration: rk = r(m+1)k mod N

So can use r to iterate modularly with a very simple result.

Where intrigues me is VASTLY simpler than what can get with the explicit form.


James Harris

Thursday, May 24, 2018

Modular simplicity is fascinating

Have a much better handle now on a lot modular where recognize much of my own success with discovery has been because of modular algebra tools. Where discussed much recently as yet again explained tautological spaces, but now with a more instructional tone. And finally emphasized how represents a new discipline where the math does algebra for you. Which is really cool.

It is worth noting how odd math history becomes then, when you realize how much EASY was waiting for anyone who could discover. Like have gone on much about:

x2 - Dy2 = F mod N

where N can be prime or composite, and nonzero. Where I don't think it gets interesting at all until you reach N = 3 or higher, where I stick with positives. And just with THAT expression was able to figure out how to calculate the modular inverse, so have my own modular inverse method! Which means I get to mention myself with Euler and Euclid. Where extended Euclidean algorithm is how Euclid gets in there. While Euler and Fermat are the others who actually came up with their own ways to figure out modular inverses. And Euler simply extended Fermat who had an idea that only worked with a prime modulus.

Was cool idea anyway. I discuss simply with post: More overview on modular inverse

Given other things from my research is not surprising am also rethinking some notions that more are from others, like the idea that integer factorization is a hard problem. I fear it is NOT actually, with a proper modular approach.

Like have seen naive mathematical discussion looking at, say xy = C mod N, where C is some composite, ironically in my opinion noting that x and y are easily calculated with the modular inverse.

Um, yeah but you have to not know what you're doing to focus on that expression. And I guess they did not as to folks who put such things in mathematical texts. But how would they know what they did NOT know, which we know now? I like that sentence.

Advances in human thinking fascinate me now, so have ranged more widely. Having success in mathematics gives massive benefits. And simplicity with modular means much can be easily checked by others, which I think is a great thing.

Mathematics should be about what works.

Human beings can think all kinds of things.

With mathematics is possible to gain absolute certainty with absolute proof.


James Harris

Thursday, May 17, 2018

More on finding quadratic residues with modular inverse

Last month started pondering if one could find quadratic residues using the modular inverse.

Copying much from my post last month will update also as answer a question raised then.

Focused on x2 - Dy2 = F, where 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

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.

Get to set some variables, like simply set F = 1 mod N and will set y, and then find D, from:

D = (x2 - 1)y-2 mod N.

Noticed before that WILL work, but also wondered if could use BQD Iterator, to possibly get different values for m, and realized that MIGHT change r, but how to know?

From the start: 2x = r + Fr-1 mod N

The BQD Iterator will give next: 2(x+Dy) = r + F(-D+1)r-1 mod N

So can just solve. Then r = 2(x+Dy) + F(D - 1)r-1 mod N, so:

2x = 2(x+Dy) + F(D - 1)r-1 + Fr-1 mod N = 2x + 2Dy + FDr-1 - Fr-1 + Fr-1 mod N, so:

0 = 2Dy + FDr-1 mod N

Assuming D is coprime to N, which is odd, have then: y = -F(2r)-1 mod N

Which surprised me. But y is free here so can be easily set. Guess you get one iteration then per try?

For example, again let N = 119, and again start with easy with let r = 2 mod 119. Then also r-1 = 60 mod 119.

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

Now there is a change from before as y = -4-1 mod 119 = 89 mod 119

And 2m(89) = 2 - 60 = -58 mod 119, so: 59m = 61 mod 119, so: m = 116 mod 119

And D = (312 - 1)(16) mod 119 =  9 mod 119

And yeah 1162 = 9 mod 119, as required.

Now though can iterate with the BQD Iterator, knowing have same r, which then is way to handle that particular thing.

So next value for x is: 31 + 2(89) mod 119 = 90, and next y is: 31+89 mod 119 = 1 mod 119.

Then 2m' = 2 - 60(-9 +1) mod 119 = 2 - 60(-8) mod 119 = 6 mod 119, so: m' = 3 mod 119

Which is just the negative of the first one. But at least it worked! Cool. There may be potential here. But now I wonder: will it always simply negate?


James Harris

Monday, May 07, 2018

Quadratics easier with more degrees of freedom

One thing that DID surprise me was when found solutions were actually easier when I focused on more variables with a quadratic case:

c1x2 + c2xy + c3y2 = c4z2 + c5zx + c6zy

where the c's are constants. Years ago, actually shied away from and went simpler by setting z=1, to get my method for generally reducing binary quadratic Diophantines. And FINALLY got curious enough to look at the more general case, where handled in two key posts:

Three variable quadratic reduction

and

Trinary Quadratic Iterator

Where yeah my BQD Iterator is my primary workhorse still, where who knows really what the TQ Iterator adds and for the moment am not working to find out.

What is really cool to me though is how you can end up with quadratics easier to solve for integers. Like copying from my post giving the reduction method, let c1 = 1, c2 = 1, c3 = 1, c4 = 1, c5 = 1, c6 = 1, so:

x2 + xy + y2 = z2 + zx + zy

Which gives:

A = -3, B = -2, and C = 4

[-3(x+y) + 2z]2  + 3m  = 16z2

Where now have two unknowns m and z determining existence of a rational solution. But by inspection has an infinite number of rational solutions with m = 0. Still shouldn't just assume that then x and y always will be integers. Easy enough to check:

-3(x+y) + 2z = +/- 4z, so: -3(x+y) =  2z or -6z

So there will be a set of integer solutions for every nonzero integer z.

Where a trivial set is: x = y = z

But you can also find others.

Here's one a little more complicated. Let c1 = 1, c2 = 2, c3 = 3, c4 = 4, c5 = 5, c6 = 6, so:

x2 + 2xy + 3y2 = 4z2 + 5xz + 6yz

Where similarly can show that you can always find solutions with: x+y = 2z or 3z

Which means can always have integer solutions with an integer z, with one reduced constraint.


James Harris

Saturday, May 05, 2018

Modular inverse anniversary and reality check on modern discovery

Today marks a year from time I found out there was a direct way to calculate the modular inverse. Where yeah for people who wonder you now have a reality check on how hard it can be to get official recognition in modern world for discovery.

Yet has been HUGE in terms of impact on blog, where I have access to the web metrics where contrarily as usual the total numbers for visits don't shift much. So I end up focusing on things like linking behavior as reported by Google Webmaster Tools as one of the more important indicators.

For me? Is just more of the same as have been witnessing now for over a decade.

Have several major results at this level, and some well beyond. So I can cross compare.

For others? Yeah can notice how much you might have to do, if you ever wish to get official recognition for a mathematical result.

Consider, I found the first direct route to calculation of the modular inverse, which is the third primary way to calculate. My competition in this arena? Euler who figured out his totient function on top of a result known by Fermat, and ideas shared by Euclid related to calculation of the greatest common divisor.

So we are the three for the modular inverse now--me, Euler and Euclid.

Primary results at this level were supposedly all found, and apparently certain people who do not understand the web believe if they refuse to acknowledge they can discredit useful mathematics.

But we know better. The web has changed so much! Thankfully is the knowledge that is important and yeah, from multiple indicators that information rocked our world. And possibly changed much across the entire web itself. Useful mathematics gets used. Believe that.

My modular inverse is a 21st century result for a computer age, where information rules.

So cool. So marking first anniversary of the discovery of the PRIMARY way to calculate the modular inverse as human mathematics seems poised to shift into fully modular, for the first time in its existence.

Modular algebra rules numbers we now know.


James Harris

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