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 - Am²   = (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

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 + 1128² + 1880² + 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 + 1032² + 1720² + 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 + 258² + 430² + 602² + 1762²) = 

            615² + 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

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

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

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

where S(x/p_j, p_j-1) is the count of composites that multiply times p_j to give a product less than or equal to x, where notice that p_j must be less than or equal to sqrt(x) or the composite count given by [x/p_j] - 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