Friday, February 23, 2018

Thoughts on math development

Throughout human experience with mathematics since the arrival of algebra, there has been algebraic manipulation which is a very human activity. Sure computers can also do it in our times, but that can be debatable as to efficacy in discovery.

So you can have mathematical arguments which are lots about manipulating algebraically. Well what if you let the math do it instead?

My own math discipline around tautological spaces is actually having the math do the heavy lifting of most difficult algebraic manipulations. Which is great. I find all that work tedious, and why bother when the math can do it, and can do it with perfect ability?

The math cannot even make a mistake! Where yes, started talking about the math in a different way when considered this reality. Where greatly enhanced further math development may depend on relying on the math as simply does algebraic manipulation far better than any humans or machines.

Which seems to be a natural progression in the development of mathematics, and possibly as needed for certain mathematical advancements as algebra itself was needed.

Like ran into the complexity of:

c1x2 + c2xy + c3y2 = c4z2 + c5zx + c6zy

where the c's are constants, which is so great has never been as simply and generally reduced as possible until my methods.

The advance was:

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

You can expand upon with basic algebraic manipulations and subtract some expression which has x,y and z from the result to get what I call a conditional residue. Then you can probe that residue with v, which is a free variable.

One of the basic things I have done is for some prime p use: v = (x+y)z-1 mod  p

That would allow me to probe for a particular prime or find that I had a result true for any, which then is forced to be a general result. And then is forced to be an algebraic manipulation.

Where the math did all the work.

And great thing? The math does not care. What is telling about us humans is how much we can miss when WE try to just manipulate algebraic expressions ourselves.

And my favorite example like to call the BQD Iterator:

It must be that if you have:

u2 + Dv2 = F

then it must also be true that

(u-Dv)2 + D(u+v)2 = F(D+1)

Easily demonstrated:

1. x2 - 2y2 = 1

2. (x+2y)2 - 2(x+y)2 = -1

3. (3x+4y)2 - 2(2x + 3y)2 = 1

4. (7x + 10y)2 - 2(5x + 7y)2 = -1

5. (17x + 24y)2 - 2(12x + 17y)2 = 1

and you can keep going out to infinity.

Have looked at that simplicity for years wondering, why new? Why didn't human beings find that just playing around?

Reality is that algebraic manipulation can have an art to it, which maybe people take too much for granted. But when the math does it, is logically perfect.

The math does not have human limitations in that regard.

Am confident that how mathematics was done, will go away. And in time will seem amazing to mathematicians that human beings ever tried to discover any mathematical truth by themselves trying to manipulate algebraic expressions. Which is really cool.

So use of tautological spaces is actually an uber-discipline as use the German word for over, and encapsulates almost all prior mathematics. There are exceptions which are of mathematical interest.

Where yeah if absorb that and can accept? Then you're facing what have considered for YEARS now trying to process. That math can simply do better what humans have tried to do for so long is not really surprising I guess.

That the math can do math itself is something I write just to read it, over and over again.

Then with tautological spaces you are more like a researcher asking the math questions.

Which means which questions you ask become more definitive in terms of what you can learn, but the primary thing is, the math answers.

Resistance to progress in this area is something I welcome to a large extent, while feeling a sense of duty and responsibility hard to explain as well, so is a mixed feeling. From my perspective MASSIVE resistance is best scenario. However also feel like must never forget consequences for other human beings and especially other human lives.

We just have to be sure as a species. And reality is, in that quest for certainty, there are going to be consequences.

James Harris

Tuesday, February 20, 2018

When primes do not care

One area where was surprised with simple was with prime numbers, as if you look at primes modulo each other, my feeling immediately was, why would they care?

Putting that mathematically is easy and did so, with my prime residue axiom. But before that had done much with my prime gap equation where notice have not talked it much since, and linking to a post made August 2006.

And next public post I find is this one from February 2010, where give my prime residue axiom:

Given differing primes p1 and p2, where p1 > p2, there is no preference for any particular residue of p2 for p1 mod p2 over any other.

Implications seem to move me usually into things dealing with probability. Will admit am not so much into probabilistic things, which is one reason have not talked as much but also is really easy to encapsulate into a small area. It also does lead to discrete results, as you can just do things like count off how many primes should be twin primes for instance within some block of primes.

Also though that ONE idea if true promptly leads to resolutions of big things supposedly unsolved. Like says Twin Primes Conjecture is right and Goldbach's Conjecture is wrong though disappointing there as we're unlikely to ever find a counter-example regardless. So yeah, plenty of reasons there to just not deal with much? Is weirdly potent an arena. SO much wrapped up there.

Am trying to stay away from talking other things, but it is sad to consider what motivations might shift certain people from a basic truth. For me? Have been lucky that my discovery has been disconnected from money. But have wondered about mathematicians who make their money a certain way, as to what might be more important to them? Truth? Or a paycheck? Well enough said on that subject.

And have focused a lot on 3; with p a prime, p mod 3, for all primes greater than is of course 1 or -1; so is the only binary prime residue in that way. Where yeah, why should greater primes have a preference for 1 or -1 modulo 3?

Am like, why would they care? Makes no sense to think a prime would to me. But realize is an area of contention based on research have done. Am NOT a mathematician so am not sure on details much, while have tried to learn in detail in the past from web search.

Talked for a different audience on my blog Beyond Mundane: No prime preference?

It is amazing how much can come with such a simple idea, and lack of prime preference for any particular residue modulo another prime has been put out there before, which I found out with web searches. My thing was to say, is an axiom.

I like that bold.

James Harris

Monday, February 12, 2018

Math, innovation and invention

Finally am happy to talk objectively about inventing my own math discipline, where have debated with myself if that even makes sense. Can anyone invent a math discipline? But of course know the historical example of calculus, where Sir Isaac Newton and Gottfried Leibniz invented at same time, and we use the version of Leibniz which is also of interest as he has far less celebrity. Better ease of use beat greater celebrity handily.

With my math around what I decided to call tautological spaces, which are complex identities used modularly, have certain ways of doing things where did look to get enough needed terminology written down. So yes, there is an invention there. Figuring out terminology is fun to me. Is work though too.

The innovation is of extreme interest considering its demonstrated power. Gist of it is so easy though:

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

Which is an equivalence written as a modular expression and introduced world's first true modular algebra in contrast to modular arithmetic. Which is about all those variables in the modulus. Actually DID go looking around to see if could find ANY math out there similar.

I write that and ponder. These are things have considered for more than a decade but often thinking quietly to myself, though leaking out here and there in posts. But now am simply stating these things completely.

Is good to be able to consider after over 18 years. And before there were fears, like maybe would just not amount to much really. And think really was 2008 when felt like, oh, is solid. And of course could check as hey, I defined mathematical proof, and knew was absolutely correct, but had done that already. Still as human beings we have emotions that can take time to settle down. So now is even better about a decade later. Am calm on much more.

There are a couple of innovations there. So there is using modular for an equivalence which is so cool, realizing is equivalent to: x+y+vz = x+y+vz

Also there is the asymmetric form. I tried symmetric forms first! Which does not surprise me. And naturally was starting with x,y and z, where point of innovation was to add an additional variable, but is intriguing that you NEED at least three variables which at one point had me contemplating if is related to why our...heading off into physics.

Could talk more about mathematical innovations have introduced where could get a VERY long post, but want a short one, so will talk just one more:

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.

Reference: Composite counting functions and prime counter

Where the innovation is explained with that last. And remember my thinking back summer of 2002 when was looking for something to count primes. And distinctly remember thinking it silly to count composites already counted by a lesser prime. Like with composites up to 10, when considering those with 3 as a factor, why bother with evens? Those are counted by 2, and I mathematized that viewpoint.

Have stared at such things through the years, and pondered what it takes. And in my case? Admittedly lots of training with modern problem solving techniques. Was trained to work to think in a way most understand with the phrase--thinking outside the box. And was trained in brainstorming techniques.

Where plenty of people are so trained--in our modern times. Guess does make the difference where that training is applied and to get lucky. For me a few key innovations helped me help our world understand vast areas of mathematics with more powerful, and simplifying concepts.

That just keeps getting better too.

James Harris

Thursday, February 08, 2018

Benefit reality of unique solitary research

Back December 1999 found myself excited about math again. Earlier had felt horrible after months with yet another failed attempt at solving an intriguing problem from Fermat. The mere mention of it seems to be a harsh negative, which puzzles me. Regardless having decided all elementary method paths available to me were checked, had wondered, how might I make more?

By December had my answer, would use identities, where could deliberately add another variable, which would call 'v' for victory...well also was available.  But could have used 'w' for win. I think I considered but liked simpler. After all, w really is double v. And had:

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

What's interesting to me is that I had it that way from the start, once had something that actually worked. Took about two weeks if I remember correctly to figure out something that did. Like I discovered I needed an asymmetric form. So yeah there was research AFTER the basic idea as asked myself: how exactly do I add another variable? And figured it out.

The specific trend that has made the difference in my research am thinking, is that I look modular, first.

Which realized recently. So am usually looking for a modular angle, I guess. As yes, thinking back when I finally realized an asymmetric form wrote it like I do now. So from FIRST useful version was the current version.

And then after talking out on math groups, when faced criticisms would switch to the explicit:

x+y+vz = x+y+vz

That is telling I realize. It is interesting the things that can make such a huge difference just based on which way people tend to go. By focusing modular, I simplified and ended up creating a true modular algebra.

Is SUCH a simple idea. You take an identity, and subtract an equation THROUGH the identity, where with my use of modular algebra, you end up with a residue, which has all the properties of the original expression, as it must. And made an absolute proof demo post where prove that:

If: x2 + y2 = z2 then(v2 - 1)z2 - 2xy = 0(mod x+y+vz), where v can be any value.

That is compact as well. Which is all about going modular. Without that modular approach would not be practical as the expressions are then too complex to be useful. And of course would play with values. Like had not realized before that 2xy then has x+y+z as a factor, and also x+y-z, but then could easily prove with simple algebra. Just not the kind of thing playing around had noticed otherwise.

That v can be any number keeps being fascinating to me. Like let v = i.

Then: -2z2 - 2xy = 0(mod x+y+iz)

Which is then easily verified, by algebra.

Still would like looking at numbers, like: -74 = 0(mod 3+4+5i) = 0(mod 7+5i)

And of course: 74 = (7+5i)(7-5i)

And back over 18 years ago it all seemed kind of odd. Would do example after example where would do with modular algebra and also would go back through and show everything explicit.

Now I simply note: invented my own math discipline.

So along with geometry, algebra, calculus and other math disciplines there is also math of tautological spaces. And yeah study of tautological spaces relies on modular algebra, but is like, calculus relies on algebra. And how did that happen really, do ponder. Guess...just is hard to process.

Should I admit that ALL attempts with that basic tautological space to probe the key equation from that famous proposition by Fermat failed? There is more to that story, as eventually expanded BEYOND the basic form, in only situation where felt so forced. Later developed a full methodology, and even considered how might work with calculus:

Tautological spaces and calculus

But yeah was a challenge for me. Worked at it for a couple of YEARS too.

Is interesting, when you keep with something even when it doesn't do what you want.

Somehow kept faith in the basic idea, maybe because it just seemed so cool. But for a bit had nothing to show for the effort. Luckily later found other more interesting things anyway. Oh yeah, with the Fermat thing eventually thought had something with a more complicated tautological space, but couldn't nail down the ending, and now agree with Gauss that the proposition is not really of much interest in and of itself.

However, yeah got me motivated to figure out my tautological spaces. Can get philosophical in this area. You do wonder about the ways you get to some place.

And have had 18 years to use as I wished against some problems, where got most of my cool results to talk about by analyzing a general quadratic, first with two variables, and more recently with three variables. Does make it easier when you are the one person you know of doing such research.

Was in complete control with no pressure. Still am.

Is weird though, but hey, is a huge thing and readily have admitted have been lucky. And introduced first true modular algebra, where is so intrinsic was no way would be using congruence sign all over.

Oh yeah technically though: x+y+vz = 0(mod x+y+vz) is an equivalence. One of my first early results which kind of feels profound.

The things that change your life. And for me a natural path, which reached for, without even thinking about much, completely changed mine. That modular reach was itself profound I now know. And thanks to it would find so much.

Having your own mathematical discipline does change you too, am sure. There is just a different way I now realize I began to look at so many other things.

James Harris

Tuesday, February 06, 2018

Solving trinary quadratic with reduced constraint

Used complex identities I call tautological spaces to analyze:

c1x2 + c2xy + c3y2 = c4z2 + c5zx + c6zy

where the c's are constants, and was able to generally reduce. Subtracting an expression like that which I call the conditional through a complex identity gives what I call the conditional residue, and THAT can be analyzed. Is like the math does all the heavy work for you. And of course, the math does not care.

Which is really cool. And my original research was in 2008, but then set z = 1 and focused on the binary quadratic Diophantine case, for YEARS until now. And this time got general reduced form:

u2 - Dv2 = Fw2

Which looks much like the binary quadratic reduced form.

With x, y and z of course have two degrees of freedom, but here is an example showing a path to solution with reducing constraint! And find it interesting as quadratics seem to be special.

So you can still generally reduce with three variables for the quadratic case AND will show a situation where also can get integer solutions for all three with setting the variable z.

Which can see with an example.

Let c1 = 1, c2 = 2, c3 = 3, c4 = 4, c5 = 5, c6 = 6, so:

x2 + 2xy + 3y2 = 4z2 + 5xz + 6yz

Which with my method for reducing gives:

A = -8, B = -20, and C = 33

So: [-4(x+y) + 10z]2 = 166z- 2m2

And I notice that m = 9z, gives me a useful substitution:

And [-4(x+y) + 10z]2 = 166z- 162z2 = 4z2

So now have: -4(x+y) + 10z = +/-2z, so: x+y = 2z or 3z

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

Let's trot through a full solution. Let z = 10, and use first, so: x+y = 20, so y = 20 - x

x2 + 2x(20-x) + 3(20-x)2 = 400 + 50x + 60(20 - x), and multiplying out:

x2 + 40x -2x2 + 1200 - 120x + 3x2 = 400 + 50x + 1200 - 60x

Where get easily enough:

x2  - 35x - 200 = 0, which is: (x-40)(x+5) = 0

And one full solution then is: x = -5, y = 25, z = 10

Where just picked some easy.

James Harris