Modular is a concept with wide applicability in a variety of domains. And was lucky that weirdly enough after Gauss, development of modular with mathematics apparently lagged. Which I know because was able to introduce first true modular algebra, which then not surprisingly, revolutionized much.
However modular algebra was not my first step into modular with mathematics, as was kind of a relief to notice was key to an earlier puzzling to me result, as SO simple. Where considered packing of spheres in a modular way. Then it makes more sense that I could easily prove that distorting that one piece from hexagonal close packing would lead to a lower density. Since full space is filled with same modules, where any distortion leads to lower density, had an easy proof of optimum packing.
Wrote up a two page paper. Having that rejected by an editor for the Proceedings of the AMS as too simple is just kind of telling to me now.
Would go on to develop a full modular algebra where like to give as easy for context:
x+y+vz = 0(mod x+y+vz)
Which is the simplest form I found with some research, and I call, a tautological space. Have created terminology over more advanced forms.
And have noted that remarkably, can use to study algebraic expressions where can give things which are algebraic manipulations, done by the modular algebra. As the modular algebra is doing all the work, is perfect and can reveal things humans never found before.
Who knew? How could anyone? Watching modular algebra do algebra? Is surreal.
My best example is what I call the BQD Iterator lots. A very simple relation which I've used quite a bit, which the modular algebra gave me, with my method for generally reducing binary quadratic Diophantine equations. My method remarkably is better than that from Gauss.
BUT can now also note that Gauss was am sure doing something I don't need to do any more--complex algebraic manipulations.
Letting the math do the work is more effective and complete. Is faster, and is of course easier.
The math easily checks infinite possibilities in a modular way. The math is complete intelligence.
Those are the kinds of things that tend to show you where human beings will go, in time.
Why do we not notice certain things? Probably is about how we are built.
Of course, how do we know what we do not know?
With the math performing better in a way that humans have tried with our intelligence, can sense are watching an intelligence, but is an infinite one which is perfect, makes no mistakes, and gets to the best answer.
Which then lets us look at what we humans did not find on our own, and wonder.
Could go on and on about it. And have deferred on what I call the social problem as to acceptance until 2028, which is not that big of a deal for me. Figured out my modular approach to packing sphere back in 1996. And had finally the evidence that the method I found around what I decided to call tautological spaces, was doing algebraic manipulations back in 2008.
And that was September 2008, with: Quadratic Diophantine Result
Figure that is decent matter-of-fact naming for that post. At THAT time did not say to myself, have a true modular algebra which can do algebraic manipulations. Didn't even realize had actually handled three variables as only focused on two. That is wild, eh? Recently talked the three variable.
So much did not know then, where would learn in time. At the time, was more like, wow, what's this thing? That's SOMETHING when you first realize that the mod can just go away. Where thank God that was years ago, and have settled down much since then.
Takes years to properly process discovering such things have learned with myself, as slowly you adjust, to what changed for you so quickly. So much a sense of things, of what I thought I knew got shattered, so quickly.
There was often elation as well, but learned to be suspicious of it. Like to say that math and emotion do not mix well. But that effort is more to maintain an even keel. As can end up with massive high's and intense low's which to me? Is weird.
Took YEARS as slowly rebuilt a firm sense of reality. And noticed was better than before, but so much work to get there. The world seems more crisp to me now, often. And appreciate so much more as well.
And web has helped me greatly and lucky was here evolving as I found these things. As web allowed me a more gradual sense of certain things, while being aware, yeah, people around the globe...what have you all been doing? Web analytics only tell me so much. Guess you would be the people reading this post though.
Where is strange yes, can have simple but powerful ideas, but one reason became fascinated with modular elsewhere. Like would stare at cargo ships loaded down with shipping containers when lived in San Francisco. And would ponder.
Noted was lucky for me that further development of modular seemed to have waited until now. Still there has been some resistance, as now over 18 years since I introduced what I call tautological spaces. With over 21 since my first paper with modular ideas and packing of spheres.
Regardless the ideas are flowing, which web metrics lets me know. So I guess something has changed for humanity allowing the possibility and simply live in the right time.
My view is the web has made the difference as notice, how are you reading these words?
James Harris
Tuesday, February 27, 2018
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
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
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, have my very useful definition of mathematical proof. Which was motivated by NEED, where am thankful there. So knew was absolutely correct, as had checked 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
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, have my very useful definition of mathematical proof. Which was motivated by NEED, where am thankful there. So knew was absolutely correct, as had checked 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
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 = 166z2 - 2m2
And I notice that m = 9z, gives me a useful substitution:
And [-4(x+y) + 10z]2 = 166z2 - 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
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 = 166z2 - 2m2
And I notice that m = 9z, gives me a useful substitution:
And [-4(x+y) + 10z]2 = 166z2 - 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
Monday, February 05, 2018
Trinary Quadratic Iterator
Finally looked at not setting z=1 to what I now know is a general way to reduce a trinary quadratic equation like:
c1x2 + c2xy + c3y2 = c4z2 + c5zx + c6zy
where the c's are constants, where was able to prove can be generally reduced.
Shows yet another type of general reduced form: u2 - Dv2 = Fw2
Which has me of course wondering what happens if THAT form is so reduced, which have done in the past to get what I decided to call a binary quadratic Diophantine iterator or BQD Iterator for short. So will use the reduction method on it, copying over the base system.
A = (c2 - 2c1)2 + 4c1(c2 - c1 - c3), B = (c2 - 2c1)(c6 - c5) + 2c5(c2 - c1 - c3)
and
C = (c6 - c5)2 - 4c4(c2 - c1 - c3)
Base result is: A(x+y)2 - 2B(x+y)z + Cz2 = m2
And some simple algebra gives:
[A(x+y) - Bz]2 - Am2 = (B2 - AC)z2
With: u2 - Dv2 = Fw2
So: c1 = 1, c2= 0, c3 = -D, c4 = F, c5 = 0, c6 = 0,
x = u, y = v, z = w
So, A = 4 +4(-1+D) = 4D, B = 0, C = -4F(-1+D)
Gives: [4D(u+v)]2 - 4Dm2 = (16DF(-1+D))w2
Which is: m2/4- D(u+v)2 = -F(D-1)w2
Where now need m. 4D(u+v)2 - 4F(-1 + D)w2 = m2
So: m2 = 4(Du2 + 2Duv +Dv2 + Fw2 - DFw2),
and m2 = 4(Du2 + 2Duv +Dv2 + u2 - Dv2 - DFw2)
Where showing all the detail for once. Helps keep me from making mistakes.
So: m2 = 4(Du2 + 2Duv + u2 - Du2 + D2v2), and m2 = 4(u2 + 2Duv + D2v2) = 4(u+Dv)2
Which is: (u+Dv)2 - D(u+v)2 = F(-D+1)w2
So the trinary quadratic iterator is just the BQD Iterator with a w2 on the end.
Shall call it the TQ Iterator for short.
James Harris
c1x2 + c2xy + c3y2 = c4z2 + c5zx + c6zy
where the c's are constants, where was able to prove can be generally reduced.
Shows yet another type of general reduced form: u2 - Dv2 = Fw2
Which has me of course wondering what happens if THAT form is so reduced, which have done in the past to get what I decided to call a binary quadratic Diophantine iterator or BQD Iterator for short. So will use the reduction method on it, copying over the base system.
A = (c2 - 2c1)2 + 4c1(c2 - c1 - c3), B = (c2 - 2c1)(c6 - c5) + 2c5(c2 - c1 - c3)
and
C = (c6 - c5)2 - 4c4(c2 - c1 - c3)
Base result is: A(x+y)2 - 2B(x+y)z + Cz2 = m2
And some simple algebra gives:
[A(x+y) - Bz]2 - Am2 = (B2 - AC)z2
With: u2 - Dv2 = Fw2
So: c1 = 1, c2= 0, c3 = -D, c4 = F, c5 = 0, c6 = 0,
x = u, y = v, z = w
So, A = 4 +4(-1+D) = 4D, B = 0, C = -4F(-1+D)
Gives: [4D(u+v)]2 - 4Dm2 = (16DF(-1+D))w2
Which is: m2/4- D(u+v)2 = -F(D-1)w2
Where now need m. 4D(u+v)2 - 4F(-1 + D)w2 = m2
So: m2 = 4(Du2 + 2Duv +Dv2 + Fw2 - DFw2),
and m2 = 4(Du2 + 2Duv +Dv2 + u2 - Dv2 - DFw2)
Where showing all the detail for once. Helps keep me from making mistakes.
So: m2 = 4(Du2 + 2Duv + u2 - Du2 + D2v2), and m2 = 4(u2 + 2Duv + D2v2) = 4(u+Dv)2
Which is: (u+Dv)2 - D(u+v)2 = F(-D+1)w2
So the trinary quadratic iterator is just the BQD Iterator with a w2 on the end.
Shall call it the TQ Iterator for short.
James Harris
Sunday, February 04, 2018
Three variable quadratic reduction
Have long talked my method for reducing binary quadratic Diophantine equations which of course is two variable, but actually simplified from a three variable analysis. Was considering:
c1x2 + c2xy + c3y2 = c4z2 + c5zx + c6zy
where the c's are constants. And found what I call my Quadratic Diophantine Theorem which I talk in this post from September 2008.
And figured out:
((c2 - 2c1)2 + 4c1(c2 - c1 - c3))(x+y)2 - (2(c2 - 2c1)(c6 - c5) + 4c5(c2 - c1 - c3))(x+y)z + [(c6 - c5)2 - 4c4(c2 - c1 - c3)]z2 = m2
where m is some integer, if integer solutions exist for x, y and z. And can see that m has an explicit solution as a function of x, y and z. But is SO hard to calculate to something simpler than given squared one. Kind of thing for which math software is good for those who know how to use, and wish to solve for it. I never bother.
Way too complicated looking so introduced some more variables to collect things.
A = (c2 - 2c1)2 + 4c1(c2 - c1 - c3), B = (c2 - 2c1)(c6 - c5) + 2c5(c2 - c1 - c3)
and
C = (c6 - c5)2 - 4c4(c2 - c1 - c3)
where neither A nor B is zero for what follows.
Then have: A(x+y)2 - 2B(x+y)z + Cz2 = m2
Which is easier to handle. And multiplying through by A, moving some things around, and completing the square:
A2(x+y)2 - 2BA(x+y)z + B2z2 - Am2 = B2z2 - CAz2
Which is easily enough:
[A(x+y) - Bz]2 - Am2 = (B2 - AC)z2
And that is simple enough. Not far at all from what I already had. Where also for certain situations would NOT complete the square, but work with the prior form.
Shows yet another type of general reduced form: u2 - Dv2 = Fw2
Maybe should come up with another cool name for iterator am sure can get from it. Maybe later.
As a first example 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 + 3m2 = 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.
Well I DO like easy. So yeah, very straightforward from my original research.
Primary point of this post was to just get a look at result without setting z = 1, which did before just to focus on the simpler case.
So now have a three variable general reduced form for one type of quadratic case.
James Harris
c1x2 + c2xy + c3y2 = c4z2 + c5zx + c6zy
where the c's are constants. And found what I call my Quadratic Diophantine Theorem which I talk in this post from September 2008.
And figured out:
((c2 - 2c1)2 + 4c1(c2 - c1 - c3))(x+y)2 - (2(c2 - 2c1)(c6 - c5) + 4c5(c2 - c1 - c3))(x+y)z + [(c6 - c5)2 - 4c4(c2 - c1 - c3)]z2 = m2
where m is some integer, if integer solutions exist for x, y and z. And can see that m has an explicit solution as a function of x, y and z. But is SO hard to calculate to something simpler than given squared one. Kind of thing for which math software is good for those who know how to use, and wish to solve for it. I never bother.
Way too complicated looking so introduced some more variables to collect things.
A = (c2 - 2c1)2 + 4c1(c2 - c1 - c3), B = (c2 - 2c1)(c6 - c5) + 2c5(c2 - c1 - c3)
and
C = (c6 - c5)2 - 4c4(c2 - c1 - c3)
where neither A nor B is zero for what follows.
Then have: A(x+y)2 - 2B(x+y)z + Cz2 = m2
Which is easier to handle. And multiplying through by A, moving some things around, and completing the square:
A2(x+y)2 - 2BA(x+y)z + B2z2 - Am2 = B2z2 - CAz2
Which is easily enough:
[A(x+y) - Bz]2 - Am2 = (B2 - AC)z2
And that is simple enough. Not far at all from what I already had. Where also for certain situations would NOT complete the square, but work with the prior form.
Shows yet another type of general reduced form: u2 - Dv2 = Fw2
Maybe should come up with another cool name for iterator am sure can get from it. Maybe later.
As a first example 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 + 3m2 = 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.
Well I DO like easy. So yeah, very straightforward from my original research.
Primary point of this post was to just get a look at result without setting z = 1, which did before just to focus on the simpler case.
So now have a three variable general reduced form for one type of quadratic case.
James Harris
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