tag:blogger.com,1999:blog-111191102018-03-20T07:20:37.587-04:00Some MathGlobal resource of innovative mathematical ideas. Discovery for the 21st century. Abstract reductionism realized. And modular rules.James Harrishttps://plus.google.com/104012861017102472975noreply@blogger.comBlogger344125tag:blogger.com,1999:blog-11119110.post-38223003710225478152018-03-19T07:38:00.000-04:002018-03-20T07:20:37.623-04:00Advanced algebraic symbol manipulationWhat I like to call tautological spaces are complex identities that can be used to do algebraic manipulation. The benefit of allowing the math to do such manipulations is easier perfection, and insights which seem demonstrably to elude human beings.<br /><br />My basic demonstration for simplicity and familiarity talks analyzing: x<sup>2</sup> + y<sup>2</sup> = z<sup>2</sup><br /><br />That can be pulled from a complex identity:<br /><br />x<sup>2</sup> + 2xy + y<sup>2</sup> = v<sup>2</sup>z<sup>2</sup> - 2vz(x+y+v) + (x+y+vz)<sup>2</sup><br /><br />Leaving a residue:<br /><br />(v<sup>2</sup>-1)z<sup>2</sup> = 2xy + 2vz(x+y+vz) - (x+y+vz)<sup>2</sup><br /><br />And since v is still a free variable, you can do equivalent of symbol manipulation by choice of v, for instance, with v = 1 find easily:<br /><br />2xy = (x+y+z)(x+y-z)<br /><br />Where notice with our original expression which I like to call the conditional, we can add 2xy to both sides. That simple move is to gives a key perfect square, allowing a factorization:<br /><br />x<sup>2</sup> + 2xy + y<sup>2</sup> - z<sup>2</sup> = 2xy, and then easily: (x+y-z)(x+y+z) = 2xy<br /><br />Easy example for demonstration to help with grasping the basic concepts.<br /><br />You can shift through every possible way to manipulate algebraically with tautological spaces using v as <i>your tool to probe</i> an expression. As you shift v, the algebra moves everything around for you. And again used a simple quadratic for simplicity and familiarity, but this approach does not care.<br /><br />Importantly, you can use modular to greatly simplify complexity that can build rapidly with the complex identities--where is to its proper purpose as you do not need all of that information.<br /><br />So the actual tautological space for analysis: x+y+vz = 0(mod x+y+vz)<br /><br />Where the result then is: (v<sup>2</sup>-1)z<sup>2</sup> = 2xy (mod x+y+vz)<br /><br />Which I call the conditional residue, which is true when: x<sup>2</sup> + y<sup>2</sup> = z<sup>2</sup><br /><br />You can rapidly zip through EVERY way an expression can be algebraically manipulated, just looking for things, and asking questions.<br /><br />The greater ease with which algebraic manipulation can be done then is astonishing, and is interesting to me that it took me finding this approach. And found it December 1999 when looking to get a better handle.<br /><br />That's strange, huh? And have wondered as what do I care really, just glad to have the tools now! Yet yeah, must be the original discoverer as once you have these ways, how do you go back?<br /><br />Fumbling with equations and trying a few things here or there which you can do indefinitely, when the math can give you huge answers in minutes? There's no way to go back for me, at least. Would be kind of like choosing to walk a thousand miles when you can just fly there in an airplane.<br /><br />So I talk to the math now, and ask the math questions, where when tautological spaces are available as an option can just get answers so much more quickly than any other way.<br /><br /><br />James HarrisJames Harrishttps://plus.google.com/104012861017102472975noreply@blogger.com0tag:blogger.com,1999:blog-11119110.post-14352051568135760572018-03-16T08:06:00.000-04:002018-03-16T15:40:55.963-04:00Working with tautological spacesReaching for more handling with mathematical expressions found out there was another level to what the math could do, which involved using complex identities I decided to call tautological spaces. Have had this approach since December 1999.<br /><br />For example the simplest tautological space: x+y+vz = x+y+vz<br /><br />Where usually show the modular algebra form, but will demonstrate like did long ago, where will show explicit. And use a demonstrative case where <a href="https://somemath.blogspot.com/2014/10/example-showing-truth-logic-and.html" target="_blank">here is reference from 2014</a> for the modular algebra form.<br /><br />With a bit of algebraic manipulation, have:<br /><br />x+y = -vz + (x+y+vz), and squaring:<br /><br />x<sup>2</sup> + 2xy + y<sup>2</sup> = v<sup>2</sup>z<sup>2</sup> - 2vz(x+y+v) + (x+y+vz)<sup>2</sup><br /><br />Where can see all the growing complexity. Now that is STILL an identity of course, so x, y, z and v are completely free variables. But let's put some conditions on them.<br /><br />Let: x<sup>2</sup> + y<sup>2</sup> = z<sup>2</sup><br /><br />Then can subtract to get: 2xy= (v<sup>2</sup>-1)z<sup>2</sup> - 2vz(x+y+vz) + (x+y+vz)<sup>2</sup><br /><br />Where now can get to:<br /><br />(v<sup>2</sup>-1)z<sup>2</sup> = 2xy + 2vz(x+y+vz) - (x+y+vz)<sup>2</sup><br /><br />With some simple algebraic manipulation. And have the result that (v<sup>2</sup> - 1)z<sup>2</sup> - 2xy has x+y+vz as a factor, when x<sup>2</sup> + y<sup>2</sup> = z<sup>2</sup>, and can also easily realize that v is still free.<br /><br />From the full explicit form can also just look at expressions that result from setting v, like v = 1, gives:<br /><br />0 = 2xy + 2z(x+y+z) - (x+y+z)<sup>2</sup><br /><br />And of course then: -2xy = (x+y+z)(z-x-y)<br /><br />Notice also are not of course limited to integers, as is a trigonometric result if z = 1.<br /><br />That is: -2cos(x)*sine(x) = (cos(x)+ sine(x) + 1)(1 - cos(x) - sine(x))<br /><br />And get a better feel for how is a true modular algebra when get to modular form, as does not care about Diophantine or not. Learned then that with an actual modular algebra is not just with integers. Should make sure to emphasize which is why am so doing here.<br /><br />The modular algebra form of course is: (v<sup>2</sup>-1)z<sup>2</sup> - 2xy = 0 mod (x+y+vz)<br /><br />So what the modular algebra form gives is compactness, and is MUCH easier to manipulate.<br /><br />Where can also notice a sense of how you get algebraic manipulations by how you set v, which can be a HUGE benefit. With explicit is just given, but with modular form you can do equivalent, as MAY find in pieces that way. Just then stitch together. But can also get an explicit result too.<br /><br />The freedom to change v in ways that help analysis, was the point of the discovery for me, wanted something I could control. And that result of also giving algebraic manipulation of the conditional expression was really cool to finally realize. Was a big surprise for me when came across.<br /><br />Here demonstrated the full complex identity where notice raised to a form where could meaningfully subtract the conditional expression which leaves a residue, which can be analyzed. With the full modular algebra form I call that the conditional residue.<br /><br />That actually is an important limitation on this approach. Like is meaningless to subtract that conditional from x+y+vz as you do not eliminate any part of the conditional expression. So had to square to get to something useful here. And depends on the conditions being set on the variables.<br /><br />In my experience, tautological spaces are effective when you maintain roughly the same number of terms or less than the conditional expression, and are ineffective if you end up with more.<br /><br />The tautological space had to be an asymmetric form I learned in order for it to work. And <a href="http://somemath.blogspot.com/2008/10/advancing-tautological-spaces.html" target="_blank">have generalized methodology</a> based on my analysis of what seemed to be necessary.<br /><br />And you can do fewer variables by simply setting for example y or z, where can consider by setting z = 1, with full case of three variable quadratics, which <a href="https://somemath.blogspot.com/2018/02/three-variable-quadratic-reduction.html" target="_blank">recently posted on</a>, as yeah for years that's what I did. Which probably helped me figure things out with a bit less complexity.<br /><br />So highlighting a bit I learned just on my own. And do wonder how much more others could figure out, as am certain only scratched the surface.<br /><br /><br />James HarrisJames Harrishttps://plus.google.com/104012861017102472975noreply@blogger.com0tag:blogger.com,1999:blog-11119110.post-4134811206994084292018-03-13T06:43:00.003-04:002018-03-15T11:38:15.709-04:00Attention reality and math usefulnessMaybe should address possibly vague notions about value of math discovery and how changed things for me. For one thing do believe there are people who think a committee of some kind of expert human beings MUST accept and promote you, or you have nothing.<br /><br />That is of course not the case and actually has not been in the past. Where history is full of instances where ideas won out over time, and can say the web speeds that process up.<br /><br />Math actually spreads on usefulness have witnessed with my own ideas.<br /><br />However, humans can simply decide to not talk about certain things in certain ways. The math does not care. Still works and humans are attracted to things that work well for them. And some will even notice who made that possible.<br /><br />So the discoverer gets attention. And in my case have not often handled it well, while decrying it not being of a certain type. I was very picky about what I wanted. And had not gotten it, and threw some tantrums in the past. Am over that now. Where also better appreciate what I have.<br /><br />Today am potentially known like to say in over 150 countries where relied on something else for baseline. That something else being downloads of an open source project wrote for Java developers and put on SourceForge, an open source repository, back in 2004. Was a reaction to a lot of emotions. Wanted something concrete in a different way. Which helped me with confidence in my math too.<br /><br />This blog with its old name though had visits according to Google Analytics from 125 countries at its peak that I noticed. Where is just what registered by however Google does those stats. And that was over a decade ago.<br /><br />That messes with your mind. Do not behave the same trying to process. So I like to say POTENTIALLY known in over 100 countries, where rely then on two sources with data from others, so is objective. And just for <i>my work</i>, with no departments or faculty or other researchers or any academics actually, with any piece of it. Is all mine, alone. Which also messes with your mind.<br /><br />There is a comfort in having other humans with you with certain things. So much harder when have to go it alone in many ways. While the shared value is still that of humanity with more useful information.<br /><br />So yeah, alone in the role of discoverer, thankfully with a world of others in the benefits of that knowledge.<br /><br />The math pulls attention in and of itself. And with good reason. Say some person wants to do something needed with some mathematical tool to solve an important problem, will pick best available if know of, and can, or not?<br /><br />If you think not, I disagree, as from what I've seen--will pick best available.<br /><br />Your discovery is not about a committee.<br /><br />It is not up to other humans to decide the value of your work with their opinions, but for your fellow humans to show they value with use. We do not talk about calculus or algebra as about an opinion.<br /><br />Obvious to most actually I think, but to the money obsessed? May believe money matters which is so much about celebrity in cultures where celebrity rules. Yet how many track those celebrities over decades? Their moments in the intensity of attention are often barely a few years, or often less.<br /><br />In mathematics we know people over centuries and even millennia. There is no comparison from modern celebrity. And often am a huge fan, but also do watch modern celebrities come, and modern celebrities go. Audiences endlessly wanting the new. And why not? Our expectations keep rising as learn what is possible.<br /><br />Was actually such a HUGE advantage for me escaping that noise. And learned what is actually necessary, and turns out it is not. Which has helped me as learned my role. My global position results from my discovery but did not come with an instruction manual. Have worked at figuring out.<br /><br />No person who understands math, or pursues for the right reasons gets lost on those details for long. Math is not a zone for instant celebrity, and soon thereafter flame-outs. It is an arena where you can help people to know, for as long as there are humans.<br /><br />Which is comforting really. And I think is important for me to emphasize. Human systems are what change with the new. Like when algebra arrived, mathematics changed rapidly for the better as our species learned more with greater efficiency. The humans back then? Even if their opinion was recorded do we care?<br /><br />The great thing: great mathematical tools with which can know more. Discovery rules.<br /><br /><br />James HarrisJames Harrishttps://plus.google.com/104012861017102472975noreply@blogger.com0tag:blogger.com,1999:blog-11119110.post-54525430040189606522018-03-11T11:27:00.000-04:002018-03-11T21:22:58.614-04:00Some science speculationHave tried to refrain from talking physics things much, especially while working at being certain of ideas. But now am more like, why not? But still will emphasize as speculation.<br /><br />So yeah, was impossible not to think of quarks when considering the entanglement have talked quite a bit, including <a href="http://somemath.blogspot.com/2018/01/basic-coverage-problem-research-overview.html" target="_blank">some recently</a> may as well give a link there.<br /><br />So we know algebraically there is an entanglement which I found first with cubics, which would maybe mean related equations help determine quark behavior.<br /><br />But ended up focusing on quadratics as EASIER but realized might mean that there is a quadratic case in physics--undiscovered.<br /><br />Best guess as to where? Electrons.<br /><br />Could explain much about electron behavior, if we think is a single particle but is actually two VERY tightly coupled particles, where is just vastly closer than with bosons. Talking as if you have physics knowledge as easier for me, but also still dealing with basics most folks should have.<br /><br />Then again, problem there is now maybe can develop better quantum chromodynamics where have been problems. Is more difficult than quantum mechanics, but of course is the physics of the nucleus.<br /><br />That means could get everything from nuclear fusion, to more advanced nuclear weapons.<br /><br />Like how casually I drop that one, where have pondered for over a decade. And for me, quarks making more sense!--Would be so HUGE is hard to really express. But does the math actually apply there? Takes the real deal physicists to be sure. Just have an undergraduate degree and from two decades ago.<br /><br />Where I emphasize is speculation, but if true, and some other countries take me seriously, and work it, the United States could wake up to a world where other nations have better nuclear weapons. Where sky is the limit there.<br /><br />Would change the global order of things, immediately. And as an American? Am not exactly excited at that possibility, of a humbled US having to change how it does all its international politics.<br /><br />But science does not care. And technology does not care. Where hesitate to note, but my emotion in wanting my situation resolved is also not relevant.<br /><br />History is full of such stupid stories. I do not want to wake up to learning should have sounded the alarm, when a nuclear explosion in my country reveals things have changed. Where may not even know who set it off. The potential technology may be there.<br /><br />We think nuclear arms need uranium, like to get to plutonium, but what if they do not?<br /><br />Could be a simple innovation maybe, if have the math you need, to get the physics, and entire world different as a result. Ad then the United States no longer the dominant world power.<br /><br />Since am speculating may as well give scarier scenario where wouldn't have to worry about delivery, if weapons didn't give off radiation. Then could just use a global shipping company.<br /><br />Could get REALLY wild speculating as hey, I like to write and could be a fiction scenario some day, but here feel like made my point, so will stop myself there, as could make it scarier.<br /><br />Where think is understandable would want to be VERY sure about my math before putting such speculation out. Think have said some before and pulled back. But do have these innovations with math which could simplify quite a bit. History shows such things cannot be simply ignored.<br /><br />Why did I spend so much space on negatives though? Just mentioned maybe nuclear fusion, which would just be cool, I think, and went on and on about other. Guess is too easy to focus on negatives for me? But maybe is natural here.<br /><br />Such things have pondered for years. The dragging from mathematicians I now believe is just typical against such stunning innovation as have found <a href="https://beyondmund.blogspot.com/2018/03/foundations-in-our-new-reality.html" target="_blank">where talk example</a> with rideshare on my other blog.<br /><br />That the math I found might be used by our reality though? Of course as someone with a physics degree find that intriguing, but also note, is speculation. May as well put down though, like with the electron, if true?<br /><br />Then THAT would be yet another thing for me in the history books. And one of the greatest as well.<br /><br />Yeah definitely need that in writing, just in case am right.<br /><br /><br />James HarrisJames Harrishttps://plus.google.com/104012861017102472975noreply@blogger.com0tag:blogger.com,1999:blog-11119110.post-55149816618625481432018-03-03T16:29:00.000-05:002018-03-04T08:50:47.076-05:00When proof rules emotionOne benefit appreciate now more than could in the past was the questioning of certainty. Before did have situations where thought that proof was about a certainty of having checked everything. And could go over mathematical equations over and over again, thinking were correct. But thankfully had that collapse more than once.<br /><br />People can talk mathematical proof, yet how do you know something is a mathematical proof? Answering that question for me became much about identities.<br /><br />But still there is that fear of being a person who can look at something believing is true, and mind can play tricks on you. Which I think is how the mathematical discipline has turned to other eyes and checking by others. But human fallibility remains possible. Especially with complex arguments where I DO think computers will handle all proof checking, eventually.<br /><br />For me though practical reality after looked into computerized proof checking, years ago, even contacting an expert in the area and getting nowhere, pushed me first with functional ideas, and then to number authority.<br /><br />So defined mathematical proof functionally. But also now like to look at things like:<br /><br /><b>(46<sup>2</sup> + 48<sup>2</sup> + 72<sup>2</sup>)(172<sup>2</sup> + 258<sup>2 </sup>+ 430<sup>2 </sup>+ 602<sup>2</sup> + 1762<sup>2</sup>) = </b><br /><b><br /></b><b> 615<sup>2 </sup>+ 3075<sup>2</sup> + 14145<sup>2</sup> + 15990<sup>2</sup> + 188497<sup>2</sup> </b><b> = 77<sup>4</sup>*2<sup>10</sup></b><br /><br />Which is my favorite one lately. It was found using the BQD Iterator. Where was easy to figure out, but the cool thing is the numerical perfection. Once calculated you know must be perfect. Checking is easy enough with modern systems.<br /><br />Compare though with:<br /><br /><b>(x<sup>2</sup> + 2y<sup>2</sup>)(u<sup>2</sup> + 3v<sup>2</sup>)(x'<sup>2</sup> + 4y'<sup>2</sup>)(u'<sup>2</sup> + 5v'<sup>2</sup>) = p<sup>2</sup> + 359q<sup>2</sup></b><br /><br />And finding integer solutions for all the variables is easy.<br /><br class="Apple-interchange-newline" />x = 1, y = 2, u = 2, v = 2, x' = 3, y' = 2, u' = 4, v' = 2, p = 358, q = 2<br /><br />First iteration is easiest.<br /><br />Both examples also should say demonstrate modular in that am using BQD Iterator with basic form u<sup>2</sup> + Dv<sup>2</sup> = F.<br /><br />So that module is used twice for first, where expanded with a technique for having as many squares as wanted, and four times for second. So went less fancy with second example and left in variables versus putting in solutions like with prior one.<br /><br />But feels so different to me regardless. There is logic and there is the feeling around looking at things. Now a step backwards to symbols, and is not so easy for me at least to feel certainty. Though mathematical proof IS there. And is interesting I think the emotion can feel despite the proof. Which in my experience?<br /><br />Is great! The proof does not care about my emotion. When I do the math, it works.<br /><br />Contrast with emotion the other way, where for instance trusting humans, you find that trust disappointed. Would rather be skeptical and find my skepticism overturned than go that route, as have done before.<br /><br />I prefer to talk to the math.<br /><br />The math will never make a mistake. The math will never be wrong.<br /><br />The math is always right.<br /><br />To me, when proof rules emotion you know as you go upside down. Trusting not your faith in your ability to look over the mathematics. Which is weird, huh?<br /><br />So I go through the checks. Like use my definition of mathematical proof to check, every aspect from beginning to end. So I know have a proof, intellectually. Logically it is perfect. Yet part of me still is hesitant, until the relief, when run some numbers.<br /><br />That burst of positive have found works vastly better to continue, than the other ever did.<br /><br />Numbers rule.<br /><br /><br />James HarrisJames Harrishttps://plus.google.com/104012861017102472975noreply@blogger.com0tag:blogger.com,1999:blog-11119110.post-31027067964737598952018-02-27T18:51:00.003-05:002018-02-28T07:49:23.933-05:00Puzzling over modular effectivenessModular 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.<br /><br />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.<br /><br />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.<br /><br />Would go on to develop a full modular algebra where like to give as easy for context:<br /><br />x+y+vz = 0(mod x+y+vz)<br /><br />Which is the simplest form I found with some research, and I call, a tautological space. Have created terminology over more advanced forms.<br /><br />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.<br /><br />Who knew? How could anyone? Watching modular algebra do algebra? Is surreal.<br /><br />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.<br /><br />BUT can now also note that Gauss was am sure doing something I don't need to do any more--complex algebraic manipulations.<br /><br />Letting the math do the work is more effective and complete. Is faster, and is of course easier.<br /><br />The math easily checks infinite possibilities in a modular way. The math is complete intelligence.<br /><br />Those are the kinds of things that tend to show you where human beings will go, in time.<br /><br />Why do we not notice certain things? Probably is about how we are built.<br /><br />Of course, how do we know what we do not know?<br /><br />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.<br /><br />Which then lets us look at what we humans did not find on our own, and wonder.<br /><br />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.<br /><br />And that was September 2008, with: <a href="http://somemath.blogspot.com/2008/09/quadratic-diophantine-result.html" target="_blank">Quadratic Diophantine Result</a><br /><br />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.<br /><br />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.<br /><br />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.<br /><br />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.<br /><br />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.<br /><br />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.<br /><br />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.<br /><br />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.<br /><br />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.<br /><br />My view is the web has made the difference as notice, how are you reading these words?<br /><br /><br />James HarrisJames Harrishttps://plus.google.com/104012861017102472975noreply@blogger.com0tag:blogger.com,1999:blog-11119110.post-55607323184411271582018-02-23T07:30:00.000-05:002018-03-01T19:51:32.148-05:00Thoughts on math developmentThroughout 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.<br /><br />So you can have mathematical arguments which are lots about manipulating algebraically. Well what if you let the math do it instead?<br /><br />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?<br /><br />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.<br /><br />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.<br /><br />Like ran into the complexity of:<br /><br /><b>c<sub>1</sub>x<sup>2</sup> + c<sub>2</sub>xy + c<sub>3</sub>y<sup>2</sup> = c<sub>4</sub>z<sup>2</sup> + c<sub>5</sub>zx + c<sub>6</sub>zy</b><br /><br />where the c's are constants, which is so great has never been as simply and generally reduced as possible until my methods.<br /><br />The advance was:<br /><br />x+y+vz = 0(mod x+y+vz)<br /><br />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.<br /><br />One of the basic things I have done is for some prime p use: v = -(x+y)z<sup>-1</sup> mod p<br /><br />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.<br /><br />Where the math did all the work.<br /><br />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.<br /><br />And my favorite example like to <a href="http://somemath.blogspot.com/2014/11/binary-quadratic-diophantine-iterator.html" target="_blank">call the BQD Iterator</a>:<br /><br />It must be that if you have:<br /><br />u<sup>2</sup> + Dv<sup>2</sup> = F<br /><br />then it must also be true that<br /><br />(u-Dv)<sup>2</sup> + D(u+v)<sup>2</sup> = F(D+1)<br /><br />-------------------------------------------------------------<br />Easily demonstrated:<br /><br class="Apple-interchange-newline" />1. x<sup>2</sup> - 2y<sup>2</sup> = 1<br /><br />2. (x+2y)<sup>2</sup> - 2(x+y)<sup>2</sup> = -1<br /><br />3. (3x+4y)<sup>2</sup> - 2(2x + 3y)<sup>2</sup> = 1<br /><br />4. (7x + 10y)<sup>2</sup> - 2(5x + 7y)<sup>2</sup> = -1<br /><br />5. (17x + 24y)<sup>2</sup> - 2(12x + 17y)<sup>2</sup> = 1<br /><br />and you can keep going out to infinity.<br /><br />Have looked at that simplicity for years wondering, why new? Why didn't human beings find that just playing around?<br /><br />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.<br /><br />The math does not have human limitations in that regard.<br /><br />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.<br /><br />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.<br /><br />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.<br /><br />That the math can do math itself is something I write just to read it, over and over again.<br /><br />Then with tautological spaces you are more like a researcher asking the math questions.<br /><br />Which means which questions you ask become more definitive in terms of what you can learn, but the primary thing is, the math answers.<br /><br class="Apple-interchange-newline" />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.<br /><br />We just have to be sure as a species. And reality is, in that quest for certainty, there are going to be consequences.<br /><br /><br />James HarrisJames Harrishttps://plus.google.com/104012861017102472975noreply@blogger.com0tag:blogger.com,1999:blog-11119110.post-53606539396076480612018-02-20T06:43:00.000-05:002018-02-22T07:35:18.984-05:00When primes do not careOne 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?<br /><br />Putting that mathematically is easy and did so, with my prime residue axiom. But before that had done much with my <a href="http://somemath.blogspot.com/2006/08/prime-gap-equation.html" target="_blank">prime gap equation</a> where notice have not talked it much since, and linking to a post made August 2006.<br /><br />And next public post I find is <a href="http://somemath.blogspot.com/2010/02/prime-residue-axiom.html" target="_blank">this one</a> from February 2010, where give my prime residue axiom:<br /><br />Given differing primes p<sub>1</sub> and p<sub>2</sub>, where p<sub>1</sub> > p<sub>2</sub>, there is no preference for any particular residue of p<sub>2</sub> for p<sub>1</sub> mod p<sub>2</sub> over any other.<br /><br />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.<br /><br />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.<br /><br />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.<br /><br />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?<br /><br />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.<br /><br />Talked for a different audience on my blog Beyond Mundane: <a href="https://beyondmund.blogspot.com/2014/07/no-prime-preference.html" target="_blank">No prime preference?</a><br /><br />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.<br /><br />I like that bold.<br /><br /><br />James HarrisJames Harrishttps://plus.google.com/104012861017102472975noreply@blogger.com0tag:blogger.com,1999:blog-11119110.post-75581254219172721322018-02-12T07:39:00.000-05:002018-02-19T07:34:02.455-05:00Math, innovation and inventionFinally 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.<br /><br />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.<br /><br />The innovation is of extreme interest considering its demonstrated power. Gist of it is so easy though:<br /><br />x+y+vz = 0(mod x+y+vz)<br /><br />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.<br /><br />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.<br /><br />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.<br /><br />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<br /><br />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.<br /><br />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:<br /><br />dS(x,p<sub>j</sub>) = [x/p<sub>j</sub>] - 1 - (j-1) - S(x/p<sub>j</sub>, p<sub>j</sub>-1)<br /><br />where S(x/p<sub>j</sub>, p<sub>j</sub>-1) is the count of composites that multiply times p<sub>j</sub> to give a product less than or equal to x, where notice that p<sub>j</sub> must be less than or equal to sqrt(x) or the composite count given by [x/p<sub>j</sub>] - 1 will not be correct.<br /><br />So the dS function is the count of composites for a particular prime excluding composites that are products of lesser primes.<br /><br />Reference: <a href="https://somemath.blogspot.com/2012/10/composite-counting-functions-and-prime.html" target="_blank">Composite counting functions and prime counter</a><br /><br />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.<br /><br />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.<br /><br />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.<br /><br />That just keeps getting better too.<br /><br /><br />James HarrisJames Harrishttps://plus.google.com/104012861017102472975noreply@blogger.com0tag:blogger.com,1999:blog-11119110.post-74950064592841677332018-02-08T07:34:00.000-05:002018-02-11T11:03:14.257-05:00Benefit reality of unique solitary researchBack 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?<br /><br />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:<br /><br />x+y+vz = 0(mod x+y+vz)<br /><br />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.<br /><br />The specific trend that has made the difference in my research am thinking, is that I look modular, first.<br /><br />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.<br /><br />And then after talking out on math groups, when faced criticisms would switch to the explicit:<br /><br />x+y+vz = x+y+vz<br /><br />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.<br /><br />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 <i>all the properties of the original expression</i>, as it must. And made an <a href="https://somemath.blogspot.com/2014/10/example-showing-truth-logic-and.html" target="_blank">absolute proof demo post</a> where prove that:<br /><br /><i>If:</i> <b>x<sup>2</sup> + y<sup>2</sup> = z<sup>2</sup> </b><i>then</i>: <b>(v<sup>2</sup> - 1)z<sup>2</sup> - 2xy = 0(mod x+y+vz)</b>, where v can be any value.<br /><br />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.<br /><br />That v can be any number keeps being fascinating to me. Like let v = i.<br /><br />Then: -2z<sup>2</sup> - 2xy = 0(mod x+y+iz)<br /><br />Which is then easily verified, by algebra.<br /><br />Still would like looking at numbers, like: -74 = 0(mod 3+4+5i) = 0(mod 7+5i)<br /><br />And of course: 74 = (7+5i)(7-5i)<br /><br />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.<br /><br />Now I simply note: invented my own math discipline.<br /><br />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.<br /><br />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:<br /><br /><a href="https://somemath.blogspot.com/2012/06/tautological-spaces-and-calculus.html" target="_blank">Tautological spaces and calculus</a><br /><br />But yeah was a challenge for me. Worked at it for a couple of YEARS too.<br /><br />Is interesting, when you keep with something even when it doesn't do what you want.<br /><br />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.<br /><br />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.<br /><br />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.<br /><br />Was in complete control with no pressure. Still am.<br /><br />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.<br /><br />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.<br /><br />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.<br /><br />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.<br /><br /><br />James HarrisJames Harrishttps://plus.google.com/104012861017102472975noreply@blogger.com0tag:blogger.com,1999:blog-11119110.post-27825964838510571582018-02-06T07:27:00.000-05:002018-02-10T11:38:46.892-05:00Solving trinary quadratic with reduced constraintUsed complex identities I call tautological spaces to analyze:<br /><br /><b>c<sub>1</sub>x<sup>2</sup> + c<sub>2</sub>xy + c<sub>3</sub>y<sup>2</sup> = c<sub>4</sub>z<sup>2</sup> + c<sub>5</sub>zx + c<sub>6</sub>zy</b><br /><br />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.<br /><br />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:<br /><br />u<sup>2</sup> - Dv<sup>2</sup> = Fw<sup>2</sup><br /><br />Which looks much like the binary quadratic reduced form.<br /><br />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.<br /><br />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.<br /><br />Which can see with an example.<br /><br />Let c<sub>1</sub> = 1, c<sub>2</sub> = 2, c<sub>3</sub> = 3, c<sub>4</sub> = 4, c<sub>5</sub> = 5, c<sub>6</sub> = 6, so:<br /><br /><b>x<sup>2</sup> + 2xy + 3y<sup>2</sup> = 4z<sup>2</sup> + 5xz + 6yz</b><br /><br />Which with my <a href="https://somemath.blogspot.com/2018/02/three-variable-quadratic-reduction.html" target="_blank">method for reducing</a> gives:<br /><br />A = -8, B = -20, and C = 33<br /><br />So: <b>[-4(x+y) + 10z]<sup>2</sup> = 166z<sup>2 </sup>- 2m<sup>2</sup></b><br /><br />And I notice that m = 9z, gives me a useful substitution:<br /><br />And [-4(x+y) + 10z]<sup>2</sup> = 166z<sup>2 </sup>- 162z<sup>2</sup> = 4z<sup>2</sup><br /><br />So now have: -4(x+y) + 10z = +/-2z, so: <b>x+y = 2z or 3z</b><br /><br />Which means can always have integer solutions with an integer z, with one reduced constraint.<br /><br />Let's trot through a full solution. Let z = 10, and use first, so: x+y = 20, so y = 20 - x<br /><br />x<sup>2</sup> + 2x(20-x) + 3(20-x)<sup>2</sup> = 400 + 50x + 60(20 - x), and multiplying out:<br /><br />x<sup>2</sup> + 40x -2x<sup>2</sup> + 1200 - 120x + 3x<sup>2</sup> = 400 + 50x + 1200 - 60x<br /><br />Where get easily enough:<br /><br />x<sup>2</sup> - 35x - 200 = 0, which is: (x-40)(x+5) = 0<br /><br />And one full solution then is: x = -5, y = 25, z = 10<br /><br />Where just picked some easy.<br /><br /><br />James HarrisJames Harrishttps://plus.google.com/104012861017102472975noreply@blogger.com0tag:blogger.com,1999:blog-11119110.post-73746767041684227622018-02-05T11:49:00.000-05:002018-02-07T07:18:55.851-05:00Trinary Quadratic IteratorFinally looked at not setting z=1 to what I now know is a general way to reduce a trinary quadratic equation like:<br /><br /><b>c<sub>1</sub>x<sup>2</sup> + c<sub>2</sub>xy + c<sub>3</sub>y<sup>2</sup> = c<sub>4</sub>z<sup>2</sup> + c<sub>5</sub>zx + c<sub>6</sub>zy</b><br /><br />where the c's are constants, where was <a href="https://somemath.blogspot.com/2018/02/three-variable-quadratic-reduction.html" target="_blank">able to prove</a> can be generally reduced.<br /><br />Shows yet another type of general reduced form: u<sup>2</sup> - Dv<sup>2</sup> = Fw<sup>2</sup><br /><br />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.<br /><br /><b>A</b> = (c<sub>2</sub> - 2c<sub>1</sub>)<sup>2</sup> + 4c<sub>1</sub>(c<sub>2</sub> - c<sub>1</sub> - c<sub>3</sub>), <b>B</b> = (c<sub>2</sub> - 2c<sub>1</sub>)(c<sub>6</sub> - c<sub>5</sub>) + 2c<sub>5</sub>(c<sub>2</sub> - c<sub>1</sub> - c<sub>3</sub>)<br /><br />and<br /><br /><b>C</b> = (c<sub>6</sub> - c<sub>5</sub>)<sup>2</sup> - 4c<sub>4</sub>(c<sub>2</sub> - c<sub>1</sub> - c<sub>3</sub>)<br /><br />Base result is: <b>A(x+y)<sup>2</sup> - 2B(x+y)z + Cz<sup>2</sup> = m<sup>2</sup></b><br /><br />And some simple algebra gives:<br /><br /><b>[A(x+y) - Bz]<sup>2</sup> - Am<sup>2 </sup> = (B<sup>2</sup> - AC)z<sup>2</sup></b><br /><br />With: u<sup>2</sup> - Dv<sup>2</sup> = Fw<sup>2</sup><br /><br />So: c<sub>1</sub> = 1, c<sub>2</sub>= 0, c<sub>3</sub> = -D, c<sub>4</sub> = F, c<sub>5</sub> = 0, c<sub>6</sub> = 0,<br /><br />x = u, y = v, z = w<br /><br class="Apple-interchange-newline" />So, A = 4 +4(-1+D) = 4D, B = 0, C = -4F(-1+D)<br /><br />Gives: [4D(u+v)]<sup>2</sup> - 4Dm<sup>2 </sup> = (16DF(-1+D))w<sup>2</sup> And<br /><br />Which is: m<sup>2</sup>/4- D(u+v)<sup>2</sup> = -F(D-1)w<sup>2</sup><br /><br />Where now need m. 4D(u+v)<sup>2</sup> - 4F(-1 + D)w<sup>2</sup> = m<sup>2</sup><br /><br />So: m<sup>2</sup> = 4(Du<sup>2</sup> + 2Duv +Dv<sup>2</sup> + Fw<sup>2</sup> - DFw<sup>2</sup>),<br /><br />and m<sup>2</sup> = 4(Du<sup>2</sup> + 2Duv +Dv<sup>2</sup> + u<sup>2</sup> - Dv<sup>2</sup> - DFw<sup>2</sup>)<br /><br />Where showing all the detail for once. Helps keep me from making mistakes.<br /><br />So: m<sup>2</sup> = 4(Du<sup>2</sup> + 2Duv + u<sup>2</sup> - Du<sup>2</sup> + D<sup>2</sup>v<sup>2</sup>), and m<sup>2</sup> = 4(u<sup>2</sup> + 2Duv + D<sup>2</sup>v<sup>2</sup>) = 4(u+Dv)<sup>2</sup><br /><br />Which is: <b>(u+Dv)<sup>2 </sup>- D(u+v)<sup>2</sup> = F(-D+1)w<sup>2</sup></b><br /><br />So the trinary quadratic iterator is just the BQD Iterator with a w<sup>2</sup> on the end.<br /><br />Shall call it the TQ Iterator for short.<br /><br /><br />James HarrisJames Harrishttps://plus.google.com/104012861017102472975noreply@blogger.com0tag:blogger.com,1999:blog-11119110.post-40982327230809745022018-02-04T09:04:00.001-05:002018-02-04T16:50:32.501-05:00Three variable quadratic reductionHave 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:<br /><br /><b>c<sub>1</sub>x<sup>2</sup> + c<sub>2</sub>xy + c<sub>3</sub>y<sup>2</sup> = c<sub>4</sub>z<sup>2</sup> + c<sub>5</sub>zx + c<sub>6</sub>zy</b><br /><br />where the c's are constants. And found what I call my Quadratic Diophantine Theorem which I talk in <a href="https://somemath.blogspot.com/2008/09/quadratic-diophantine-result.html" target="_blank">this post</a> from September 2008.<br /><br />And figured out:<br /><br />((c<sub>2</sub> - 2c<sub>1</sub>)<sup>2</sup> + 4c<sub>1</sub>(c<sub>2</sub> - c<sub>1</sub> - c<sub>3</sub>))(x+y)<sup>2</sup> - (2(c<sub>2</sub> - 2c<sub>1</sub>)(c<sub>6</sub> - c<sub>5</sub>) + 4c<sub>5</sub>(c<sub>2</sub> - c<sub>1</sub> - c<sub>3</sub>))(x+y)z + [(c<sub>6</sub> - c<sub>5</sub>)<sup>2</sup> - 4c<sub>4</sub>(c<sub>2</sub> - c<sub>1</sub> - c<sub>3</sub>)]z<sup>2</sup> = m<sup>2</sup><br /><br />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.<br /><br />Way too complicated looking so introduced some more variables to collect things.<br /><br /><b>A</b> = (c<sub>2</sub> - 2c<sub>1</sub>)<sup>2</sup> + 4c<sub>1</sub>(c<sub>2</sub> - c<sub>1</sub> - c<sub>3</sub>), <b>B</b> = (c<sub>2</sub> - 2c<sub>1</sub>)(c<sub>6</sub> - c<sub>5</sub>) + 2c<sub>5</sub>(c<sub>2</sub> - c<sub>1</sub> - c<sub>3</sub>)<br /><br />and<br /><br /><b>C</b> = (c<sub>6</sub> - c<sub>5</sub>)<sup>2</sup> - 4c<sub>4</sub>(c<sub>2</sub> - c<sub>1</sub> - c<sub>3</sub>)<br /><br />where neither A nor B is zero for what follows.<br /><br />Then have: <b>A(x+y)<sup>2</sup> - 2B(x+y)z + Cz<sup>2</sup> = m<sup>2</sup></b><br /><br />Which is easier to handle. And multiplying through by A, moving some things around, and completing the square:<br /><br />A<sup>2</sup>(x+y)<sup>2</sup> - 2BA(x+y)z + B<sup>2</sup>z<sup>2</sup> - Am<sup>2 </sup> = B<sup>2</sup>z<sup>2</sup> - CAz<sup>2</sup><br /><br />Which is easily enough:<br /><br /><b>[A(x+y) - Bz]<sup>2</sup> - Am<sup>2 </sup> = (B<sup>2</sup> - AC)z<sup>2</sup></b><br /><br />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.<br /><br />Shows yet another type of general reduced form: u<sup>2</sup> - Dv<sup>2</sup> = Fw<sup>2</sup><br /><br />Maybe should come up with another cool name for iterator am sure can get from it. Maybe later.<br /><br />As a first example let c<sub>1</sub> = 1, c<sub>2</sub> = 1, c<sub>3</sub> = 1, c<sub>4</sub> = 1, c<sub>5</sub> = 1, c<sub>6</sub> = 1, so:<br /><br /><b>x<sup>2</sup> + xy + y<sup>2</sup> = z<sup>2</sup> + zx + zy</b><br /><br />Which gives:<br /><br />A = -3, B = -2, and C = 4<br /><b><br /></b>[-3(x+y) + 2z]<sup>2</sup> + 3m<sup>2 </sup> = 16z<sup>2</sup><br /><br />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:<br /><br />-3(x+y) + 2z = +/- 4z, so: -3(x+y) = 2z or -6z<br /><br />So there will be a set of integer solutions for every nonzero integer z.<br /><br />Well I DO like easy. So yeah, very straightforward from my original research.<br /><br />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.<br /><br />So now have the three variable general reduced form for the quadratic case.<br /><br /><br />James HarrisJames Harrishttps://plus.google.com/104012861017102472975noreply@blogger.com0tag:blogger.com,1999:blog-11119110.post-37053786319658694762018-01-29T06:55:00.002-05:002018-01-31T10:57:20.248-05:00Some facts as baselineWell may as well baseline current somewhat. While also pondering what may be doing going forward.<br /><br />One of the coolest things early was figuring out a simple way to count prime numbers, which I talk endlessly; relies yes on previously known ideas, but with a slight but powerful tweak. And not only had my own way to count primes, eventually realized lead to a partial differential equation. Which did have me at one point actually reading Riemann's notes, in original German as studied some. Was an odd feeling there but that was over a decade ago.<br /><br />My find of a coverage problem showed the efficacy of my tautological spaces which also helped with a new way to reduce binary quadratic Diophantine equations. Have expanded on tautological spaces so are general so can tackle other things. And have my ring of objects. Actually may as well admit am usually doing things in the ring of objects, though not declared.<br /><br />There is my prime residue axiom. And also have some additional tools for probing numbers where the BQD Iterator is one of my most widely used. Amazes me how versatile it is.<br /><br />Where weirdly played with VASTLY more after giving a cool name. Had had for years. What's up with that? Naming things. Who knew?<br /><br />There are plenty of things with which can occupy my time when I feel like it. Or simply review, thinking to myself--I found that.<br /><br />Oh and of course still trying to emotionally grasp having my own modular inverse method. That just came like a bolt from the blue last year.<br /><br />Am lucky. The research has simply been rewarding. Am a solitary researcher where don't even pretend otherwise but maybe in the past didn't so boldly declare. Now find that comforting to state clearly. And don't have to do things would never have wanted to do, like be an academic.<br /><br />Is so perfect a situation in certain ways from my perspective. Yet at times that makes me wonder.<br /><br />The math isn't going anywhere. Once found and shared, is simply available for our species.<br /><br />Clearly have to a large extent relished controversy around my research, where guess that hasn't been secret. And been grateful to quit even bothering with trying to talk to mathematicians, despite occasional things here or there where feel a duty. If up to me? Would be done with them for good. As well as emphasizing am not one and will not become one. The math is the draw. Web enables people to get it when they need it.<br /><br />Other things? Not needed. And now would simply be tedious, as going through motions needed in a world that has gone away.<br /><br />Even now I work more to handle the constant global attention. More aware I know than others.<br /><br />Which actually is true. Means have to totally let go of blaming math people for other problems which are just about me, like with money. The web does free you up from needing all that academic machinery including journals if you're NOT into it and NOT an academic, so don't need the silos which were needed in the past.<br /><br />Yet it also pushes you to rapidly figure out how to then do ok, without the financial support. But at least the lack of approval from mathematicians have deemed irrelevant. Public does NOT notice or care. Just can't really talk math to them anyway. And why bother? Let interest drive people here.<br /><br />Well enough for a basic baseline. Finishing up things has me better defining my research as abstract reductionism realized. And much of what I do is dominated by modular. And having introduced procedures that have given the world's first true modular algebra, there is so much can do both there and beyond.<br /><br />Is SO cool.<br /><br />Still do have responsibilities feel like to mathematical industry is how I like to phrase. And have stated will revisit in 2028, God willing. And if necessary.<br /><br />Now though with so much settled can get back to just enjoying a continuing conversation with the math. More contemplating what I already have. NO plans on active research.<br /><br />Have had enough fun there.<br /><br /><br />James HarrisJames Harrishttps://plus.google.com/104012861017102472975noreply@blogger.com0tag:blogger.com,1999:blog-11119110.post-37941534479621182102018-01-27T18:50:00.001-05:002018-01-28T07:06:13.072-05:00Social problem evaluation complete for a decadeAs much as I think information travels is not necessarily the case that academic mathematicians are aware of my research. Even with sending by email am not sure was read.<br /><br />Which is ok. If academics do not know that would of course put academics off the hook! And at this point in time is not readily something can evaluate.<br /><br />My past evaluations are pending waiting for more information. So they will remain in place.<br /><br />So will let more time pass. And God willing, will return to this concern in a decade if necessary.<br /><br />Which lets me return blog to usual type postings, so will let this post end current investigation into the social problem.<br /><br />So next posting on that subject, noting to self, should not be before January 2028.<br /><br />Blog will now return to math subjects as interests me, without worrying on such matters.<br /><br /><br />James HarrisJames Harrishttps://plus.google.com/104012861017102472975noreply@blogger.com0tag:blogger.com,1999:blog-11119110.post-4720872687593159882018-01-26T10:16:00.001-05:002018-01-27T08:25:30.578-05:00Social implications from modular inverse discovery receptionOne of the more important recent events for me and also for mathematical research was my find of a new primary way to calculate the modular inverse, which was posted here May 5th 2017. With such a major research find was able to settle any number of questions in the social domain as well which helped me evaluate modern academia.<br /><br />What is easily established is: is a major discovery without concern of validity, is a surprising discovery as a primary in this area is a surprise, and there is no justification possible for mathematicians of any type, either applied or pure to ignore.<br /><br />Also the result in and of itself has pure and applied mathematical characteristics, so in essence, is a perfect litmus test across multiple categories.<br /><br />And the mathematical academic world failed that test.<br /><br />Will note my diligence as a researcher and discoverer did mean again at least sending to some academic mathematicians and also to the NSA, twice. No point in naming the few mathematicians though will admit at least some were at a University of California school.<br /><br />There was no response from academics to emails. NSA responded form letter to first effort submitted through their website. Did not respond to second effort.<br /><br />Regardless as is usual with my mathematical discovery, web search has been more revealing, and there are searches which have trended is the appropriate phrase telling myself, higher. Will not post the searches am checking in this post as that can impact their behavior which am still studying. Though have noted elsewhere as also experimented by posting on Reddit, which have discussed on this blog. And primarily attracted trolls.<br /><br />This failure of the mathematical community is fascinating to me though. And DID explore more what I call the social problem as considered it. However, is kind of easy to understand from an evolutionary and anthropological perspective. Am not sure on that phrasing. Is easy if you consider humans as very much an ape species, which we are. Is too bad the truth revealed by science also has political tones. And our evolution beyond other ape species is shallow in many ways. Where intellectual behavior is involved with highest brain function, but am looking at purely naive territorial behavior.<br /><br />Like it ignores my ability to just talk out situation on web. Also ignores likely impact across academia which will likely be vastly larger than just for mathematicians. That ignores potential impact for mathematicians for generations in academic world as well.<br /><br />Academic process arrived to a large extent with Sir Isaac Newton in its more modern form. Though also much remains from prior am sure. Am not surprised a revisiting from top to bottom may be in order. Leverage from this result could be significant in that process.<br /><br />Yup, may be an opportunity to remake the modern university. Cool! Time will tell.<br /><br />To me as a mathematical discoverer though is an interesting challenge to mathematical truth.<br /><br />Which definitely indicates shows behaviors rooted in lower brain function because higher thought of course realizes how this story will end. The mathematics will win.<br /><br />In the meantime will admit situation allowed me to relax considerably. Cleaned up a few things including settling my 14 years or so leisurely consideration of coverage problem with algebraic integers, which also lets me champion my fascinating quadratic factorization number theoretic probing tool.<br /><br />While also continuing other activities which actually consume most of my time now. Is not like there is much pressure on me from a research perspective. Now I have more practical concerns often. So I just get to enjoy discovered things. Is cool.<br /><br /><br />James HarrisJames Harrishttps://plus.google.com/104012861017102472975noreply@blogger.com0tag:blogger.com,1999:blog-11119110.post-50136758582828394152018-01-25T07:19:00.001-05:002018-01-28T07:46:49.855-05:00Probing number properties with quadratic factorizationFeel extremely lucky that a simple quadratic factorization was available for checking a number of things, and was around 2010 started studying. Will talk how that basic factorization allows probing deeply into numbers.<br /><br />In the complex plane:<br /><br /><b>P(x) = (g<sub>1</sub>(x) + 1)(g<sub>2</sub>(x) + 2)</b><br /><br />where P(x) is a primitive quadratic with integer coefficients, g<sub>1</sub>(0) = g<sub>2</sub>(0) = 0, but g<sub>1</sub>(x) does not equal 0 for all x.<br /><br />And is VERY important that the g's are normalized. While studying with algebraic integer solutions may have been done, came up with a generalized way to do that--and beyond.<br /><br />Introduce k, where k is a nonzero, and new functions f<sub>1</sub>(x), and f<sub>2</sub>(x), where:<br /><b><br /></b><b>g<sub>1</sub>(x) = f<sub>1</sub>(x)/k</b> and<b> </b><b>g<sub>2</sub>(x) = f<sub>2</sub>(x) + k-2</b><br /><br />Multiply both sides by k, and substitute for the g's, which gives me the now symmetrical form:<br /><br /><b>k*P(x) = (f<sub>1</sub>(x) + k)(f<sub>2</sub>(x) + k)</b><br /><br />And introduce H(x), where: <b>f<sub>1</sub>(x) + f<sub>2</sub>(x) = H(x)</b><br /><br />Then just solve for one of the f's, and substitute so I can find:<br /><br /><b>f<sub>1</sub><sup>2</sup>(x) - H(x)f<sub>1</sub>(x) - kH(x) - k<sup>2</sup> + k*P(x) = 0</b><br /><br />Which gives me <i>total power to force a monic quadratic with integer coefficients</i> by how I choose k and H(x). And then you can solve for f<sub>1</sub>(x) using the quadratic formula:<br /><b><br /></b><b>f<sub>1</sub>(x) = (H(x) +/- sqrt[(H(x) + 2k)<sup>2</sup> - 4k*P(x)])/2</b><br /><br />And H(x) is a handle for every possible factorization with the g's, with k = 1 or -1, you just get algebraic integers if you have H(x) be a polynomial with integer coefficients.<br /><br />However, you now have that if k is some other nonzero integers besides 1 or -1, then one of the g's can be blocked from being an algebraic integer by: <b>g<sub>1</sub>(x) = f<sub>1</sub>(x)/k</b><br /><b><br /></b>That blocking of course occurs when non-rational.<br /><br />But the product then is still allowed to be an algebraic integer: <b>f<sub>1</sub>(x) = k</b><b>g<sub>1</sub>(x) </b><br /><b><br /></b>The entanglement with non-rationals is FASCINATING. And yet, k is proven to be a factor.<br /><br />Here should note is where you can introduce contradiction! If instead of declaring in the complex plane, could get contradiction or not if declared in ring of algebraic integers just by choice of k.<br /><br />The main point there is simple: one of the g's provably cannot be an algebraic integer if nonzero integer k is not 1 or -1, and the g's are not rational, when H(x) is a polynomial with integer coefficients. While one of the g's IS being forced to be an algebraic integer.<br /><br />Where simplicity in the proof comes from the f's being roots of the same quadratic. H(x) is key here. Depending on how is chosen you are then pushing back to pick the g's.<br /><br />Which is how the analyst is choosing from infinite possibility.<br /><br />Should admit was EXTREMELY surprised that the algebra DOES recognize as distinct roots of monic polynomials with integer coefficients i.e. algebraic integers.<br /><br />And did not fully appreciate against my biases until <a href="https://somemath.blogspot.com/2015/11/non-polynomial-factorization-short.html" target="_blank">this post November 2015</a>, when finally considered k = 1 or -1. Where had been considering algebraic integers as just kind of a mistake since my own research from 2003, so had around 12 years with that idea in mind. But the math actually has them as distinct and is curious in what ways versus other members of my ring of objects.<br /><br />Where at least know that there are other members NOT algebraic integers where multiplying times some non-unit, nonzero integer gives a product that is.<br /><br />Oh so yeah, bringing up my ring of objects which includes all integer-like numbers as well as integers themselves. And my ring of objects excludes any numbers not integer or integer-like, which was the approach I used to include them all. Switching to abstraction of exclusion to include versus attempting to get all with rules of inclusion.<br /><br />So much from a basic factorization! And also feel lucky and amazed with the power of abstraction shown with this approach! Finding is one thing, but is great for that mathematical logic to be there.<br /><br />These are mathematical pieces giving a handle on some aspect of infinity.<br /><br />That is wild. To me H(x) is fascinating with that ability to handle. It is meant to cover an infinite number of possible expressions where as a human researcher am focused very narrowly. Getting a handle on infinity in this way intrigues me as a powerful technique.<br /><br />Having a quadratic factorization available then leads to an easy way to thoroughly check on number theoretic properties.<br /><br />Reference post: <a href="https://somemath.blogspot.com/2017/10/simple-generalized-quadratic.html" target="_blank">Simple Generalized Quadratic Factorization</a><br /><br /><br />James HarrisJames Harrishttps://plus.google.com/104012861017102472975noreply@blogger.com0tag:blogger.com,1999:blog-11119110.post-34230885588471431692018-01-24T07:36:00.001-05:002018-01-24T12:13:14.640-05:00Pragmatic research perspective worksComing into doing my own looking for mathematics for fun was a great thing for me. So yeah there was never any reason to think would be collaborating with others as for all I knew would never find anything. And am NOT a mathematician, but am someone interested in numbers who has had a lot of of modern problem solving training.<br /><br />When I DID have a profound find and dutifully went to mathematicians was kind of thinking certain things would happen which did not.<br /><br />And I got to do solitary research exploring my own major research finds and gain a different perspective not just on the mathematical community, but also academia.<br /><br />With a need to know, the pursuit of truth was fun for me, and still is. So no problem there. Actually am type to do solitary research, so it just went well.<br /><br />However DID learn some things. And in my opinion many modern academics are desperate for that one thing or other which will give them a burst of attention, usually with others in their field, but SOMETIMES you can get news headlines, on which to build a career.<br /><br />And academics are terrified of looking bad, or upsetting the majority so work very hard to be sure to never get on the wrong side of their colleagues, especially with research interests, which I think is sad and maybe problematic. The truth does not care. What isn't discovered as a result?<br /><br />Which to me is the celebrity mentality. Could replace academic with singer or actor, and is the same dynamic.<br /><br />Looking for that Big Break. Wanting to exploit for as much as you can to get status. Then milk that status for all its worth where academics DO get an advantage there in terms of longevity with it. Though for celebrities can kind of get equivalent, like with a Grammy Award or an Academy Award, which can tout for a lifetime.<br /><br />That to me explains what I call the social problem--like full mathematicians with advanced degrees, behave as if social opinion is all that matters.<br /><br />Yeah for THEIR careers.<br /><br />So I learned functionally reality of what it is like to be a mathematician. And no, do not want to be a mathematician.<br /><br />Am thankful escaped ever thinking I might want to be.<br /><br />Almost became a physicist, and yeah, glad simply escaped modern academia.<br /><br />The ideal of academia is giving people freedom to pursue best ideas to further human knowledge. And much of that does get done am sure. My support of academia remains though more is with love of the ideal will admit.<br /><br />In my opinion, modern academia presented to the public is more often defined by people looking for a career than people searching for truth. And it shows.<br /><br />Public is ever more skeptical, as am I, as am part of the public too. Am NOT an academic and have no interest in becoming one. And am far more testing of the reality now.<br /><br />The wannabe celebrity types? Are so easy for me to spot now.<br /><br /><br />James HarrisJames Harrishttps://plus.google.com/104012861017102472975noreply@blogger.com0tag:blogger.com,1999:blog-11119110.post-36913306132880948142018-01-22T11:32:00.001-05:002018-01-24T07:05:57.094-05:00Basic coverage problem research overviewMy most challenging result thought would have help from mathematicians, which is key, as am NOT a mathematician as I routinely emphasize. And publication seemed like a good start, back about 14 years ago when a key paper highlighting the ability <i>to appear to prove</i> something not true with valid methods, disprovable by established number theory finally went live. The problem I'd found there was the declaration--<i>in the ring of algebraic integers.</i><br /><br />And talked much in detail in 2017 with post: <a href="https://somemath.blogspot.com/2017/10/publication-does-matter.html" target="_blank">Publication does matter</a><br /><br class="Apple-interchange-newline" />Paper is simply correct declared in the complex field without even apparent error. And importantly is also correct declared in my ring of objects.<br /><br />Unfortunately the rest of the story made things more challenging in some ways, while being a boon for me from a singular research perspective. As have discussed in detail and chase the link to get more links to full story, the chief editor tried to withdraw my result, by simply deleting out of electronic file of an electronic journal. EMIS maintains the original, and I moved on.<br /><br />This post will cover my primary basic research in that area primarily, as without help from the mathematical community had the vastly more difficult task of personal verification.<br /><br />One area where focused was: how could I know for sure on my own? So I found a functional definition of mathematical proof and promptly put to use.<br /><br />Importantly simplified also from cubics to quadratics. And first public post on this blog I note for here where that has been shown to have happened is with my <a href="https://somemath.blogspot.com/2007/07/wrapper-theorem-and-ring-of-algebraic.html" target="_blank">wrapper theorem post</a> in August 2007.<br /><br />Wasn't until September 2008 that I further tested my use of what I call tautological spaces, which are complex identities, with my <a href="https://somemath.blogspot.com/2008/09/quadratic-diophantine-result.html" target="_blank">Quadratic Diophantine Theorem post</a>, where used conditional:<br /><br />c<sub>1</sub>x<sup>2</sup> + c<sub>2</sub>xy + c<sub>3</sub>y<sup>2</sup> = c<sub>4</sub>z<sup>2</sup> + c<sub>5</sub>zx + c<sub>6</sub>zy<br /><br />where the c's are constants, and importantly, realized could set z=1, and got my method for reducing binary quadratic Diophantine equations. So yeah, there is a research path with three variables which have not pursued.<br /><br />It was by 2010 that I had the idea of focusing on a simple quadratic factorization:<br /><br class="Apple-interchange-newline" />P(x) = (g<sub>1</sub>(x) + 1)(g<sub>2</sub>(x) + 2)<br /><br />Is interesting then had about six years to figure out how should be constructed, where yeah that 1 and 2 are benefits of research, though also kind of obvious to pick?<br /><br />But looks like don't have a public post discussing until February 2015, with <a href="https://somemath.blogspot.com/2015/02/blocking-algebra-with-algebraic-integers.html" target="_blank">post talking blocking</a> with algebraic integers where had to correct recently. However I also did my most exhaustive stepping through of original approach in the paper now using quadratics in September 2011, with post: <a href="https://somemath.blogspot.com/2011/09/under-hood.html" target="_blank">Under the hood</a><br /><br />And am skipping other research areas related as trying to be more focused for this post. And is interesting that gap I think from having a second path to proving the coverage problem, to what found public now on this blog.<br /><br />But even then didn't really push it much until last year.<br /><br />So have roughly a span of research in this area of about 14 years from point of publication which is not extraordinary I don't think. And was allowed to figure things out at a rather leisurely pace, while was also doing other research.<br /><br />But that was just my own validation. I wanted to be thorough.<br /><br />From perspective of a solitary researcher really cannot think of it flowing any better. Though it is intriguing was given the opportunity, which is outside scope of this post.<br /><br /><br />James HarrisJames Harrishttps://plus.google.com/104012861017102472975noreply@blogger.com0tag:blogger.com,1999:blog-11119110.post-61363226077620817362018-01-19T10:08:00.001-05:002018-01-21T09:18:35.746-05:00Some overview early 2018Last year was great for me as managed to settle some things, and now am focused on posts that assume people who read can check math versus me focused at all on any other assumption.<br /><br />My mathematical approaches are focused simply, with most requiring only knowledge of algebra and modular algebra, which aids in checking. And with me is a true modular algebra versus what I notice primarily elsewhere is modular arithmetic.<br /><br />And have had a powerful technique since December 1999, which involves subtracting conditional expressions from a tautology which is a complex mathematical identity, like:<br /><br />x+y+vz = 0(mod x+y+vz)<br /><br />Of course then there is no doubt about correctness as long as get each mathematical step correct. And made an <a href="https://somemath.blogspot.com/2014/10/example-showing-truth-logic-and.html" target="_blank">absolute proof demo post</a> where prove that:<br /><br /><i>If:</i> <b>x<sup>2</sup> + y<sup>2</sup> = z<sup>2</sup> </b><i>then</i>: <b>(v<sup>2</sup> - 1)z<sup>2</sup> - 2xy = 0(mod x+y+vz)</b>, where v can be any value.<br /><br />With much discovery using this technique and following from it, have gained confidence with it, and there is that troubling area of why others have not cheered to my knowledge.<br /><br />But had a clue from decades ago when a modular approach to packing of spheres was summarily rejected when I sent to the Proceedings of the AMS back in 1996. Long before I was very much aware of a coverage problem with ring of algebraic integers in 2003. Best explanation is that modern number theorists long ago started resisting valid mathematical discovery! Which explained much last year.<br /><br />Now that is so great a resistance they are not acknowledging a new primary modular inverse method I found last year, and my analysis indicates their focus is on: control the mathematics by controlling public opinion of the discoverer. Where originally I dismissed that approach as meaningless.<br /><br />My focus is on the math. For a long time I scoffed at the notion that my credibility mattered as it does not as to correctness. But DOES matter as to people checking! As yes, can be a problem if assertions are simply dismissed so mathematical proof is not considered.<br /><br />The idea of a democratic aspect to mathematical industry I find distasteful. But then again, mathematical industry is dynamic not just on what works, but on people knowing what works.<br /><br />That number theorists, who I think are the pure mathematicians who are the primary culprits, can fight mathematical discovery is not as surprising to me now, as human behavior in this area is well-known. In this case my best guess is that government funding for research that is invalidated by valuable and valid discovery is a major part of it. So the monetary motivation provides easy explanation for much.<br /><br />Which seemed obvious to me over a decade ago, but is wacky enough was worth carefully checking that assessment. Is remarkable though, how a system designed to facilitate pursuit of human knowledge was turned upside down.<br /><br />It is of interest that going to applied mathematicians does not break that as they are cowed and outnumbered. Beaten down I think deliberately by people who knew somehow before their research was usually worthless. And now know mathematically why, from me! That story is SO wild. I do love it, do admit.<br /><br />Is now of interest how things play out as time marches on.<br /><br />Mathematics contained? Intriguing to even think possible, eh? I think so.<br /><br /><br />James HarrisJames Harrishttps://plus.google.com/104012861017102472975noreply@blogger.com0tag:blogger.com,1999:blog-11119110.post-44169781228155426892018-01-18T12:33:00.000-05:002018-01-19T08:11:34.615-05:00Finding knowledge rulesWe get born and get told things. And to me knowledge is useful information and folks in the modern era get told PLENTY. However, humans found that knowledge, right? And the process can have problems, where we have humans at the front lines called scientists as well as others who keep figuring things out.<br /><br />To me is an awesome process and definitely respect while also find it fascinating when people are told facts and believe you are trying to convince them of something.<br /><br />In mathematics there is mathematical proof. And proof is definitive.<br /><br />If knowledge is available, yet you believe things not true, where that can be determined by fact, then I suspect you have problems with determination of fact, or have not been informed.<br /><br />With those able to access the web, who have training will admit my belief is: <b>ability to determine fact when available represents a valuable skill.</b><br /><br />And I also believe that some people lack ability to determine fact and simply rely on what they are told. Which is NOT necessarily a terrible thing! Most people must simply rely upon what they are told for MUCH. Like with medicine, am going to listen to that medical doctor. Probability can challenge with authority and better my own health versus endanger? Very small. Or nonexistent.<br /><br />Our human reality is vastly complex with too much knowledge for any one human to have much of a grasp of it all. However, it is still possible to determine fact, if you have the skills. While some apparently require the majority to believe fact, when is determined or they may instead believe false information which to me is simply, not knowledge.<br /><br />This blog audience focus from now on is to be directed towards those who can determine mathematical fact.<br /><br />The ability to find knowledge should be cheered and supported. And in my analysis people who depend primarily on the social group to know, who either lack ability to check for fact or refuse to check, can only be convinced by that social group they believe. And effort in their direction is simply wasted.James Harrishttps://plus.google.com/104012861017102472975noreply@blogger.com0tag:blogger.com,1999:blog-11119110.post-36568316632460869192018-01-16T08:04:00.000-05:002018-01-16T12:50:34.710-05:00When people work harder reducingThe burden on me to establish certainty with my own research should be vastly hard, especially with ideas that challenge the status quo. We have had a lot of great humans who have done great things, which work great too. And people flying all over with all kinds of cool technology, in countries with great freedoms, along with so much else in support of my views there. Greatness works.<br /><br />So I must take my time and accept great difficulty in establishing unimpeachable certainty. Which is fun! It pushes me to understand my own research and has even aided further discovery. But also there is so much demonstrative now, done.<br /><br />One of those ideas that demonstrates has to do with Diophantine equations like:<br /><br /><b>c<sub>1</sub>x<sup>2</sup> + c<sub>2</sub>xy + c<sub>3</sub>y<sup>2</sup> = c<sub>4</sub> + c<sub>5</sub>x + c<sub>6</sub>y</b><br /><br />Where is binary because you have unknowns x and y, and is also called two-variable. Diophantine just means looking for integer solution. I figured out a best way to reduce to something simple.<br /><br />For example:<br /><br /><b>x<sup>2</sup> + 2xy + 3y<sup>2</sup> = 4 + 5x + 6y</b><br /><br />Where I just tried something simple and was lucky there ARE integer solutions. My method for reducing gives:<br /><br />(-4(x+y) + 10)<sup>2</sup> + 2s<sup>2</sup> = 166<br /><br />And turns out you can easily solve from there and find integer solutions:<br /><br /><b>x = 4</b>, <b>y = -2</b>, or <b>x = 5</b>, <b>y = -2</b><br /><br />More recently played around and had to work harder for this example, as wanting something simpler in ways:<br /><br /><b>x<sup>2</sup> + y<sup>2</sup> = xy + x+y + 102</b><br /><br />Which my method reduces to give:<br /><br class="Apple-interchange-newline" />(-3(x+y) + 6)<sup>2</sup> + 3s<sup>2</sup> = 3708<br /><br />And figured out solutions: <b>x = -10</b>, so <b>y = -8</b>, or <b>x = 3</b>, <b>y = -8</b><br /><br />Copying from prior posts.<br /><br />Reference posts: <a href="http://somemath.blogspot.com/2011/05/reducing-binary-quadratic-diophantines.html" target="_blank">Reducing binary quadratic Diophantines</a>, <a href="https://somemath.blogspot.com/2016/08/reducing-quadratic-diophantine-to-find.html" target="_blank">Reducing a quadratic Diophantine to find solutions</a><br /><br />My method for reducing is I think the best in the world, and supersedes methods that trace back to Carl Gauss who is a HUGE hero of mine. And his methods now include wasted effort finding something called a discriminant, which is not needed for reducing these equations.<br /><br />People wasting their mental energy though is not surprising to me, or a great concern. Others might value their efforts more highly than I do, as is a telling failure, in my opinion.<br /><br />From my perspective, is more telling when people work harder than necessary, when they could do better than those who find better.<br /><br />Those who seek truth best, interest me more than those who do not.<br /><br />May seem cold, but many may wish to be something they are not. To me is important how to figure out those with potential. Finding the truth when is so easily available? When easy to find on the web?<br /><br />If you can't do that easy then how can you find the things waiting to be discovered, when that can be so hard?<br /><br />Challenges are met by people with ability. Why I deeply appreciate the challenges in front of me, as I learn so much more about what I can do.<br /><br />Everything testable and each statement of mine checkable though I wonder if am reaching in claiming is the best possible. But I think not. Like consider, how I got something important.<br /><br />Use my method to reduce on: x<sup>2</sup> - Dy<sup>2</sup> = F<br /><br />Gives the BQD Iterator.<br /><br /><br />James HarrisJames Harrishttps://plus.google.com/104012861017102472975noreply@blogger.com0tag:blogger.com,1999:blog-11119110.post-87033412575150782042018-01-15T11:04:00.002-05:002018-01-17T12:34:04.350-05:00Thoughts on infinite depthCorrect mathematics will check out correct over infinity. There is no error or possibility for error. And that is a perfect test. I like to call it--infinite depth. As incorrect mathematics on the surface can look ok, but will lead to contradiction.<br /><br />Questions of whether or not mathematics can have rules that lead to contradiction were handled most famously with the work of <a href="http://en.wikipedia.org/wiki/Kurt_G%C3%B6del" style="background-color: white; background-image: none; color: #0b0080; font-family: sans-serif; font-size: 13px; line-height: 19.1875px;" target="_blank" title="Kurt Gödel">Kurt Gödel</a> being of signature importance. Lots of people talk him, but the essence of the desire with mathematics is what I stated.<br /><br />People can though get excited over the question of: can you have valid mathematics which leads to something wrong? And I'd say simply: no.<br /><br />But I see it as, if mathematics leads to something wrong, then is NOT valid. So that idea of a check possible, to me answers the question.<br /><br />But how can you know? We can't check infinity with infinite tests one-by-one. Where I rely much on tautologies for my work.<br /><br />Like: <b>x+y+vz = 0(mod x+y+vz)</b> is logically a tautology and mathematically is called an identity.<br /><br />Is equivalent to: <b>x+y+vz = x+y+vz</b><br /><b><br /></b>Much of my mathematical research reduces back to validity if that identity is valid.<br /><br />Actually have a reference post from 2014 that is demonstrative:<br /><a href="https://somemath.blogspot.com/2014/10/example-showing-truth-logic-and.html" target="_blank">Example, showing truth, logic and absolute proof</a><br /><br />Identities by definition are valid; therefore, a result that so reduces is perfectly checked.<br /><br />Functionally that means that a person like myself can make a mathematical statement, and find that statement implies something else. Where readily admit have come across such or had them brought to my attention and even when I know have a mathematical proof can be that emotion of FEAR.<br /><br />Then test the implication and look at a new result. Discovery rules. Fear turns then to elation. But regardless, emotion is irrelevant. The math behaves perfectly. Infinite depth means there can be no mistake. And I marvel to myself or chatter a bit about it, like maybe here on this blog.<br /><br class="Apple-interchange-newline" />So have given the logic. Emotion can mean something else. One phrase like to use, over and over again is--math and emotion do NOT go well together. Like one way I check people is to ask, if they say absolutes do not exist, if they believe: 2+2 = 4<br /><br />Number authority is so useful. With questions of truth can still be of interest to ask a person, why do you so believe? Most people will of course not debate you over such a thing, but why not?<br /><br />Simple enough, yes, but there are humans who will debate you over it. I find it to be a telling human check.<br /><br /><br />James HarrisJames Harrishttps://plus.google.com/104012861017102472975noreply@blogger.com0tag:blogger.com,1999:blog-11119110.post-34389261072154923962018-01-11T08:08:00.001-05:002018-01-15T07:55:50.821-05:00Some math perspective and pep talkThe math sustains me like to say, and getting going into a new year is useful to reflect on just a bit, some of how that works.<br /><br />Like one of those results will use for comfort:<br /><br /><b>P(x,n) = [x] - 1 - sum for j=1 to n of {P([x/p_j],j-1) - (j-1)}</b><br /><br />where if n is greater than the count of primes up to and including sqrt(x) then n is reset to that count.<br /><br />My way to count primes in its sieve form.<br /><br />That is SO compact! The math has a succinct efficiency I appreciate. One of my favorite posts talking is actually on my blog Beyond Mundane: <a href="http://beyondmund.blogspot.com/2013/05/simple-and-fast-prime-counter.html" target="_blank">Simple and fast prime counter</a><br /><br />One of the last results where I got lots of criticism over a decade ago, where what I saw as specious comparison to known methods ignored the new. Yes, basic concepts were same, so?<br /><br />The innovation involved is how is simply displayed and compact, which is also obvious in comparison.<br /><br />Love considering as the time passes. As over 15 years now and yeah, look over what math people <i>still teach</i>, as if a world of students cannot also find easier on the web.<br /><br />And something that so floored me only a few years ago, realizing can easily find all integer solutions for:<br /><br /><b>x<sup>2</sup> + (m-1)y<sup>2</sup> = m<sup>n+1</sup></b><br /><br />From: <a href="https://somemath.blogspot.com/2014/12/squares-and-nth-powers.html" target="_blank">Squares and nth powers</a><br /><br />Not surprisingly to me, using this result, which follows from my BQD Iterator could make for popular shares. Like:<br /><br /><b>1<sup>2</sup> + 3<sup>2</sup> = 10</b><br /><br /><b>8<sup>2</sup> + 6<sup>2</sup> = 10<sup>2</sup></b><br /><br /><b>26<sup>2</sup> + 18<sup>2</sup> = 10<sup>3</sup></b><br /><br /><b>28<sup>2</sup> + 96<sup>2</sup> = 10<sup>4</sup></b><br /><br /><b>316<sup>2</sup> + 12<sup>2</sup> = 10<sup>5</sup></b><br /><br />From: <a href="http://somemath.blogspot.com/2017/08/sum-two-squares-to-power-of-10.html" target="_blank">Sum two squares to power of 10</a><br /><br />Simplicity has HUGE benefits as if people get curious can check you on EVERY math detail. Web enables a lot as well, as folks can know. They can just check you on the facts.<br /><br />Can also check what they thought they knew, about others in a high profile area. For me was lots of surprise for years.<br /><br />Now these results are comforting, while less exciting, for me.<br /><br />There is that thing about the math where when you discover you get a different emotional connection to your discoveries, which I know from how I look at my math discoveries versus math from others.<br /><br />The math does not care. That actually IS a comfort. Because the knowledge is more important to me.<br /><br />So yeah, this pep talk is working for me like usual. And what discoveries to pick? Is just about my mood. Could talk my better than Gauss's way to reduce binary quadratic Diophantine equations. Could go on again about my modular inverse method, but am on pause there. Still relishing it though.<br /><br />Or lots of other things.<br /><br />Pep talk complete! Well I feel better now. Who knows why humans who come and go might fight against mathematical discovery, and do we really care?<br /><br />The math feels like a friend to me. And I know the math does not care. Convincing myself, I should not either. We humans have choice.<br /><br />Knowledge should be about reinforcing choice. The math has given me more choice. Am thankful.<br /><br /><br />James HarrisJames Harrishttps://plus.google.com/104012861017102472975noreply@blogger.com0tag:blogger.com,1999:blog-11119110.post-7897964783585057292018-01-10T11:40:00.002-05:002018-01-10T21:00:57.823-05:00Power of mathThat correct mathematics has a power of its own sustains me. And what I call the social problem is much about people who clearly do not believe it does.<br /><br />Lies? Have to be supported constantly. That human energy can only last so long. And besides, the truth can crush lies even when LOTS of people keep trying to push them.<br /><br />The valid math wins over time. And I have a unique opportunity, as NO major discoverer at my level has ever faced such opposition.<br /><br />My predecessors would be jealous am sure. But they're long dead. Gives me different perspective on their lives as well. They had to fight their own battles too. History tends to be glossed over as to full reality. Still realize how much better it is, here am thankful. I get to know so much more than they ever could in ways, I think. Yeah it is different between understanding possibility and living in this modern reality.<br /><br />Gist of it is, opportunity like no other as in challenges, I simply learn more useful.<br /><br />From my best perspective is simply an interesting exercise.<br /><br /><br />___JSHJames Harrishttps://plus.google.com/104012861017102472975noreply@blogger.com0