Friday, November 28, 2014

Binary Quadratic Diophantine Iterator

It must be that if you have:

u2 + Dv2 = F

then it must also be true that

(u-Dv)2 + D(u+v)2 = F(D+1)


In general true binary quadratics, except for the cases D = 0 or D = -1, F = 0, are connected to an infinity of others, as all can be reduced to this form, so it's what I like to call an infinity result.

By true binary quadratic I mean one that does not have a form like this example:

x2 + 2xy + y2 = c4 + x + y

is: (x+y)2 - (x+y) - c4 = 0

So I don't consider that to be a true binary quadratic because it behaves like a single variable equation, with x+y as the single variable.

One of my favorite discoveries I've decided to call it a binary quadratic Diophantine iterator.

The beauty of the iterator is that an already reduced equation simply gives you another of the same basic form, as:

(u-Dv)2 + D(u+v)2 = F(D+1)

Is equivalent to the original as notice, with u' = u-Dv, v' = u+v, and F' = F(D+1), you have:

u'2 + Dv'2 = F'

You get the iterator when you use my method for generally reducing binary quadratic Diophantine equations on the already reduced form, and to me is one of the coolest things ever. Looks like I first gave it in its general form back October 2008. This thing conceivably could have been discovered 2000 years ago, but somehow it wasn't so I had the honor of finding it.

While it is derived it can be verified to be correct simply by using the equations given.

Given:  u2 + Dv2 = F

Verify: (u-Dv)2 + D(u+v)2 = F(D+1)


Expand out: (u-Dv)2 + D(u+v)2 = F(D+1),

which gives:

u2 - 2Duv +  D2v2 + Du2 + 2Duv + Dv2 = FD + F

Notice -2Duv cancels out +2Duv, and then it's just a matter of grouping:

u2 + Dv2  - F + Du2 +D2v2 = FD

And: u2 + Dv - F = 0, so:  Du2 + D2v2 = FD, and dividing off D, gives:

u2 + Dv2 = F

Verification complete.

Example using iterator:

Let F = 1, u=x, v=y, and D = -2.

1. x2 - 2y2 = 1

2. (x+2y)2 - 2(x+y)2 = -1

3. (3x+4y)2 - 2(2x + 3y)2 = 1

4. (7x + 10y)2 - 2(5x + 7y)2 = -1

5. (17x + 24y)2 - 2(12x + 17y)2 = 1

and you can keep going out to infinity, but I'll stop with 4 iterations.

Notice now you can use the simple case of x=1 and y = 0:

1. 12 = 1

2. (1)2 - 2(1)2 = -1

3. (3)2 - 2(2)2 = 1

4. (7)2 - 2(5)2 = -1

5. (17)2 - 2(12)2 = 1

And you have answers to x2 - 2y2 = 1 and x2 - 2y2 = -1.

When I found it years ago back in 2008 I extended the symbolic version further. Copying four of the five I see in that post to here for convenience. Here I'm using u = x, v = y.

1. x2 + Dy2 = F

2. (x-Dy)2 + D(x+y)2 = F(D+1)

3. ((1-D)x-2Dy)2 + D(2x + (1-D)y)2 = F(D+1)2

4. ((1-3D)x + (D2 - 3D)y)2 + D((3-D)x + (1-3D)y)2 = F(D+1)3

While the symbols go out to infinity, when using actual numbers you may be able to divide off factors from D+1 routinely. So for instance with D = 1, every other iteration 4 can be divided off if you so choose. So it still can be iterated out to infinity, but that then seems trivial to me, and when 4 is divided off the series simply oscillates.

James Harris

Monday, November 24, 2014

So what results

My recent post covering an attempt at a research perspective on considering the value of web search results is useful to talk about another more troubling issue which I think is a reasonable concern when you hear some person claiming to have important results that aren't being properly accepted. After all, such a belief can simply be a hallmark of delusion, and I think it of interest to talk about that issue and in my case, how deep that delusion would have to go.

Turns out my first result where I ponder its importance where if it is correct there should be validation from established mathematical institutions is from 1996. I finally wrote about it on this blog May 2008:

And re-reading what I posted I can see the ambivalence but also I didn't want to talk about my own doubts. While here it is to note that I have a result from my mid-twenties which if correct would have been one of the greatest improbable finds in all of human history. So I naturally doubt it.

One of the biggest things that really bugs me about the concept of the result is that the problem is so old and Sir Isaac Newton tackled it. I find it just so difficult to imagine that he would have missed such a simple thing.

But I had a paper, sent it to a math journal, which rejected it as too simple, and I lost the paper. That is, I had it typed up or something on actual paper, while now everything is online as of course the web revolution happened, so the original paper as far as I'm concerned is lost. And I don't like talking about this thing. It irritates me. I can't find anything wrong with my approach or its logic, and no one else has ever told me anything wrong with it--too simple is not an actual mathematical error--but if that mathematical argument is correct its existence annoys me greatly. It should never have been left for me to solve.

But you know the best and worst thing about having such a result? It's the doubt.

I've had over 18 years now of doubt with that thing. Doubt can be such a treasure. In mathematics, doubt can push you to understand why you think you know something.

Rather than endlessly debate its correctness with myself or others, I gave myself a simple challenge: find something else.

No major discoverer at the highest levels in the area of mathematics to my knowledge has only a single major result.

It's like this tired refrain I'd taunt myself with through the years--find something else--no matter what it was always there until I just let it go.

But it's amazing to realize that 26 year old me, saved my future.

Moving on to something else was the best thing to do in that situation. Don't sit and argue with yourself or others, find something else. And there is always something else to do with your time and attention.

So I went from spherical packing to Fermat's Last Theorem, looking to find something else, where for a long time I've refused to discuss that most infamous, frustrating, and demeaning effort, as it is a lightning-rod for insults. And I think a lot of that pursuit was misguided. Chasing a famous problem can come from the wrong motivations I think. Great thing was I quit caring about that kind of stuff, and started just having fun.

But it's interesting to me that the stigma of that thing is so great. It stalks you, but that's not that big of a deal now.

For others pondering what they may see as indications of delusion, I think there is a natural concern. Especially if you worry about encouraging a person with some kind of mental disability or illness. But I think people naturally tend to focus on what they see as emphasized, with a continuing focus on that thing as somehow proof of something: like, why can't this person just let it go?

For me that's the fun part, I can. My spherical packing approach irritates me. Pursuit of Fermat's Last Theorem was humiliating, and turned into a continuing stain so great I hesitate to even mention the thing. But I stopped working on it over a decade ago. Was done in 2002.

So what about other results then?

It's odd to realize that I can idly throw away what I don't like as what I could consider my first major result. Dismiss my work on the next thing I spent so much time and effort on, and not even touch the areas of current potential controversy.

But I did develop mathematical analysis techniques from tackling Fermat's Last Theorem which I used with my next move on, which lead to my one and only mainstream publication in a formal math journal.

Oh yeah, so I DO have a published result. Turns out that is a lifetime thing which can't be taken away from you, as I checked! I'm listed as a published mathematical author. Where? I forget. It was over a decade ago that I checked. Actually called them up and chatted on the phone. It's funny the things that seem important until they don't. I think it was MathSciNet. Those who care about such things might be able to find it.

So do I hold on to any results?

No. My view is that I chase ideas, and chase certainty. And I pondered mathematical proof maybe because I had the spherical packing thing: How do we know what's true or not?

Correct mathematical results need no defense.

If every person believed just that one thing then you'd remove all of the heated argument out of mathematics.

People might still argue but it'd only be in teasing out the actual logical details and verifying them.

So I don't hold on to results, nor do I defend them. I present them. And thanks to the web that's enough.

James Harris

Web acceptance research perspective

Obviously mathematical research stands on its own but it is of interest how others react to it. My position has given me a unique perspective where it actually helps to try and step away from it objectively, which is something I've valued trying to do for years, and I think it worth it to talk about some conclusions.

The acceptance of the web for a particular content author can arguably be considered reflected in web search rankings. One concern is in choosing a metric simply because one is available, but I think there is substantial quantitative data that supports this metric. And in fact my own experiences give both a subjective for me reference point and potentially objective for others reference.

The benefit of this research perspective has to do with best routes to sharing research information. The 'social disruptive' nature of the web is much discussed and the potential ideal of the web is to remove gatekeepers, like math journals, or force them to re-define their role.

Specifically recently I've noted the emergence of my own mathematical research in the specialized area of number theory which considers binary quadratic Diophantine equations. That is, I have a high ranking across web search engines, with the search: reduce binary quadratic Diophantine equations

The search is important in that it is asking how to do something. That would naturally seem to reflect a demonstrated need which can be satisfied by the results of the search. To my knowledge I am the only person doing that particular search string though I have suggested it now to others.

But also importantly the actual mathematics helps solve a highly particular type of problem and absolute correctness is required, that is 100% correctness, as the mathematics has to be perfect. So it's a great example that the web can find such things.

However, I have other search results, like the name of this blog. At one point I found it necessary to change the name of this blog. My choice of the rather generic "Some Math" was not just because I thought it appropriately descriptive but also to aid in considering this kind of research as to whether or not the web would rank it highly. This social experiment was telling in that the term--some math--gave results at first not related to my content, but over time my content rose in search results. And I could easily compare across search engines.

Another contrary result has been with the search: definition of mathematical proof

That has been extremely informative as the ranking of my page where I define mathematical proof has shown a negative reaction to my discussion of it. For instance emphasizing its search ranking in the past actually lead to it dropping completely from web search across web search engines. I will note using its search position in emotional tones that are not relevant for this post, but considering what is known about web search it involves people linking to your content. So what causes people to link, or de-link is of interest.

With years of experience now in this area I think I have a very clear idea about how the web behaves with regard to certain types of content, also which content has a weaker link attribute, like my definition of mathematical proof, versus content which has a very high link attribute.

It's telling that when I changed the name of this blog to its current one, the link for the definition of mathematical proof promptly disappeared from web search, but the web switched to other research on Usenet math newsgroups with other results!

Talk about validation of a research path, when one source was removed as changing the name of this blog broke web links to it, the web first switched to an alternate source of the same information, which were posts I had made on Usenet, and then switched back, as revealed by the new link name rising in web search.

The web simply went and found the research in a dynamic process.

From a research distribution perspective it appears that math journals may be irrrelevant: people will find useful content regardless which is reflected in web search results which adjust dynamically even if where that information can be found is artificially manipulated.

Presumably if I found a way to destroy all sources that I provided, others might simply put the content up again, which is an experiment I don't see necessary. The web reflects a continuing need for information, but presumably any number of people who needed the content have copied it off the web.

Looking at this situation as objectively as I can I will admit to a subjective reality, and maybe as an independent researcher will note a certain amount of relief. An independent researcher like myself can put up content directly without gatekeepers. Web validation can come from search engine ranking, which is dynamic in nature meaning that it is a constant evaluation, which can even shift to chase the research, if one source is made unavailable, indicating a real need.

Less dynamic behavior can also be associated with content that presumably is deemed less important in the dynamic process like my definition of mathematical proof, which presumably is less of a direct need for people who might be interested in this information.

All search results considered for this evaluation were at #1 in more than one search engine at one time or another, and this position was confirmed to be global, as in, those searches would come up at #1, anywhere in the world. Though confirming that has been easier or harder at times. Primary reference has been Google, which thankfully also during the time of this research inquiry has been the dominant search engine worldwide, which it is at the time of this writing.

There was some limited checking by trying web searches in different geographic areas though I have used primarily states in the United States including Hawaii.

James Harris

Friday, November 21, 2014

Crowd approval

My focus is on knowing things absolutely. So I made a post where I talked about how you can check for absolute proof. There is a value though in seeing that what you've found is appreciated by others.

For a sense of some of my math research rank, anywhere in the world you can go to a web search engine and you can get an objective measure by a simple test which tells you about the interests of other people besides myself.

Web search: reduce binary quadratic diophantine equations

Turns out my research dominates on that subject as amazingly enough I have the privilege of having improved upon methods that Gauss previously best developed. And Gauss is a hero of mine so I think that's really cool. And the point then is to see that's not just some personal opinion.

Look at the other search results besides my content that come up to see the competition, as the web is highly efficient for helping you with such comparisons. I guess that's the future.

My way of reducing binary quadratic Diophantine equations is one of the best in the world. That's the kind of objective assessment the web can give you. Don't trust me, trust it. Or don't, but it's not like that changes things. That pull is so strong that it's like a tidal wave. The pressure is actually more intense than possibly faced by anything but results at the highest level.

So the good news is that my research has to survive intense pressure and scrutiny.

Getting the attention of the world requires your math work.

Interesting to state it, which can sound so far-fetched--which is why the web search is so potent. It removes my opinion from the mix. All I'm doing here is pointing out what is verifiable. It's not even hard to check.

Unless possibly you think there's a vast conspiracy with the major search engines of the world desperate to fool you, it's an objective test.

There is no such vast worldwide conspiracy. I checked.

And if you think maybe there is, do you really think you're that important? Why should they bother fooling you?

Yet, it's one of my minor results which is part of one of the more fascinating advances in human thought that took over a century to occur.

And to me that's the kind of thing that should be cherished by the mathematical community, but instead it's supported by the world as revealed by web search. Um, between those two, which should concern me the most? Yup, the world.

So there is this world support for me and my ideas. That it comes from new technology? Should that bother me? Actually that should give more confidence to others. But no official support of which I'm aware, which I admit is probably more about lack of information. If you think I'm just some individual railing against the mathematical establishment, then that perception can control your response, which I think is not good. Why can't mathematical proof matter more?

Reality can defy perception.

But the situation has been helpful as a part of me needed to see what people would live up to a certain standard, in terms of truth.

But I've accepted I can't hold up everything while I push that kind of standard. I just wish it had worked. My goal was to find people for whom absolute mathematical proof was all that mattered. Possibly I was misguided in that search.

For fans of my research, yeah, there are lots of positives. So yeah, it's clearly been recognized as revealed by web search. That's better than having some people claim you're right. Let the crowd give its approval.

Turns out binary quadratic Diophantine equations are really important and not just for "pure math" so people want the best, regardless of who discovered it.

That is, world is weirdly efficient.

James Harris

Wednesday, November 19, 2014

Finding solid ground

There is an extra to the extraordinary issue of thinking you've found a problem where the mathematical field has an important error in that the field prides itself on not only not having error, but being the one major discipline where such a thing is impossible. Worse, if you "prove" the problem you also have to deal with where do you go talk about it? To the very people who couldn't see it before? Some of whom may have relied on it to build their careers?

But the personal side of it was far bigger for me for a long time. I grew up solidly within the system, went to a top college on an academic scholarship. Got my degree in physics. Was thoroughly indoctrinated in the mainstream, and didn't expect to find a problem with it. It took me a long time to work through things, and at times I'd try to discuss, but I now realize that the "personal" part means it isn't helpful to others.

So what about someone who comes across my mathematical research and now has to struggle with its implications? I'm ready to finally help there.

And I think that is a responsibility I should accept. For those who just dismiss my ideas of course it's irrelevant. Such people will not be convinced by what I say here. But this post is not for them.

First off, is the question of--how do we know anything? And in mathematics you can hear a lot of talk about mathematical proof and axioms, but also, especially after my research came out, unfortunately, you can see pushback where mathematicians are claiming that "mathematical proof" is actually about the weight of the opinion of mathematicians! So I gave a post, stepping you through an absolute mathematical truth for demonstration purposes:

And the BIG thing that should jump out is that I rely a lot on identities. Like x = x is an identity. And of course that's true! So like if I'm thinking about arguments someone might make against my research, it's like there's a point where you can jump off. If some person wishes to debate whether or not x=x is true then you really aren't going to be able to reach common ground, you know?

But for those who will accept it, then it's not very far to solid ground. Then I have arguments that rely on identities with conditions. Like, if x2 + y2 = z2 then under those conditions these things that follow from identities must be true. Not complicated.

When you're pondering "truth" or what some person claims is truth, then you have to find that thing you accept from which all else must follow and I like that I can say: it's identity. If you believe in mathematical identities then you can get the rest.

It's important to have that linchpin.

So what about acceptance? Why can't you go to some mathematical conference on say, my work on the problems with using algebraic integers? Or why can't you read up on my prime counting function from others? Or the partial differential equation that follows from it from others?

There I have to hedge that maybe you can find others discussing such things, but to my knowledge the only other people I know of who talked about them publicly were people years ago arguing with me against their importance online.

It's up to other people where they choose to engage and what they choose to discuss publicly.

So the short answer is, I don't control what other people choose to engage upon publicly.

They do.

So the choice tells you more about them than me.

From a personal perspective though, it can be a lot about who you are. So it's like, if you conclude that something is true, like that identity is key, and everything else follows, but then get stuck on--but what about these experts in the field who aren't talking about this thing?--then you're dealing with two different issues.

And I've been there. Spent a lot of time there. That's one of those personal things I realized wasn't doing anyone any good by discussing publicly, like on this blog.

My research has drawn the attention one would expect, which means that on the social side I can put forward ideas that promptly and rapidly zip around the planet. That does take a while to really feel ok. Maybe it will never really feel ok to me, but that's not a good reason for me not to help other people.

After all, coming across these ideas, you can have a real concern about how to know if they are correct or not.

And that I can understand and feel I have a responsibility to give that help.

James Harris