## Sunday, May 29, 2016

### More product of sum of squares

With the basic set for doing a product of sum of squares you can build bigger ones easily.

And went for LOTS of easy with this example as could make something that looks harder but is extra work for nothing. That would be an illusion as is so easy to do.

Like:

(x2 + 2y2)(u2 + 3v2)(x'2 + 4y'2)(u'2 + 5v'2) = p2 + 359q2

And finding integer solutions for all the variables is easy. Coming up with variables is harder. But yeah you can just keep going as far as you want.

x = 1, y = 2, u = 2, v = 2, x' = 3, y' = 2, u' = 4, v' = 2, p = 358, q = 2

Using first iteration and using positive as all will get squared. So:

(1 + 8)(4 + 12)(9 + 16)(16 + 20) = (3582 + 359*4)

Which is:

(9)(16)(25)(36) = (3582 + 359*4)  = 129600

Which worked. Used easy first iterations. Of course iterators work out to infinity.

Just playing around.

James Harris

### Real digital divide

Years ago when was still talking out ideas on Usenet, was also discussing what for me was a mystery back then as this blog, then with a different name, was getting visits from 125 countries/territories as Google Analytics put it, which shocked me.

Also have mentioned here at some point decided that established definitions of mathematical proof were not sufficient for me to determine if a mathematical argument I had was one, so I came up with a functional definition, and posted here.

That roared up web search, which I found out from web stats, and again was befuddled. But talked that out and got a surprising reaction. Math people on the math newsgroup where was talking things, went to great lengths to dismiss.

But web search is relatively new, and for certain people is something they clearly can easily question. While for those growing up with the web is something they have had their entire lives.

So I realized the digital divide from those who really are still 20th century, for whom they can question web things and dismiss as if meaningless. And those who grew up in the 21st century who use and rely on those same things as important.

For people growing up in the 21st century the notion that I can simply fix such things is ludicrous.

So yeah I changed the blog name, and watched with interest as lost that search position for my definition of mathematical proof which has not returned to it since, which was a way to check certain things. It was going back up and I talked it here, and that seemed to end that which I think is interesting.

Have talked more recently about some of the things I face, like how can have results here, which I can check with my own definition of mathematical proof of course, and not seem to get anywhere with the mathematical establishment, as can't force math people to acknowledge, of course.

But you know? That digital divide explains something. I won't elaborate, but it's not a concern really. I really think that past people who grew up in the 20th century seem confident in situation that they can control it, which does disappoint me. I've watched things over the years often wondering. But reality is you just don't control information that way. And never did really. Modern web just makes it more visible I think.

To understand though, you have to be fully 21st century. So cool.

I find that comforting as those people can be mean. So they are out of it, based on what they don't understand in our modern world.

My interest has been in having important information available to the world. And I feel confident that has been done, and there are people who find it helpful.

That's also a huge comfort, and a unique benefit of the modern web. I can see that reality, and don't have to just rely on faith, thanks to the web.

James Harris

## Saturday, May 28, 2016

### Problem with confidence

One of the best aspects of the sciences is endless challenge, when ideas face relentless scrutiny and if found wanting, are replaced by new, better and more effective ones. But theoretically mathematics is on a different system, and is, with mathematical proof.

Because a mathematical proof is a perfect entity, it cannot be wrong, so it will never collapse or fail.

However, confidence in mathematical proof while logical, does not mean confidence in a mathematical approach is the same thing, and today the world relies on some conjectures for public key encryption, a popular security system with mathematical underpinnings.

The problem is that any public key can be broken if you can factor a certain rather large, by our standards, number, where the understanding for some time has been it is hard to factor such numbers.

But xy = P, where P is a product, may look hard, but is it? There is no mathematical proof that it is, and the belief that if P is sufficiently large then it is difficult to find x and y is at this point in time just a matter of opinion and faith in the abilities of people who have been known to have worked to solve such problems.

Historically such faith has been found to be wanting. It has NOTHING to do with mathematics. It is JUST about confidence in human ability.

I will note that this area is covered by binary quadratic Diophantine equations, as xy = P is one. So yes, I have lots of research and for awhile focused on integer factorization but stopped when I became concerned that maybe I could find much easier ways.

I worried, if I did, could the world handle the consequences? That faith in people shattered in a moment could have unfortunate results. So I moved on to other areas and NOT saying I could find an easier way, as then just emphasizing more faith in a person.

Weirdly enough in this area the notion that the problem hasn't been solved more easily is based on the belief that if anyone had done it, the world would be told.

But if you could easily break into computer systems around the globe using such systems and peruse endless web traffic encrypted with them, with people endlessly believing you couldn't, and even maybe got away with astonishing things as their confidence refused to be shaken, would you tell anybody?

During World War II, the British broke encryption systems, along with my country the US, and did they tell? Yeah, after they'd won the war.

James Harris

## Thursday, May 26, 2016

### Some thoughts on math politics

Occurs to me maybe should be more helpful for others who come across this blog and need a decent synopsis of state of acceptance of my research, as far as I know, and I have a good example I think to illuminate.

Back January 5, 2015 put a post where I show a method for finding integer solutions in general for:

(x2 + ay2)(u2 + bv2) = p2 + cq2

Which looked like something on which I could try out something I decided to call a Binary Quadratic Diophantine iterator or BQD Iterator for short.

An example from the post shows a solution I found:

(x2 + 2y2)(u2 + 3v2) = p2 + 11q2

And here are some possible solutions for the variables:

x = 1, y = 2, u = 2, v = 2, p = 10, q = 2.

And what makes the story informative is I was answering a question I saw on some math site, so I put up my general solution. Later I checked back and saw it had been deleted by the site.

And I'm NOT a mathematician which I gleefully and routinely note. In my world you take a correct answer, but in my experience in the world of math people it's not so simple to them. They often clearly care about the source. Who found it.

And I do wonder, but maybe some of them really see it as defending the mathematical field? From that perspective maybe even a correct answer can seem worthless in comparison to letting someone they see as the wrong person gain credence?

But it's like with that example, does established math world have an answer? I don't know. I looked over that math site, which is linked to on the original post and you can too. I didn't see one. There were several attempts though.

Problem is these are esoteric things! Who actually needs the answer?

If someone does? Then my solution will get picked up.

But for that to happen would probably need to be some practical problem in the real world which needs it.

But yeah to me shows the problem--I can put things up, and even show to math people, but can't make them acknowledge it. Ok.

Regardless I have answers, which found myself! And can put up my findings on this blog which draw attention from around the globe.

And guess could be comforting to others to get more information, so last year according to Google Analytics blog had visits from people from 68 countries. And I even checked cities this time, and visitors were from 423 cities. To me seems kind of abstract really.

So yeah maybe a bit frustrated with that behavior of math people not properly acknowledging results of mine I think are important and valid but if you get mad? They use that against you. Problem is the behavior is the opposite of what I think most people would expect which actually protects it. If you challenge you sound wrong! The faith in the mathematical community is so solid. Ok.

And why bother if the information is publicly available regardless, so people who do need it? Can find and use it. The math politics is just kind of weird to me and hard to explain which is why I like the example above. You can see a real world example of a research finding, simply rejected, leaving a question unanswered at the math site source, while you can see the answer on this blog.

But that kind of covers situation and to me this post is for those people who might wonder. And short answer is, I can't make math people acknowledge these results. And wouldn't if I could. It's that simple.

James Harris

## Friday, May 13, 2016

### Going back a bit

Behavior of numbers is endlessly fascinating to me, and math gives an infinity of tools to study them! Helps to consider the simple and well known not surprisingly, and a post from 2009 covers some basic facts about the equation:

x2 - Dy2 = 1

Where traditionally you find integer solutions for it, which goes back hundreds of years, but it's actually trivial to solve for rational solutions:

x = (D + t2)/(D - t2)

and

y = -2t/(D - t2)

Here are some easy examples. I like easy:

Let D = -11, and t = 1, then: x = -10/-12 = 5/6, y = -2/-12 = 1/6,

And as required: 25/36 + 11(1/36) = 1

Advance to t = 2, then: x = -7/-15 = 7/15, y = -4/-15 = 4/15,

And as required: 49/225 + 11(16/225) = 1

Expressions give a well known parametric equation for the circle with D=-1.

So yeah, you can actually use those equations to graph hyperbolas or ellipses. And D is related to eccentricity, which is a calculation I've not done, though I've seen someone give the expression showing how they relate. Though I do wonder, what if people had seized upon this way, instead of the way with eccentricity to graph?

Those solutions definitely look simpler. And in our age with computers, not hard to code, though haven't coded. So have no idea if is much worse than traditional ways or not. I DO know I don't typically see those equations and only found out they've been known for centuries when got excited when I re-derived them on my own, and then went looking and found out I didn't have an original discovery. Were actually known to Fermat himself.

I sat down and wrote up a bunch of other things about it back then. Like, for any positive integer D, if D+/-2 is a perfect square, then D+/-1 is the first solution.

82 - 7*32 = 1

Where 7+2 = 9, so 8 is the first solution, and 3, is second. Yeah. So was thinking to myself need to focus on numbers again.

That rule works out to infinity so I might as well make one up instead of just using the example from the page. So, um, say 169, which is 132, so D = 167 and 168 should work.

1682 - 167*132 = 1

And that expression x2 - Dy2 = 1 is SO weird in a way in that there are SO many ways to get trivial solutions for x and y as math people usually describe the easy things, as trivial, so math people would only focus on the harder ones! Years ago I just figured out all the rules for any integer ones, which are known as Diophantine solutions. But now it feels like ancient history to me.

So thinking should talk numbers more as can get bogged down in other things, which I realize interest people less.

James Harris

## Friday, May 06, 2016

### Appeal of useless research to some

Was amazed to realize that my need for a functional way to determine if a mathematical argument was a proof or not, without worrying after if was a figment of my imagination, lead me to a functional definition, which lead to the concept of a functional definition versus traditional descriptive ones.

Then I found myself founding functional definitions where I needed them, including for entertainment which transformed my entertainment experience, while also helping me with social media theory.

That to me is useful research. However the mathematical field became dominated by pure mathematics some time ago, which prides itself when research is valueless to the public.

But why should that be cheered?

Supposedly frees researchers to focus on valuable intellectual pursuits without looking for profit, but modern pure mathematicians profit all the time, from charitable funding from a public that is giving them the freedom--to pursue information they think is useless to the public because it is "pure".

Prestige in mathematics was and is built on value to the public. Luckily the value of mathematics itself will never be called into question as mathematical ideas are intrinsic to our science and technology.

However those mathematicians of today with work that is not relevant to the public necessarily rely on efforts of other people who proved value.

But what does that do for the future of those who are to come. When maybe a public has tired of the charity?

My distancing from mathematicians is telling. To me? It's safer.

Why pay for useless research? Why reward for things that don't do anyone else any good on an endless faith that maybe someday they might in some hypothetical refused to be given by those who might claim it beneath their dignity to so speculate?

To some the appeal of money for nothing of value to the public may be obvious! Why produce something of value to them?

That could be harder I think.

Luckily such shortsightedness is irrelevant. There are others to take up the mantle of mathematical discovery, and show that mathematical logic, even at its purist is extremely useful.

My functional definition after all is pure as well. But useful? Most definitely.

To me that shows a faith in mathematics and logic that many may lack as the best ideas should work in the real world because reality is really cool.

I actually like being able to get things done--in the real world. Not just in the mind's of mathematicians.

But of course, I am NOT a mathematician, thank God.

James Harris

## Thursday, May 05, 2016

### My kind of functional definition

Years ago I found myself looking at a mathematical argument wondering how I could know for certain it was correct. Theoretically mathematics offers the ability to have an absolute certainty with mathematical proof, but how do you know if you have a proof?

Checking available definitions of mathematical proof and then looking at my mathematical argument, I found I still didn't have enough to be sure! And long ago got tired of thinking I had something which was correct, only to discover later I was wrong! And under that pressure found myself switching to what I decided to call a functional definition. Through the years have refined it. And now have something I think is quite formal:

mathematical proof (noun): a mathematical argument that begins with a truth and proceeds by logical steps to a conclusion which then must be true.

So yeah, I made that up. The original form was good enough for me to look over my mathematical argument and determine, yes, it was a proof! Which was a huge relief. I call that a functional definition as unlike typical what I call descriptive definitions, you can use it to see for certain with certain kinds of abstractions like mathematical proof, if you have the thing. Rather than talk a lot on that subject will give one of my latest:

entertainment (noun): any socially accepted activity chosen in order to alter mood in a desired way which is unlikely to bring harm in any way.

Where I was considering theory on social media, and concluded that I could explain certain social media metrics relating to what I see now as audience levels, by considering entertainment value others must see in public presentations. But what was entertainment? So I sat down and came up with a functional definition, which notice covers everything from a roller coaster ride, to a sad movie or play, to dinner with friends. While it also excludes dangerous things like taking illegal drugs.

Functional definitions are remarkable to me in contrast to descriptive definitions and my most powerful one is not needed to be given for this discussion, but will note I have it as an intriguing point on which to end.

Faced with various issues I concluded I needed a functional definition for science.

And it is by far my most prized.

James Harris