Thursday, July 30, 2015

Sums of squares for same value

Handling sums of squares can be mostly automated using some of my mathematical research. For instance consider:

52 + 202 = 17*52

82 + 192 = 17*52

132 + 162 = 17*52

These were found using what I decided to call a Binary Quadratic Diophantine iterator, or BQD Iterator for short, which lets you find in general solutions for x and y with:

x2 + (m-1)y2 = F*mn

When F = x02 + (m-1)y02, my research proves there are non-zero integer solutions for x and y, where n is a count of iterations.

To get my examples used x0 = 1, and y0 = 2, and picked m = 5 so that I had 4 which I could pull into the square, to get a sum of two squares.

At each iteration you get a split point, where you can go positive or negative, which means as you iterate you may generate extra solutions for the same sum.

For my example I had a duplicate in the second iteration which is why there are only 3 distinct solutions instead of 4.

And the 5 being squared in this example is not an accident, as that's the second iteration. The first iteration has 17*5.

So yeah I could have kept going, and would have had a maximum of 6 cases of two squares summing to equal 17*53, but less if there were duplicates.

I have an earlier post which shows how the solutions were found.

And I have additional research using the BQD Iterator showing how you get a desired number of sums of squares.  Which could mean for math hobbyists, maybe could be useful in generating magic squares of squares?

James Harris

Wednesday, July 29, 2015

My support of academic rules

Things can sound different depending on how you present them, and it occurs to me that I should note my surprising success following academic rules for presenting research results.

So yeah, I got published. And always add that the editors then got weird on me by trying to pull the paper--against established rules. And that didn't matter. The original paper is held online by EMIS and I'm a published mathematical author.

Theoretically I could simply put forward certain positions as formally peer reviewed, but defer to the reality of the weirdness from the editors, where one admitted the chief editor doubted his peer reviewers! They used two, unlike most journals from what I've heard.

So yeah, my published research passed a higher standard than most math papers ever see.

But clearly the editors maybe had less faith in the peer review system than I did.

So of course I support academic rules.

When followed they can only help me.

That shouldn't be a surprise. If I believe my own research then naturally I'm a supporter of the global system. I need it.

If I'm correct, then people following the rules will support that conclusion, which is what happened, when rules were followed.

My greatest interest is in the health of the global intellectual community not just as a concerned human being but also as a practical matter. And journals are an important part of our academic world, which is key to our intellectual community.

There are winners and losers in these kinds of things. And my place in mathematical history was set years ago. Not that I didn't keep building on it.

But you see, I have faith in the system.

In contrast to the well ordered world of academia consider the mess that is all over the web, like I had a fun time years ago for a while talking out some ideas on a Usenet newsgroup called sci.math where you had to keep people entertained. I got creative.

Some of them were nasty though, and quite a few apparently had little if any real mathematical knowledge. Wading through the useless postings could get tedious.

But some actually did know math and there were some actual professors including one of the most notable a professor from the prestigious Hamilton College that I posted about, who helped a lot. You have to grab those opportunities. Better than paying tuition! And I got a lot of useful info there as I'm not a mathematician. However the egos were out of control and some had an irritating habit of routinely proclaiming they had destroyed you, which it turned out involved trying to smear you across the web. Some of those people firmly believed in insults as a tool of power, in their bizarre imaginations. So absurd.

Of course I knew they didn't matter. When time came to get things done, notice I got a paper published.

Turns out I have only one mathematical discovery that really needed formal peer reviewed publication, otherwise I wouldn't have needed to bother with it at all. But I didn't know that before. Regardless I like the idea of at least one published result and do have a lot of respect for the system.

You have to know what you're doing, you know?

And don't take yourself too seriously! Life's to be enjoyed. Gotta have fun! And follow your own schedule I say. Don't let others pressure you onto theirs.

People can think what they want. Doesn't make it true.

James Harris

Saturday, July 25, 2015

Simply more effective

Check out:

x2 + 2xy + 3y2 = 4 + 5x + 6y

Are there any integer solutions? Yup.

Mathematical techniques where Gauss had a big role with them give one way to reduce it to a simpler form, but I found another.

That reduction to the simpler form using my own research gives:

(-4(x+y) + 10)2 + 2s2 = 166

It's easy to see how I made my example as I just counted up from 1 to 6, and now you can see the reduced form, and can if you wish click the link to see how it was done.

There's a hard to explain satisfaction in mathematics when you can find a simpler way, especially when your hero had the earlier techniques, as Gauss had a big role with early methods, while I've had the luck of being able to innovate and find there was something further along.

And it's fun to discuss, of course, so will do that here.

A reason to reduce to a simpler form is to aid in finding solutions, so let's keep going.

Subtracting 2s2 from both sides the equation above is:

(-4(x+y) + 10)2 = 166 - 2s2 = 2(83 - s2)

And got lucky as no, didn't know years ago when I first thought to use it as demonstration but it DOES have an easy answer, and you can see that 83 - 81 = 2, and get a 4 on the right side. God knows I like lucky!

So you can see that s=9 works, giving -4(x+y) + 10 = 2 or -2, so x+y = 2, or x+y = 3.

And can also easily determine these are the only integer solutions that will work, as s = 9 is as high as you can go without going negative, and only an odd s can work, and none of the other lesser odd values do. Way cool, eh? Just from reducing to another form.

You can solve for x and y also by substituting with the original equation. However you do it, you get two answers.

And x = 4, y = -2, will work or x = 5, y = -2.

Will show with the first which gives:

42 + 2(4)(-2) + 3(-2)2 = 4 + 5(4) + 6(-2)

You can actually fully solve for s as a function of x and y, which was never high on my to-do list. Others have done it in general using math software, and may as well go ahead and give it versus being silly:

s = (c2 - 2c1)x - (c2 - 2c3)y - (c6 - c5)

With this example c2 = 2, c1 = 1, c3 = 3, c6 = 6, and c5 = 5, as I did that on purpose.

So s =  4y -1, and -9 works, to give y = -2. Am slipping in here as it's one of  the few results I have where others figured it out first, but they just plugged everything into math software, so it's not like they did any real work. Like to keep this blog exclusively my findings.

There's an odd feeling when you see the numbers behave as your research says they should. Will admit find it comforting, but also feel a bit odd. I found it. Wonder what Gauss would say to me if he were still alive.

If you think my claim of a better approach ludicrous then reduce the given equation using other techniques.

To see more you can check out my post on concepts in binary quadratic Diophantine equations.

That method for reducing to a simple form does something interesting when confronted with an equation already in a simpler form:

u2 + Dv2 = F

as it gives you back an equation that is also in that form:

(u-Dv)2 + D(u+v)2 = F(D+1)

And I've used that a lot and talk about it a lot, where you may not know it comes from using a method to reduce binary quadratic Diophantine equations that improves upon techniques pioneered by Gauss. And Gauss is a major hero of mine in the mathematical field.

One social problem I've seen is when people who lack even the most basic math knowledge cast judgement because they have no clue how these things work, but can react to a statement on a feeling. Math does not care about your feelings.

Actually some things I do really relate a lot back to him, for instance, I read that he kept up with news by relying on lots of newspapers. And it is remarkable to consider this man of a different time, using the tools available, having all these physical newspapers mailed to him from all over. Clearly it was important to him.

While today I can rely for lots of news on Twitter, as well as other web sources, where yeah, part of the reason for making more effort there is trying to emulate Gauss, and I have it so much easier than he had it.

Originally I was frustrated with why a better and more simple way to do something, like reducing binary quadratic Diophantine equations wouldn't be cheered loudly, as it took over. And you know? The mathematical community DID apparently pick it up quickly. And is using it.

And why wouldn't they? Can you think of a reason? I can't.

Can you think of why some people though, might prefer less effective, more complex techniques? I can.

Simplifying mathematics is a joy. And apparently has endless power. I could go on and on, with things I've done with just this one thing. Oh yeah, I have, on this blog.

James Harris

Friday, July 24, 2015

Feel better remaining cautious

Took me a while to start really considering things from the perspective of someone else, who might look over my mathematical ideas and come to certain conclusions and then I keep talking for over a decade like so much is up in the air.

Reality is I've always preferred an extraordinary amount of caution, while also at times getting into arguments with some people in the past, and more recently expressing frustration at times when what I expected didn't happen.

But what about for those who have worked it all out? Who know the math? Know the story?

Going forward I've decided to accept the things proven as proven without endlessly giving out warnings about who accepts what or not, though it feels better to remain cautious.

And even more importantly realize that the "mathematical community" is made up of people around the globe using valid mathematics. In the past I'd use that phrase in a way that didn't fit the reality. Wonder if I should go back and clean that up. Was how I used the phrase really that bad though?

For someone who knows the story then, I can say that our world consumes and uses a vast amount of mathematics, without which modern civilization would not be here.

A tiny sliver of some of the mathematical ideas out there are negatively impacted by some of my research. So much of it all involves esoteric areas. And the mathematical community is healthier than ever.

Working mathematics is growing rapidly, and my ideas help. Where they help, people are using them.

I could waste my time worrying about worthless things, or focus on the positives while remaining cautious. I like remaining cautious and sticking with the positives.

These are the kind of posts where I debate with myself, about what the point is. But yeah, for some people these things are just settled. While to me there is a bit of a sense of something not done until the established experts acknowledge.

But that's me stuck in the past. Was born into a different time and really a different world as a person born in the 20th century. This one is so much more fun though.

These things matter to a lot of people all over the world and I need to respect that reality. I can't just go with, oh, I was taught things have to happen a certain way so forget you. I need to accept that it's a great thing. Our world is moving forward. I need to move with it.

James Harris

Thursday, July 23, 2015

Some important facts and observations

Recently I have begun noting that the mathematical community can be presumed to have certain characteristics like use of correct mathematical results. That puts me firmly within that community, while I also note I'm not a mathematician, as my educational specialty was actually in the sciences. I do consider myself to be a mathematical discoverer.

What I think is my most controversial result at this time apparently pulls in others things which I think can be broadly considered to be a social problem, which is the label for this post and others like it. That social problem includes issues around my find of a coverage problem where the ring of algebraic integers can be shown to leave out certain integer-like numbers.

Explaining the coverage problem can be done with a simple factorization:

P(x) = (g1(x) + 1)(g2(x) + 2)

where P(x) is a quadratic with integer coefficients, g1(0) = g2(0) = 0, but g1(x) does not equal 0 for all x.

At the trivial level, it's very easy to get started, as of course:

P(x) = (x+1)(x+2) fits the requirements.

But if you look at non-polynomial g's then it's easy to find examples where algebraic integers are excluded, where you do not need a field.

The result here backs up a previously published result, where I should note an attempt by editors to remove the paper after publication, doing so from an electronic journal, while the original remains available where I've discussed the situation many times, where here I think is a decent reference post.

That paper served two purposes, both to highlight the coverage problem, and to give me the distinction in human history of being I'm sure the only person to be published with a formally peer reviewed and correct result, under established mathematical views, which nonetheless leads to a contradiction.

I doubt any other human will get that chance, so seized it.

I guess in some ways it's a dubious accomplishment, but I think I had a duty to grab it anyway, as no other human being will probably even have the opportunity.

Supposedly that was impossible, but this weird coverage thing made it possible. So yes, the paper IS correct by accepted standards of rigor, and it does lead to an apparent mathematical contradiction. Resolving that apparent contradiction is easy though. So it actually invalidates "established mathematical views". Human beings got some things wrong, but the math is still perfect.

In general with all of my mathematical results have taken the long view, where early on with this result and others I leaned heavily on simply reporting to mathematicians, making sure it was in their field of interest as best I could, and leaving it up to them to do the right thing. Which included journal submissions.

However, historically except for one case already mentioned, all submissions to journals have been rejected by their editors. Which was fascinating, curious, and intriguing as to the various reasons given. It is of little interest to expand upon further here.

At some point if I chose I could again send papers to journals. It just doesn't seem necessary as I can make the information publicly available myself, like on this blog.

Given that I do have a formally peer reviewed result, with the one published paper, even if the editors tried to withdraw, am officially a mathematical author, and also have the right to present as an established result. There has been no formal refutation of the paper which fulfills the requirements of such, which are that same be itself formally peer reviewed and published. There were attempts in the past at attacking these ideas, which are not worth elevating in this discussion as did not fulfill minimum requirements for mention at my level.

Any attempts, however, to refute these ideas are of continuing interest to me.

Members of the mathematical community can then by the rules consider them valid. And I can present them as valid, while I also like to caution that to my knowledge the results are not accepted by mainstream mathematicians, which is actually irrelevant.

But in terms of the social problem I find the situation intriguing, and have noted I have little motivation to end the situation.

Members of the mathematical community are presumed to have very high standards, rely on mathematical proof, and act on the basis of their own conscience and judgment.

Members of the mathematical community will of course be held by the absolute standard of the discipline that correctness rules, which is true through its long history, with no excuses.

This blog will continue as a reference for much of my mathematical research. And I like it more as a reference rather than dealing with these type issues which I see as more social and political. However, I continue to reserve the right to discuss what I see fit here, as it is my blog.

Other research results are far less controversial than the one dealing with the coverage problem, but much of the above still applies. Members of the mathematical community who may wish more from me are simply not going to get it at this time.

My assessment of the global mathematical community is that it is extremely healthy at this time, with working mathematics helping to drive a tremendous amount of global progress. I expect that will continue with little reason to believe anything of importance will interfere.

James Harris

Wednesday, July 22, 2015

Reality check with Ramanujan

Last year I was gratified to see I could expand upon something noticed by Ramanujan.

He noticed that 2n - 7 = x2 had integer solutions for 5 values of n, and you can read up on that at MathWorld. And Euler had considered something more general. While I could generate those solutions with my own research.

Using my research I can show how to find integer solutions for x and y, when:

x2 + 7y2 = 2n

where with Euler's result, n is 3 or greater, and x and y are odd integers. He also found a really cool solution for x and y, while with my research you can generate solutions with my BQD Iterator.

The applicable Binary Quadratic Diophantine iterator here, or BQD Iterator for short is:

u2 + 7v2 = F

means that

[(u-7v)/2]2 + 7[(u+v)/2]2 = 2*F

where the iterator has been adjusted by dividing 4 off, which requires u and v odd integers. Letting u = v = 1, gives F = 8, and you get Euler's result, as the iterator goes out to infinity.

So you just plug the number in and watch it go. I actually did that with a prior post last year when I discussed these things as I found them.

For me it's a maybe minor result which is so awesomely important because it relates to Ramanujan and Euler. Which is the kind of thing that is remarkable for someone. I wonder how many people in history will ever be able to explain something beyond what was known by those two when they were working on the same problem?

Is there anyone else in mathematics?

And there you go. It's a reality check with Ramanujan. And I guess Euler but that would make too long of a blog post title.

I have no clue how many people have managed such a thing, but as one of them I'll admit that it was a confidence booster. And it was me heading off in that direction after working at generating solutions for equations with Mersenne numbers, and I just picked one to see, and found that Euler and Ramanujan had been intrigued by similar things.

To me that's just awesome.

Got a bit of a reaction last year as apparently that find zipped around the globe. Didn't notice? If so, that's not my problem. My duty was to present publicly, which I did, on this blog.

The mathematical community cheers mathematics, its discovery, and the profundity of a discipline that spans all of human civilization.

Sometimes you can get a reality check that points out the obvious. But for the discipline as a whole, its history speaks for itself.

James Harris

Monday, July 20, 2015

Weird power of the individual

Years ago I realized I needed to get comfortable with drawing attention from, well, from possibly every country on Earth. It is an odd thing as an individual to step through a door knowing there is no return, but that is the potential of mathematics.

One of my favorite results highlights an oddity to knowledge which I guess is inescapable:

It turns out that if you have:

u2 + Dv2 = F

then it must be true that

(u-Dv)2 + D(u+v)2 = F(D+1)

Rather simple algebra, and I found it, but why wasn't that found thousands of years ago? Why was it there for me to discover?

We don't know what we don't yet know! It is inescapable.

Yet the information has always been there. It just took someone to find it. But why me?

But it is just one of so many now I find it hard to keep count. For some reason discovery in mathematics doesn't stop you at one but seems to require an individual have many huge results. I do wonder why.

While it is a derived result it's verifiable by just simplifying the second expression by expanding out and using the first. I've stared at it so many times through the years wondering why it was for me to find. That can mess with your sense of sanity.

Recently I was jarred yet again when I found it even allowed me to explain things that had been considered by Ramanujan and Euler. And that was just one thing. If either were alive today, he'd probably chuckle in appreciation at finally learning the 'why'. I've been using it since 2008, here and there to figure things out.

Information travels fast in our world, and did before, while now the web means it moves faster. For years I've watched bursts of attention with fascination, knowing I have the best seat in the world. Others can just guess.

Mathematical discoveries are like perfect engines of attention: incorruptible, indestructible with no emotion, nor any care about human things like race, sex, nationality or anything else human. They don't care if people wish them to be true. And useful ones roar across the planet, whether you see that happen or not.

It really does feel like being pulled by something that is inhuman. It will not stop either.

And I get to be at the center of it all, contemplating the oddity of knowledge, how it does not seem to care about human social needs or structures and does not fit into the plans of man.

The world starts changing you immediately. It's like being pulled into an intense school like no other as yes, I can accidentally impact all kinds of things, if I'm not careful. You learn rules of politicians, consider economic impact, and worry about destabilizing things.

The attention gives you the influence. These words will be read in many countries. That gives them an impact just from that reality by itself.

But the best thing for me is to realize the weird power in the potential of the individual human, beyond what we can explain or predict. And I know what so many people around the planet may not realize: how much potential they have.

My focus is not just on world systems, but also on the individuals that make up our many human societies. God only knows what they will discover. We get to wait in anxious anticipation as the discovery engine continues to roar around our world. It's so exciting to contemplate.

The additional attention starts immediately. Information moves fast in our modern world. Very fast.

James Harris

Tuesday, July 14, 2015

Something about numbers

For those who are fascinated by them the absolute perfection possible with numbers is a thrill.

To others it might seem strange but for me there is something electric:

(462 + 482 + 722)(1722 + 258+ 430+ 6022 +17622) = 

            615+ 30752 + 141452 + 159902  + 1884972   =  114*74*210

My research let me easily find that perfectly balanced result. Just like to stare at it.

Yet the techniques for finding it traceback to a simple cool little result:

u2 + Dv2 = F

requires that

(u-Dv)2 + D(u+v)2 = F(D+1)

Which is one of my favorite discoveries.

For me there is a thrill in the absolute logic, knowing something that is perfection, and in the beauty of numbers following rules with absolute precision. Nice to find incorruptible things, you know?

Connections. And it's interesting seeing them separated for this post, and wondering how might it look to me without knowing how one leads to the other?

I've accepted you cannot just assume that thrill about numbers and logic exists in someone.

Some love it.

James Harris

Monday, July 13, 2015

When math surprises

Years ago I thought it simple enough if you actually found some important math--contact leading mathematicians who specialized in that area, notify them of it, and sit back and watch it get picked up, if it were valid. If it weren't then of course, no.

Not a mathematician myself, when that scenario didn't play out, I was at a loss. Check and re-check, see that it is correct and then got a bit angry, until I wondered to myself if I were antagonistic to math society itself. But of course that's not possible if my results are valid! Why not?

It's kind of weird, but by definition valid mathematics is important to the mathematical community. That is, the global mathematical community is actually focused on valid mathematical results! Obviously it doesn't willingly waste tons of time focused on false results, eh? Of course not.

But there can still be a challenge if you find people who may for whatever reason believe they are members of that community which may not be as simple an assessment as you might think where I have some ideas worth highlighting here and now.

For instance I was published in a troubling story, where recently I noted how much support has been required for that published paper to remain available from an official source despite the attempt to remove it after publication by editors of a math journal which soon after decided to simply roll over and die. I love that story. It's so wild.

Kind of story that just has to give me a chuckle. I readily admit.

For those who may think that math must be wrong, or way too complex, I've spent more time going through the argument, and actually got I think a slightly simpler angle recently, focusing on the factorization:

P(x) = (g1(x) + 1)(g2(x) + 2)

where P(x) is a quadratic with integer coefficients, g1(0) = g2(0) = 0, but g1(x) does not equal 0 for all x.

The conclusions reached from that expression validate my earlier paper, and I can comfortably say there really is no doubt, mathematically. But it's a surprising result! And for some who currently think they are mathematicians, may seem to be a troubling one.

Of course I have other results as well, where am gratified that my work with counting prime numbers seems to have a recent surge in popularity. I not only innovated slightly on ideas that go back to Legendre, I found a way to get a difference equation in a prime counting function that works by calling itself. What was my innovation? I focused on a function that counts composites by each prime excluding counts by smaller primes.

Weird how such a simple thing could have such profound implications! Leading to a far simpler prime counting function which uses a function I decided to call P(x,y), where using my approach I can also get the most succinct, fast prime number counter available. You can get faster with more involved algorithms, but it's a powerful and simple introduction to counting primes, with very simple ideas.

I love that thing.

But it's also surprising mathematics, maybe giving a route to answering some of the great problems in mathematics that some people who the world thinks are mathematicians do not want.

These are not issues that should concern me, however. My joy is in the discovery itself.

You see, I'm a member of the mathematical community--though I don't consider myself to be a mathematician--who greatly prizes valid mathematical results, which I assure you, is the dominant quality of the mathematical community through thousands of years of human history.

More so than any other field I think that mathematics has a problem with people who cannot measure up to its standards for truth.

But thankfully there will always be those like me, who would prefer to stand with the greats in one of the greatest intellectual disciplines of all time. What good is it, if not correct? Through thousands of years it is those who echo the truth who define the discipline. A mere hundred or so with some delusion or other pales in comparison.

Those who prize truth in mathematics are its community.

Our record speaks for itself.

James Harris