## Tuesday, December 13, 2016

### Summing squares to powers

Sometimes I just like to play with numbers:

12 + 9*12 = 10

82 + 9*22 = 102

262 + 9*62 = 103

282 + 9*322 = 104

Which goes out to infinity, where I used a simple rule.

If: u2 + 9v2 = 10a

Then: (u - 9v)2 + 9(u + v)2 = 10a+1

Which is just using my BQD Iterator. And notice that u and v can be positive or negative, while I like to show positive as is easier and looks prettier, which allows some selectivity, which I used behind the scenes to get my series above. Some choices made it more boring.

Like yeah so if you use a = 4, u = 28, and v = -32, then next is:

3162 + 9*42 = 105

So yeah, every power of 10 can be written as a sum of two squares.

And was fascinated a couple of years ago that the general result is, for an integer n equal to 0 or higher, and an integer m equal to 3 or higher:

x2 + (m-1)y2 = mn+1

I have m raised to n+1 so that n is a count of iterations. And if m-1 is a square then every power of m can be shown as the sum of two squares.

And talk it all out in this post. So I just used m = 10 above.

James Harris

## Sunday, December 11, 2016

### More thoughts on coverage

This post is labeled behind the scenes as maybe isn't so interesting for those looking for other things, like math. But is useful for me to record things in place I like to say.

Have realized some posts which talk the coverage of the blog in terms of registered visits could be really useful later. So with year closing out found myself in Google Analytics this morning running the numbers so far, focused on location.

And according to Google Analytics blog had visits from 57 countries so far this year, consistent with what posted earlier on this subject. Have a map of the world can look at and see whitespace where no visits, and as expected most of that was across continent of Africa. And looks like cover about half of South America.

The city counts kind of fascinate me now, and had visits according to Google Analytics from 346 cities, where can switch now to languages, and has a number of 38 languages. Is simpler for me to just read that number off from the Google Analytics page than dig deep into it.

To me the numbers of country counts are down a bit. I know not why.

But then again, how does it matter across the board? Well reality is means I have a direct connect which exists by objective measures. I'm just reading off information conveniently provided by the vast resources of Google.

Will there be a shift for me based on these numbers? Probably not because of them but because of changes in my use of the blog as is no longer primarily a research blog as have stopped new mathematical research. That freed up a lot of mental energy for other things. Math research is fun yes, but EXHAUSTING.

Now it's more interesting for me to consider what I've learned. And yes, continue to learn, but not chasing possibility wherever the math might take me.

Now it's more like sitting with the math and chatting conversationally. Which is a lot more relaxing.

James Harris

## Sunday, December 04, 2016

### When simplicity bemuses

One problem I have had for two decades now is when I look at something that seems too simple to me that I discovered. Which is not just a problem I have as get a weird pleasure out of noting that my first paper sent off to a math journal was covering packing of spheres and was rejected as too simple. Here's a link to a post I finally put on the blog in 2008 discussing, and yeah I lost the paper. So I had to recollect the argument.

Far as I know the original paper which was on paper is gone. Probably tossed it into the trash or something, which doesn't matter. My problem with it was, how could there be this simple approach to a problem that was over two thousand years old?

I'm just not able to maintain confidence on that one as I tell myself, but Sir Isaac Newton worked on this problem, how could he not have noticed this simple approach? Then am like, but I don't need it anyway. I say that about lots of things. Is like, who cares, and I look at something else I have. And now? Have TONS.

Like take a look at this one. Copying over from a post on my Beyond Mundane blog though LOTS of posts about it on this one:

With natural numbers--means use ints or longs--where pj is the jth prime:

P(x,n) = x - 1 - sum for j=1 to n of {P(x/pj,j-1) - (j-1)}

It counts primes when n equals the count of primes up to sqrt(x), so if n is greater than the count of primes up to and including sqrt(x) then n is reset to that count.

There is nothing simpler which is as fast for counting prime numbers in ALL of mathematics. And nothing faster for its size. But is SO simple. For years will admit bugged myself by wondering why Archimedes doesn't have it, and how would human history have changed if he had?

And that's not even expressing it pretty as Blogger doesn't make that easy far as I know. If it IS doable without a bunch of extras would like to know! Would switch to showing that way. Is MUCH prettier with the summation sign used.

These days am not as bemused about it as in the past, but still am to an extent I rarely admit. It uses mathematics that is easy, as you really only need division. And in a more general form you don't even have to know what a prime is, as it will select them out.

The math knows what a prime number is. There is nothing else like it in all of mathematics and it even leads to a partial differential equation.

That the math does know what a prime number is without some human guiding it used to mess with my mind massively. And I realized, the math does not need us.

So why is such a simple discovery mine? I could go on. I could fill this post with simple results that bemuse me. I even improved upon work by Gauss. Found results in areas where Euler and Ramanujan both considered something. And what of it?

When I need a pick me up though can just do Google searches. And even at times will make myself look for math things done by others related to my results. I wish they were as interesting though. That's better for the human species.

One thing will admit, I don't know how anyone who knows mathematical history thinks a person has just one result. Or even a couple and is a big deal. You want to compete in mathematical discovery? Start talking with at least half a dozen major results, maybe. It's an incredibly competitive arena. Maybe most competitive of all human intellectual endeavors.

I shrug at people claiming a single result. Am like, maybe you're getting started? But probably not or you'd know better. Come back when you have a dozen.

And have thought now for years about how these are mine, but reality is I'm just some guy who figured out some math.

Maybe the reality is we really will never know why about some thing.

But how do you know?

James Harris

## Sunday, November 20, 2016

One result I stared at for quite a bit is the start of an infinite progression using what I now call the BQD Iterator:

The series starts with

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

5. ((D2 - 6D + 1)x + (4D2 - 4D)y)2 + D((4-4D)x + (D2 - 6D + 1)y)2 = F(D+1)4

and that goes out to infinity.

Here is my reference post discussing it, wow back in 2008. So yeah back then hadn't named the BQD Iterator. Just kind of informally talked it for a long time. Naming it worked better, and think I came up with a cool one.

It's interesting looking through the post seeing my efforts to try and use that series to solve for x and y. Through the years have had mixed feelings about such efforts.

James Harris

### Our picky world

Thankfully years ago I learned to very wary of trusting myself when it came to my mathematical ideas. Sure I could tell myself that searching for the new can lead to false positives but it's another to live it. And have that gut-wrenching moment when realize something you'd convinced yourself to be true, was utterly false.

With the rise of the web I was also really lucky in that web allows me to not rely on myself, which is my continuing strategy.

Yeah but eventually that lack of trust in self can just get silly.

But am SO skeptical I like as the years go by to check, re-check, and check again. And at times I've done things to try and test the web interest.

The web flexes and shifts with any such efforts. It actually, well it can kind of mess with you as the new is so different than before.

The web behaves like an intelligent entity though I realize am seeing reflected the interest of LOTS of human beings. But for me often is like am dealing with one being. Can be, um, remarkable can say.

The world grabbed hold of my research as soon as it went public. I'd rationalize as watched the evidence. Tell myself things I realize now are not true so I could just mostly continue my life as if nothing had changed. Now I simply talk about things as the mood moves me. It's not necessary for me to say anything else about it, ever. There is relief in making that hard statement. But part of me wants to believe I still need to be working at it.

But it IS knowledge. Turns out I can't take it away even if I wished.

And had some thoughts on how I use the web to test things in a post here almost two years ago. But now realize I was still only JUST beginning to grasp how the world treats information.

That post fascinates me as it is remarkable to me how much more I know now compared to when was so sure had most figured out. And also now am free to focus more on things will admit might interest fans of the research! Which is LOTS more fun. So my posting behavior has shifted.

Finally as I stopped thankfully doing my own basic research started looking around and realized more and more our world is very picky. It is odd to consider that you can survey any number of people effortlessly just with a few searches.

Now will admit more and more I assess others. And web lets me do it with minimal effort. That's also more fun.

For years I convinced myself certain things had to happen or certain things should happen around my research, but I was wrong. The world does what it wants. The web though lets me watch in awe.

James Harris

## Friday, October 28, 2016

### Why focus on two conics

Was fascinated to realize that an ancient expression gives quite a few conic sections as it in fact gives 2 of 3:

x2 - Dy2 = 1

Where depending on the sign of D, you can get an ellipse, which includes circle for D=-1, or a hyperbola for positive D. So I like to call it the two conics equation.

But have noted it as giving 3 conics as some consider the circle to be one in its own right, but circles are ellipses. And am sure is something I considered years ago before focusing as I do.

And that's with fields of course. The equation got lumped into the category of Diophantine where the hyperbolas got all the attention, though only looking at integer solutions, which is a historical oddity I think which traces back to Fermat as to how that emphasis stuck. The equation has been known for thousands of years though.

For a bit got excited as wondered if I should really go with three conics but I thought over so many things carefully years ago. And have learned the hard way to trust myself. There are times it takes me days to remember why things are a certain way. Luckily here was a matter of minutes.

James Harris

## Sunday, October 23, 2016

### Naming things is hard

One of the best things to me about what I've learned is that I can unequivocally say that people don't need to mess with what is commonly called Fermat's Last Theorem, as Gauss was right about it. And I think in mathematical history it will be seen as an oddity more important for the effort around it, as the result itself is trivial for the reasons he gave.

And for me it is relevant because it was SO frustrating that I invented my own mathematical discipline to tackle it which I eventually called modular algebra symbology.

For a while was still looking at it with that then was like, oh my.

And I realized I needed the object ring.

And modular algebra symbology explores what I decided to call tautological spaces, where you do analysis on what I decided to call the conditional residue.

And with modular algebra symbology I came across something I decided to call the binary quadratic Diophantine iterator or BQD Iterator for short.

Also years ago decided I had found an axiom previously not accepted to be one! And decided to call it the prime residue axiom.

Oh yeah and renamed something the two conics equation as can give hyperbolas or ellipses, and its generally used name, at this writing, is considered to be a historical error.

And remembered that I found what I like to call a quadratic residue engine which is key to derivation of expressions to count quadratic residue pairs.

I think that's it. Not interested in continuing to update as have a couple of times, and still maybe have named a few other mathematical things here or there but I think I got my most important namings.

James Harris

## Saturday, October 22, 2016

### How much a global resource

So far this year according to Google Analytics the blog has had visits from 298 cities in 50 countries, by people with 36 languages, so suspect Google Translate does help.

Those numbers show a global interest reality which is greatly appreciated.

Oh yeah, guess may as well give last year for reference.

Last year it shows visits from 423 cities in 68 countries and 43 languages.

To get these numbers went into Google Analytics and set the dates, and then clicked options to get information and read off what it showed.

Thanks for the interest.

James Harris

## Tuesday, October 18, 2016

### How do I know?

The appeal of human authority is great and for most people that's all they have. They are told what is true or not, and in many cases have no means to check. Human authority is the ultimate authority for them, but in mathematics that is not the case.

But MOST of what most people know of mathematics IS on human authority while I found myself metaphorically in the wilderness with new mathematical approaches which was VERY distressing. So I found absolute truth in identities.

The identity which changed so much for me I call a tautological space:

x+y+vz = x+y+vz

Is weird to talk how much I considered that, and thought about it, to convince myself that what I was taught by human authority and common sense must be true, actually was! Identities are true in and of themselves, like x = x. Is just true. But I was doing different things with it.

x+y+vz = 0(mod x+y+vz)

Wrapped up into a modular expression is the SAME THING but presented differently. Is weird though. Some may think that 'mod' means the two expressions are different. Puzzle that one out then. (If it is new to you, I explain more on this page.) Why are the expressions equivalent?

From there it was just a matter of believing in logic, and over time I relied on mathematical authority, and human authority? Well that can just be plain wrong.

And I learned a love of mathematical proof.

James Harris

## Tuesday, October 11, 2016

### Why proof matters

Mathematics and logic are the human disciplines which can rely on absolute proof. And I think it interesting to consider the possibility of people who don't even believe that exists. But it is a cool thing which theoretically removes the problem of conflict with mathematical ideas. Or as I like to say, proofs don't fight.

However there can be things subject to interpretation, though functionally I went ahead and presented a method for always determining mathematical proof which can be logically determined to be perfect, which is to begin with a truth, and proceed by logical steps. At each logical step, you have a truth, so my functional idea was to connect truth to truth with logic connectors. Oh, you may wonder, but how do you know you began with a truth? Good question. Um, asking myself and noting.

And I made an example to demonstrate just so can point to it when discussing:
Example post demonstrating absolute proof

Which I think is handy.

The example shows that with the condition that x2 + y2 = z2, then

(v2 - 1)z2 - 2xy = 0(mod x+y+vz) must be true, where v is a free variable.

So, since v is free, let's say I let v=10, then I know that 99z2 - 2xy has x+y+10z as a factor, when x2 + y2 = z2.

Like if x = 3, y = 4, then z = 5, and 99(25) - 2(3)(4) = 2451, and 3 + 4 + 10(5) = 57, and 2451 divided by 57 equals 43.

You get trivial results like that here but can use this approach to probe into LOTS, where v is a tool of your mood. I like that, and in fact picked 'v' for victory, mostly.

That result is absolutely true where those expressions are available, without regard to ring. I like that.

Oh, and I made up that mathematical analysis path, and like to call the mathematical area--modular algebra symbology. For some reason now that naming...well is accurate. Naming things is fun but at times I wonder at my choices. Where use what I call tautological spaces, where advanced the concepts enough to cover all the mathematics interesting to me.

But yeah to me modular algebra symbology is SO cool, and it helps that I made that up, so is my own personal mathematical discipline, but others can use of course but for me will always feel different. And yeah I pioneered a functional approach to determining if you have mathematical proof, so it feels different using it too.

Web search is an AWESOME way to check a person on bold claims. Easy is to just search on modular algebra symbology, as yeah I did make that up. Also you can search on tautological spaces, as I made that up too! While checking me on how people check mathematical proofs is where the hardest work may be, but search on that too where will not suggest searches. If interested, figuring out how to search to check me on functionally defining mathematical proof is useful as well.

Because mathematics can allow absolute proof it does make it kind of boring to argue it, and I quit that years ago, though will admit that the utility of arguing about mathematics is about finding error. So it DOES have value, arguing about mathematics, but in my opinion, the only value is in finding error which means a mathematical argument is NOT proof.

Sure someone can CLAIM absolute proof, but does that same person actually have it?

Turns out I figured out how to check, and yup, do use it! Is fun.

And yes, to me saying absolute proof is redundant but is useful for emphasis and clarity.

Proof lets you move on and do other things, like find more mathematics! Or hang out. Do something entertaining, or some other kind of work. Or, whatever.

James Harris

## Saturday, October 01, 2016

### When derivation is surprisingly easy

One of my favorite and more popular results is a derivation of the already known way to count things called quadratic residue pairs, but with a derivation can go beyond what was known before it.

For those curious about what a quadratic residue pair looks like, here's a list of quadratic residues for 31:

1, 2, 4, 5, 7, 8, 9, 10, 14, 16, 18, 19, 20, 25, 28

And there are 7 pairs: {1,2}, {4,5}, {7,8}, {8,9}, {9,10}, {18,19}, {19, 20}

And that count is given by: floor((31 - 1)/4) = 7

Deriving that is easy, and fun!

Turns out MAYBE though I took a slightly long route as you can look at:

x2 - Dy2 = 1

Which I like to call a two conics equation as it gives hyperbolas or ellipses depending on the sign of D, and look at it modulo D-1:

x2 - y2 = 1 mod D-1, so:

x2 = y2 + 1 mod D-1

And you can notice easily enough that a quadratic pair can be there though the full set of rules for when it is, are not obvious there. Like for D-1 = 12, there are none. So how does the math do that? I think y always has prime factors of 12 or something, and um, it isn't very simple. Check out my post linked above for the derivation where I explain a LOT, as just sitting here now am like, it's not THAT simple.

If you can, you might want to try yourself! See if you can derive a quadratic residue pair count from here, or even more fun, find a derivation out there in the wild. That's kind of a trick thing to put though as I found the first, and to my knowledge only derivation. Do some research to see how the result was found before which if I remember correctly involved something called the pigeonhole principle.

Oh yeah, so I used "mod" so much I quit using a congruence symbol years ago, and don't think I lost anything. Also for those who'd like a primer for me on the subject, wrote one in a blog post years ago:

Focus on modular arithmetic

I really think modular is one of the coolest things ever, and of course can show up beyond mathematics. In mathematics though modular concepts lead to astonishing simplifications. I like to say, modular algebra gives a handle on infinity.

James Harris

## Thursday, September 22, 2016

### Why focusing on usefulness works for me

A few years ago found myself more and more focused on usefulness of ideas, whereas primarily before was focused on discovery, which can lead to a lot of effort and time spent on validation once something is found.

Turns out it can take a LOT of work to evaluate an idea, and in mathematics at first I was focused on established authority. But that turned out to be frustrating, and you can feel bad bugging people not interested. The web though let me focus on usefulness as to how useful something appeared to be based on interest.

Is so wild, when you share some ideas and can see them loop around the world.

Whether we realize it or not, mathematics is a massively huge subject where few can bother to get a good grasp of it, if that's even possible now for a single human, where what is known and used determines the focus of the human species. If you dig deep you can find mathematical ideas that just don't get a lot of people using them for whatever reason, and they stay buried. The attraction isn't there.

But like back in 1999, I decided to use x+y+vz=0(mod x+y+vz). And eventually decided to call it a tautological space as is an identity. Eventually I decided to call a result I derived with it a binary quadratic Diophantine iterator or BQD Iterator for short, and you can search on that and find out LOTS.

External validation from the establishment would be ok, I guess but for me? Too much work. And besides, I don't need it. Discovered more probably thanks to not having it, which is perspective learned. And then don't have to feel guilty bugging not interested people just because they happen to be mathematicians.

Now I've ended actively searching for new ideas completely which is such a relief. Finding new ideas is easier than checking them, and easier than validating them, but is still HARD.

But thanks to the web, seeing use is easier than all of that.

James Harris

## Thursday, August 18, 2016

### Reducing a quadratic Diophantine to find solutions

Will readily admit do some posts to talk things where it also helps when I just get curious as well. As found myself thinking should do something new showing my method for reducing binary quadratic Diophantine equations, and eventually decided to go with:

x2 + y2 = xy + x+y + 102

There is a LOT of deliberate easy there and found out could still get more difficult with that last.

Where it took me a bit to pick that where I was too ambitious at first and tried 1000 which had NO integer solutions! Played around there for a bit and decided to use smaller numbers and 100 and 101 had no solutions. Luckily 102 did, as get tired of such things quickly.

(If you want to test your math knowledge, find solutions before reading further.

Can you prove that with 1000, 100 or 101 instead of 102, no integer solutions?)

Oh, and first thing is to standardize to the form I used with my method to reduce! As forgot to do that at first and was confusing myself:

x2 - xy + y2 = 102 + x+y

What follows relies on my post showing how to reduce and you can reference it to see how I get A, B and C as well as the rest.

Ok, so then A = 9 + 4(-3) = -3, and B = -3(0) + 2(-3) = -6, C =-4*102(-3) = 1224

Which means:

(-3(x+y) + 6)2 + 3s2 = 36 +3672 = 3708

And dividing off 9, gives:

(-(x+y) + 2)2 + s2/3 = 412

And -(x+y) + 2 = 20  works, as gives s = 6, as a solution. I like to use the positive solutions as just want an answer. This thing is picky though! Was maybe a bit surprised had to search a bit, but glad 102 wasn't too far from where I started.

x+ y = -18, so y = -x-18, and substituting with original:

x2 - x(-x-18) + (x+18)2 = 102 +x - x - 18

So:

x2 + x2 + 18x + x2 + 36x +324  = 84

3x2 + 54x +240 = 0, which is: x2 + 18x +80 = 0

And (x+8)(x+10) = x2 + 18x + 80

So have two possible solutions and will use x = -10, so y = -8. Oh there's a symmetry thing going there.

So original was: x2 + y2 = xy + x+y + 102

And: 100 + 64 = 80 -18 + 102 = 164

And that's just one set. Noticed also that s = 33 works. And can get:

x = 3, y = -8

9 + 64 = -24 + 3 - 8 + 102 = 73

Is interesting to me that coming back to play with the math often for me is an adventure, especially as get some distance from the discovery.

Here it also kind of intrigues me that there are NO integer solutions for:

x2 + y2 = xy + x + y + 1000

Had a vague feeling that some combination would work, and that vague feeling was wrong. And then there weren't any integer solutions with 100 or 101 either.

Those integers can be SO picky.

Numbers just stay interesting to me.

James Harris

## Friday, August 12, 2016

### Simply interesting across centuries

Will admit one of my more gratifying finds can be shown with some really simple examples, where luckily noticed that both Ramanujan and Euler had shown interest in the area.

Here are some expressions:

12 + 7 = 23

Next : 32 + 7 = 24

And next is: 52 + 7 = 25

And then: 12 + 7(3)2 = 26

One more: 112 + 7 = 27

Notice that 7 is bare, in all but one case.

And according to MathWorld, Ramanujan has a result called Ramanujan's square equation:

x2 + 7 = 2n

Which they say only has solutions for n = 3, 4, 5, 7 and 15.

And I've shown the n = 3, 4, 5 and 7 cases, but am not interested in iterating 8 more times to get the last one.

And also on MathWorld I found that Euler had an interest in this area as well with his own solution which was NOT using my BQD Iterator. Will admit have no clue how he found it, as is something more general.

Where I explain more in a previous post where I note these as validating historical connections.

My methods generalize to allow you to solve a much larger variety, ok an infinite variety of similar. Will admit that being able to explain with something that Euler probably didn't know and that Ramanujan probably didn't know in an area where both showed an interest feels weird. Am doing better with it now as time has passed. You just kind of feel good after a bit and lose most of the overawed feeling.

That celebrity aspect does make me ponder a bit, but will admit is good fun.

James Harris

## Thursday, August 11, 2016

### Simplifying understanding works best

Years ago I learned the hard way just how important simplifying, simplifying, simplifying is when you're trying to figure out your own math. Repeatedly I'd had mathematical arguments I thought brilliant, until after making them simpler I'd watch fall apart with what then looked like silly errors.

So I learned to love simplicity, and also to dread it.

It is an odd thing when you look at something you found, maybe even years ago, and realize it has simple implications you have not checked.

Mathematics is brilliant in that everything has infinite consequences some kind of way.

Infinity creates the highest standards.

That would fascinate me at times, looking over mathematics I found knowing it has to work over infinity. And scare me if I dared check something I hadn't before. And then that odd feeling when the mathematics works perfectly.

There is nothing like it.

If you can simplify? Do it, if you dare.

James Harris

## Wednesday, August 10, 2016

### Covering the blog with interest

Feel like I do need to emphasize am NOT a mathematician, and there are some difficult aspects to this situation, where feel like have important mathematical ideas without establishment support. But so glad for the interest, and trying to be more appreciative of people who really have supported me with it.

So this blog which started March 20, 2005 has people STILL reading blog posts throughout that history. And some of the most popular posts according to Blogger are years old! That is amazing.

Though yeah most interest is concentrated more recently which is also good, as like to think have learned a few things over time, and more refined results are more recent as well. Before was spending MOST of my effort at basic research.

And will note have ended basic research as of July 31, 2016. Which I posted about on one of my other blogs.

Being a very curious person have been gratified to find blogs help me structure around my interests and keep up with things! And also of course blogs help you efficiently share with the world

Ending the basic research process lets me focus more on refining ideas, or just enjoying the ones I have.

Thanks for continuing interest. Yes, I do notice. Much appreciated.

James Harris

## Sunday, July 10, 2016

### So much from one thing

From my method to generally reduce binary quadratic Diophantine equations, I found the following when using on an already reduced form.

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)

Which I eventually decided to call a Binary Quadratic Diophantine iterator, and you can do SO much with it. Thought I'd highlight a few things here.

You can find sums of square for the same value with it.

52 + 202 = 17*52

82 + 192 = 17*52

132 + 162 = 17*52

Using it you can get an infinity of expressions to approximate the square root of 3, where posted about one which will show here:

sqrt(3) approximately equals xn+1/yn+1, where:

xn+1 = 362xn + 627yn

yn+1 = 209xn + 362yn

and x0 = 1, and y0 = 0;

Next, x1 = 362 and y1 = 209, and (362/209)2 approximately equals: 3.0000228

Iterate, and you get, x2 = 262087, y2 = 151316

And: (262087/151316)2 approximately equals 3.00000000004367

Iterate again: x3 = 189750626 and y3 = 109552575

(189750626/109552575)2 approximately equals 3.0000000000000000833

And 189750626/109552575 approximately equals 1.732050807568877 showing only digits that match with sqrt(3).

You can find a sum of squares to equal a square with as many as you like.

To demonstrate I posted an example doing 5 sums of squares to get a square.

Where I found, two sums of fives squares that give a square:

42 + 6+ 10+ 14862 = 882

862 + 129+ 215+ 301+ 8812 = 9682

You can do lots of things of that type and I guess I played around quite a bit. Also though found could explain something noticed by Euler and Ramanujan which was cool.

It is interesting that I found a burst of things after coming up with name Binary Quadratic Diophantine iterator or BQD Iterator for short. Why would naming it make such a difference for me?

Am sure I did things with it before naming it as have had it for years now. Like I found could connect ellipses and hyperbolas back before I had the full current form. There's just so MUCH so will just go ahead and stop there. Feel like covered enough things for this post.

James Harris

## 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

## Wednesday, March 16, 2016

### Why bother with sqrt(3) approximation

Turns out a simple method for calculating the square root of three is to use the following.

sqrt(3) approximately equals xn+1/yn+1, where:

xn+1 = 362xn + 627yn

yn+1 = 209xn + 362yn

and x0 = 1, and y0 = 0;

Which is just for fun, as have noted every time I put up one of these that of course you can just use a calculator. But it is interesting to me, if I just came across that myself, maybe could make me a bit curious. If you try simplest case x1 = 362 and y1 = 209, so 362/209 approximates sqrt(3). And can see how well it works there.

(362/209)2 approximately equals: 3.0000228

Where clipped a bit as no reason to show all of what pc calculator gave.

Iterate once, and you get, x2 = 262087, y2 = 151316

And: (262087/151316)2 approximately equals 3.00000000004367

Where arbitrarily clipped again. And that method goes to infinity, so you can iterate as much as you like.

Will show one more.

Iterate again: x3 = 189750626 and y3 = 109552575

(189750626/109552575)2 approximately equals 3.0000000000000000833

And 189750626/109552575 approximately equals 1.732050807568877 showing only digits that match with sqrt(3).

For a mathematician? Probably a trivial thing. Is easily explained how it works, and goes back to number theory that is considered rather old. Though I used my own tool to find it, so for me that part is cool. Does my approach work further? Nope. What I did works for sqrt(2) and sqrt(3) and nothing else. Though I do wonder if I were more clever if could figure out how to get something from it to work further. Tantalized by the clues.

Am confident that if you program it into a computer you can quickly exceed its capacity to check the result. But still, is just the sqrt(3).

So why bother with it.

Not even a question really. For me? I like playing with numbers. For some maybe there is some mystery, like why does it work? How does it work?

I think that's cool.

Not every one wants to be a math expert and look at only advanced mathematics that pushes the limits of even the greatest minds on the planet to understand.

I know I don't. Sometimes I just like to play with numbers, just for fun. Sharing that? Why not?

So it's easy mathematics. Guess what. I like easy.

James Harris

## Sunday, March 13, 2016

### True power of correctness

Talked about my disruptive approach to mathematical discovery, where was more into brainstorming ideas where meant many were useless or wrong, but would put up public which helped me have them critiqued. So different am sure from what most math people do, but am NOT a mathematician, where like to emphasize, as frees me to do things different.

Still some people can ignore even the simplest disclosures and try to hold you to rules they think apply. And with disruption of established ways? Yeah that word "disruption" can mean some will react badly, and the insulting names game can seem appealing to them, but to me is childish. I call it a playground mentality. Which to me emphasizes is talking a behavior and not labeling a person or group of people.

And so long ago seems now, when was actively arguing out math ideas can remember distinctly when something remarkable happened. And yes had times when was convinced some idea was correct only to have it fall apart in shambles, worst after MONTHS. But at some point I flipped on my own ideas as realized the math does not care.

If you have a correct mathematical approach it does not care. It does not know you exist. The math has no opinion on you. It has no feelings whatsoever it is just correct. It will work.

At some point arguing with others, which just don't bother now, realized an odd feeling as the slings and arrows of words kind of made me flinch but was like a shield was there. Eventually it's like this odd feeling as you realize nothing is getting through. The math deflects all attempts against it. Blocks all attempts at reason to find fault, as there is none.

And the words ceased to have meaning.

The math does not care. It protects without thought. Defends without effort.

I was shielded by correctness not because the math defended ME but because I relied upon it.

It is fun then, as you attack it. Try to take down your own ideas. Get creative at approaches to debunk it.

The true researcher should be the most virulent in testing against own ideas. Best ones don't care what you think. Flawed approaches can reflect your need or mental limits, or those of others if you rely upon ideas handed to you when wrong.

There is no point in defending correct mathematics, as trust me, if it IS correct it is unassailable. Perfectly shielded. I try to think it defends me but must fight my emotion.

The math does not care.

James Harris

## Thursday, March 10, 2016

### Disruptive approach to simpler math

One of the things I realized over a decade ago with my approach to mathematical discovery was my leverage of the web to check ideas. So I'd brainstorm for ideas, put them up public, and get rapidly critiqued, usually meaning the idea failed. But sometimes would rapidly innovate to something different, robust and perfect.

Like my way to count prime numbers which is SO CLOSE to what was known but is more compact because of innovative approach.

Will give the sieve form as is smallest to show, and actually is MUCH prettier with the summation sign but not so sure on how to do that with Blogger or if I can. Interested readers can just write it out to see it that way.

With integers ONLY, where pj is the jth prime:

P(x,n) = x - 1 - sum for j=1 to n of {P(x/pj,j-1) - (j-1)}

The P(x,n) function will count prime numbers up to and including x, if n equals the count of primes up to sqrt(x), and if as you iterate you never let n be greater than the count of primes up to and including sqrt(x), if the function receives an n greater than that value it just needs to reset it to that count.

That's mine. I found it.

For example P(100,4) = 25.

(Wish to pursue more? I use labels on most posts, and beneath this one should see the prime counting label. Click on it for more related posts!)

My way of doing research was very disruptive as I wasn't so excited about mathematical errors, as much as idea generation, and tossing of erroneous approaches rapidly. Where I used the web for feedback to help figure out when things were wrong. And now finally with the web can distribute perfect ideas which withstood all challenges.

As mathematics is awesome in a fascinating way: correct ideas can't be broken, cannot fail, cannot lose against challenge. So why not hit them with everything you've got?

The math doesn't care.

James Harris