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Monday, December 18, 2017

Two squares and powers of 50

12 + 72 = 50

 482 + 142 = 502

 1462 + 3222 = 503

 21082 + 13442 = 504

 115162 + 134122 = 505

 823682 + 940242 = 506
____________________

Result follows from a simple rule.

If:  u2 + 49v2 = 50a

Then: (u - 49v)2 + 49(u + v)2 = 50a+1

Which is just using my BQD Iterator. And u or v can be positive or negative, so I selected to eliminate trivial solutions. Let's divide off shared factors where can, and highlight a pattern:

12 + 72 = 2(25)

 242 + 72 = 252

 732 + 1612 = 2(253)

 5272 + 3362 = 254

 28792 + 33532 = 2(255)

 102962 + 117532 = 256


So yeah, found myself curious again, where have already shown with powers of 10 and powers of 5 from those, and realized hey, can do 50 as well. And I do find it interesting looking back at overlap between lists, without seeing any different values.

The more general reality is that there always exists nonzero x and y, such that for an integer n equal to 0 or higher, and an integer m equal to 3 or higher:

x2 + (m-1)y2 = mn+1

Where my BQD Iterator is tool for getting those values. Where copied from this post from 2014.

And am endlessly fascinated by it, even as I use it in what I see as marketing ways. So here use it to do sums of two squares to powers but have also used it to do arbitrary sums of squares, as it shows that squares and exponents have this deep relationship.

These relationships control integers, whether you know of them, or believe in them, or not.

The math does not care.

The math has infinite knowledge. And the math can do so much where I like what has been shown to me. But I DO appreciate the knowledge.

The math knows.


James Harris

Wednesday, December 13, 2017

Simply lucky is best am sure

ENDLESSLY fascinates me that in the late 1800's mathematicians got lost trying to finally solidify mathematics into an established discipline. Which is also remarkable to learn as before was more of a hodgepodge of tools. And in our times I got to figure out what went wrong with an argument so simple, a quadratic will do.

And talk much as IS important, and they seized upon roots of monic polynomials with integer coefficients as being a good idea for members of a ring. Where monic just means 1 or -1, so for instance something like: x2 - 4x + 3 = 0

To show a flaw with their reasoning I need a generalized quadratic factorization.

In the complex plane consider:

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

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

Simplest example is: P(x) = (x+1)(x+2)

Where I throw a wrinkle into the mix by doing something no respectable mathematician would do--I add an extra factor I call k.

g1(x) = f1(x)/k and g2(x) = f2(x) + k-2

Multiply both sides by k, and substitute for the g's, which gives me the now symmetrical form:

k*P(x) =  (f1(x) + k)(f2(x) + k)

That extra factor is JUST to force one of the f's to have k as a factor, while I can force the f's to be roots of monic polynomials with integer coefficients. And if all seems very distant, consider means can show that 3-sqrt(-26) has 7 as a factor.

Have talked in detail with this post: 

Easily explaining historical miss

Have given a more dry and purely mathematical explanation with:

Non-polynomial factorization short argument

Which is useful for well-trained math people, and it IS all very easy though. Was simply lucky that something this HUGE could be handled with a quadratic and a trivial leap that I force an extra factor which is something math people are trained NOT to do, and in so doing can blow up a hundred years of mathematical thinking.

So they did NOT solidify the mathematical discipline, but instead weakened it, but our species was ok, because most math we use was developed long ago, as I recently explained.

There I note that certain folks are simply not needed for valid mathematics. And yeah, when realized that years ago knew had a bit of a slog to getting these things properly recognized.

Of course have plenty of the demonstrated ability NOW for being the kind of person to figure such a thing out. Actually first came across problem with an entirely different approach using tools I developed, as created my own math discipline. Those same tools also let me do lots of remarkable things. And I found my own way to count prime numbers with a clever tweak on old ideas. Get tired quickly trotting out a list of things, so will stop there. Wow, yeah I have a lot can trot out there for being the person would discover such a thing, I guess.

But also has been lots of luck am sure in it all, though also, fundamental results often DO come from simple, is a good thing. Simple is so cool.

But we humans are clever by much I guess, while also can fall into odd traps, like NOT adding extra factors! While I'm NOT a mathematician, I proudly note, so to me? Why not? So I did.

When will situation resolve? Did you read that part about certain folks NOT being needed?

I'm just the discoverer. I do not make the knowledge. I simply found it, and often wonder.

When humanity feels like it, I guess. Meanwhile, I enjoy the knowledge.

There is such a thrill with discovery. Do appreciate knowledge gained.

Mathematics is endlessly fascinating to me.


James Harris

Monday, December 11, 2017

Working discovery reality

There are those things which can simply challenge points of view, like one of my favorite discoveries as connects to so much across number theory. Apparently it just connects integers to each other across infinities and with years to ponder, still find myself wondering. Is much about an iteration.

Given:

u2 + Dv2 = F

then it must also be true that

(u-Dv)2 + D(u+v)2 = F(D+1)

------------------------------------------

My reference source says I found it back in 2008, but just started calling it a Binary Quadratic Diophantine iterator in 2014, where like to call BQD Iterator for short. And it has a unique pedigree, as results from using my method to generally reduce binary quadratic Diophantine equations on:

x2 - Dy2 = F

You switch signs by shifting the sign of D. Where I found out that is the most important expression for integers in number theory, partly because I could generally reduce to it! Where of course can generally reduce:

c1x2 + c2xy + c3y2 = c4 + c5x + c6y

Where am going over old territory quickly as have brought up all before. So yeah, you have this simple relation at the heart of number theory, and can do SO much focused on one expression, like recently found a new primary way to calculate the modular inverse using:

x2 - Dy2 = F mod N

And key here is noticing how discovery propagates. So I finally used tautological spaces against a general binary quadratic Diophantine equation, and found a better method for reducing! Then that method of reducing at some point I used on a fundamental equation and got my BQD Iterator. And then I began to play.

One of my favorite demonstration results like to use lots now to just show:

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

            615+ 30752 + 141452 + 159902  + 1884972   =  774*210

Found with it.

Number theory is supposed to have numbers.

The BQD Iterator behaves like the core equation of number theory, clearly commanding much. Still I find I ponder.

Such power in a result draws GLOBAL interest and not surprisingly. And as the years go by will admit have thought more about a responsibility. Should I have it? I say, yes. Discovery doesn't make me but I can feel better that way. The math may not care, but I should. You end up feeling a responsibility for the knowledge which doesn't need me beyond discovering it, but feel like can help others in appreciating properly.

If you do not think a planet of humans notices useful working mathematics then I think that you do not believe in math. Web enables. Knowledge moves very well in our times.

Believing in math is an interesting thing. Either you do, or you do not. Actions tell.

There is a sharp divide there that intrigues me. If you believe it IS the math which is important then focus is there. If you believe actually other things are more important, others can tell.

For me contemplating a global attention reality has been daunting for some time. I wonder how I am to know how, as grew up just outside of Tifton, Georgia, USA. Growing up in a small town rural area of South Georgia has given me much, and I ponder how much works for me now.

It amazes me global attention. Like to talk about the math as like a friend as that is comforting for me, and my imagination makes it feel real. The math is the initial why of the attention and the math stays with me, no matter what.

To my mind when you discover? You present. And let others do THEIR jobs.

Today am a person known around the globe who while struggling with the attention has kept my appreciation for what drives it.

Is interesting that belief in mathematics though, even if does not matter to use. People can rely on math tools without understanding them, or face consequences without believing in them, but some also believe in the beauty of truth. And if you are such a person, is a good thing am sure.

If you believe in math, then you know what discovery can do. Not just for what we know now. But for all our species may ever know.

As someone who is thankful for the knowledge, am working harder to focus on being a responsible bearer of the truth. Besides, it also is SO much fun!

Discovery? Works.

I discovered the BQD Iterator, and am thankful for all it can bring, even though will never know it all. Can just be grateful for more opportunity opened up to our world.


James Harris

Saturday, December 09, 2017

When math is correct

Our math has developed with our species and greatly enabled much. Without math humanity could not exist as it does today. And thankfully much of what is useful? Was developed long ago.

Was very hard for me in many ways when began absorbing that the mathematical field had gone astray in the late 1800's as seemed so impossible. And am thankful for over a decade now to consider, which allows me to easily explain the mathematics part at least.

However, I think it is natural to wonder how that could be possible in terms of the human side of a very advanced world we think, where mathematics is so important to just about everything technologically related. Where I found a problem arrived in the late 1800's, when MUCH of the math that is used in science and technology came long before.

Like modern calculus was developed by Sir Isaac Newton who died in 1727, and Gottfried Leibniz who died in 1716 just did a web search. And what they were doing is WAY beyond what most people learn today, if they even get to calculus. Where like if you THINK you are mathematically advanced, can you look at a cable hanging between two electrical poles and calculate the mathematical curve you see there?

Pondering was very humbling for me. And consider, you can hold a rope loosely between your two hands, yet calculating that curve so many know? Requires advanced mathematics, which pushed my limits to understand, as a teenager.

Turns out that curve is a catenary. Figured out back in, yup, the 1600's thanks to the Bernoulli brothers. Where here's a link to article about Jacob Bernoulli. Where came across the problem as a teenager and was humbled when I struggled with learning a solution to the problem with variational calculus, trying to work through to understand. Which only made sense to me many years later. Felt weird to struggle again with math.

We're lucky that so much in math got well developed long ago.

When we look at mathematics from the late 1800's turns out gets specialized in very specific ways while yes, there has been much relevant to technology but that is applied math which has no problems of which am aware. But you find that MUCH of the prestige in modern math field is attributed to pure mathematics and MOST of it goes to number theorists who primarily claim their research is useless in the real world.

Is very weird situation folks, as when you find out they are full of error, you start noticing things. Like who was one of the most acclaimed recent mathematicians who tragically died in a car accident with his wife? John Forbes Nash Jr.

An applied mathematician who gained quite a bit of celebrity including a movie made about him, check his mathematics only awards. Yes, he got the Nobel in Economics but famously there is no Nobel prize for mathematics things directly. (Some think maybe Nobel had something against mathematicians, I think. Where did I get that from? Regardless, he didn't set up any awards for them.)

Used to rip on mathematicians about not giving him ANY awards and then they gave him one.

Related to my criticisms? I'd think not. But yeah applied mathematicians are immune from the problem I found, as it cannot help them. It actually means number theorists can gain success with results which appear to be true, which actually are not. Such a thing could only be useful in exploiting your fellow humans, if you knew of it, with certain kinds of things. Applied mathematicians do not have option of using it, even if they wished.

Studying behavior towards applied mathematicians from others in mathematics is useful.

It does not take much to get a grasp of the actual situation, once you get suspicious.

There is a muddled picture I think for most people as to them is just mathematicians and what is with this pure versus applied and useful versus not or whatever? And there is how people can get away with mathematics which is NOT correct! Which I studied first over a decade ago.

So how do YOU know?

Well I get suspicious as soon as a mathematical result is emphasized as being worthless for anything practical, does not lead to any follow up results, and is incomprehensible supposedly except to a few leading mathematicians in the world. They've switched up the value of human knowledge.

Today many otherwise sensible people take it for granted that knowledge can be abstruse, useless and only understood by a select few, which those who know their human history know in the past was attributed to priesthoods. The Catholic Church at one point I think emphasized to people that reading the Bible could mess up their minds and they should only rely on Catholic priests for guidance. One of the things Martin Luther brought back I think, as going by memory, is the simple notion of reading the Bible again. Which was part of the Protestant revolution where was recently celebrated over a half a millennium since his famous Ninety-five theses.

But are distant from math now, now aren't we? If I got anything wrong above maybe will catch it with editing later. Main point though is the religious aspect to emphasizing trust from the people.

Reality is math developed for its usefulness. And continues to so develop so humanity is doing just fine. Esoteric reality of number theory means most people don't know and don't care about it, even though there has been a practical aspect with computer security.

Which is a very different subject and I tend to shy away from THAT subject lately.

So yeah, turns out is easy to explain how there can be this bizarre error that entered into mathematics in the late 1800's which did not derail science and technology, but did shift mathematical community, to focus on people who behave like a modern priesthood. With true believers trained to simply accept their proclamations, they can gain adulation, reward and continuing employment, when people who understand how things went wrong in late 1800's can reasonably suspect they have few if any correct mathematical results. Is so wild. But they are simply not needed for valid mathematics.

Good news is mathematical industry is doing just fine, regardless. Our species continues to progress and no reason to expect that will change from this particular problem I found.

When math is correct? It works for us. And vast amounts of useful mathematics is doing so much for our world, no worries.


James Harris

Thursday, December 07, 2017

Explanation and unary two conics equation

Much to my surprise came across a simple explanation for size of integer solutions to:

x2 - Dy2 = 1

I like to call it the unary two conics equation, as depending on sign of D, you can have ellipses or hyperbolas. And traditionally people in mathematics like to attribute it to a guy they also say should not get credit, which is SO wacky to me. Why not just correct the error? I like name I give it better. And traditionally its integer solutions are of interest as solving it rationally was known to Fermat, who liked to play intellectual games looking for integer solutions.

And integer solutions can seem kind of random, if you don't know the why, like consider:

98012 - 29(1820)2 = 1 with D = 29

But next one:

112 - 30(2)2 = 1 with D  = 30

Giving smallest positive non-trivial integer solutions.

And turns out that key to size is D-1, and not D unless is prime. Which is the kind of mathematical wrinkle that can confuse humans I guess? Which is why wasn't just noticed centuries ago? But even if they noticed, how would they explain? I can explain why.

Also people have to be curious. I've tried to research this subject and apparently number theorists gave up on finding an answer long ago so claimed there was NOT one. So no way they'd ever find one, eh? So primitive to me, how some people choose to think a certain thing, in order to not learn, satisfied with a belief. So was declared no answer existed. Interesting.

Where lots of the easier solutions are going to be when D-1 is prime. And hard starts if has small prime factors, and harder the more of them you have. Like if D-1 has 4 as a factor and is not square? You start getting the bigger ones like example I showed, where also is key that D is prime.

There are also other things make for easiest answers, like trivially found solutions when D-1, D+1, D-2, or D+2 is a square, as well as when D is divisible by 4, for when D+4 or D-4 is a square, where kind of didn't even wish to give all that but figured why not for completeness. Oh yeah, so smaller integer solutions for those more boring cases as well.

The general result around factors of D-1 is vastly more interesting to me.

With explanation in hand just jumps out at you if you scan through tables of integer solutions. For me? Was so wacky wild years ago. But of course for me now is old news. And also, I discovered so my emotional relationship with the information is different.

The result then is that D-1 is the control in general for size of solutions, and its prime factors as to how small and how many are what matters. Turns out the math has to work harder for integer solutions if D-1 has lots of small prime factors, especially if D is prime. And hardest if D is prime, D-1 has lots of small prime factors and 4 is a factor, except for the exceptions with various squares, like if D-1 is square. Which then was the previously unknown explanation. Easy answer which is backed by mathematical proof!

Is slightly convoluted though when all the details are laid out. So yeah, D-1 is most important, but D impacts also if prime! And yet there are all these exceptions when you get easy, like D plus or minus 1 or 2, and yeah, I guess it could be hard to tell by looking over solutions trying to guess. And results could look random to a human without all the rules?

Where copied from this post used as reference. Have another post soon thereafter talking two conics equation size which is way detailed.

Is very interesting to me these relationships between integers which control things. There is that mathematical precision, of course, to the machinery of integers in relation to each other.


James Harris