Wednesday, March 22, 2017

Valuing discovery and quadratic residue pairs

Years ago discovered there were these powerful control equations that force things like distribution of quadratic residues. Learning that happened at some point where was long enough ago not sure when, but for sure one of the important revelations was found fortuitously from staring at:

(n - m)2 = m2 + 1 mod D-1

which follows from:

(n-m)2 - Dm2 = 1

Should note don't use ≡ because I use modular algebra SO much and why add some extra line under equals all over the place? Seems like useless work to me. Oh yeah, to try and help those who might need the info wrote a post explaining mod years ago. 

So was pondering a slight variation from the usual x2 - Dy2 = 1. And had been playing around with the different form and became fascinated by the forced quadratic residue there.

Very glad I noticed it!

Which lead me to finding out about quadratic residue pairs. Which I'd never heard of, until started searching to see if I'd found something new. So there, yes existence had been noticed before. Ok. Is cool to learn more historical number theory to research your own discovery. But then also I figured out how to prove some things, which was new.

You can kind of see things from usual form as well:

x2 = y2 + 1 mod D-1

And you can easily solve for unknowns modulo D-1 with either form.

Using my alternate: 2m = n-1(n2 - 1) mod D-1

Which you need for a proof, which is how to derive the count of quadratic residue pairs which for a long time was one of the most popular posts on this blog. And yeah can use it that way as IS one of the most powerful influences across integers, and yeah can use it to count. It is one of the infinity tools of number theory, which helps rule over quadratic residues.

Prior explanations for the count I found, using web search, relied on the pigeonhole principle. They had no clue.

For people into number theory, you can just go check! Web search. And is how I found much so I know information in this area is readily available. That reference reality is a major plus of our times.

So far as I know I gave the first direct derivation which not only lets you know why, it allows you to do more. Will admit am more into the 'why' as exciting, so just talked more you can do in my post as can expand to counting in more situations! But didn't do myself. But is a good example to see how a derivation can give more.

Especially is that odd feeling when you notice also an easy path that somehow escaped others. Why not figured out a century ago? Who knows. But probably because Gauss pioneered modular in his time, and I picked up where he left off, in ours.

Is odd too. Looks like in past, mathematicians knew more of modular arithmetic, but didn't realize a full modular algebra, which I did. And for me to get there had to create an entire mathematical discipline and pioneer abstract reductionism so maybe is harder than I wish to accept!

For me? Is just what I did. Guess that doesn't give me the perspective of looking at from a distance.

The joy of discovery I guess can be hard to describe but it definitely can motivate to find more, which I think is great. There is that thrill, for what YOU find, with your mind, and mental effort. Is a thrill...hard to describe.

Here again my modular approach leads to a short, powerful answer and in this case, a first derivation.

And it dawns on me that modular as a first reach in problem solving may be something very new to the human species, but why?

James Harris

Tuesday, March 21, 2017

Supporting faith in the system

Role I want to play can seem mysterious or even a bit complicated until you consider what I see as strange things. And to me the strangest things have driven others that seem so dramatic, like why I put forward a definition of mathematical proof, which I noted at the time back in 2005, was motivated partly by behavior from mathematicians.

And it's in the post what I found troubling, where had routinely seen in the news mathematical arguments presented as possibly proofs, which were called proofs, and mathematicians talking as if mathematical proofs were delicate. Huh?

Such a bizarre contention was about trying to explain to press, when things called proofs were later shown to have error, and I say, should call mathematical arguments being checked by the system to see if are proofs instead of calling proofs to the public before that is even done.

That behavior upset me. And surprised me too! Why talk as if mathematical proof was fragile or delicate? Why? Why? Why?

Do try to feel empathy for academic mathematicians desperate to make a career or just keep one going, but there have to be limits on selfish behavior, and constant focus on what is good for the system as a whole.

Political behavior from mathematicians or even worse, chasing fame with a claim is not an excuse for such disruption.

It irritated me. So I wrote a definition of mathematical proof, which has a functional basis.

Some of these things are SO wild, but true. It's like thinking about it later though, seems extraordinary. The explanations are worth nothing I guess. Usually I DO try to explain.

And, also note that gave me the benefit of checking my own mathematical results with it. Which I also needed as why wouldn't I?

With my one published paper, also note I've championed the system, as it passed anonymous peer review, and actually faced two reviewers.

Questioning their own system by mathematicians is probably why things stand as they are. If they just followed the rules would be so much easier for them. And I'm NOT a mathematician so I can critique that while wryly noting my detachment from that system is a plus.

Will also note my disdain for what have seen as quests for celebrity from certain mathematicians, and a disappointing amount of hero worship, which muddles the quest for truth from the established community. It's like math people want to believe based on SOURCE instead of argument. Naive hero worship worries me.

Yes, I have my own ways of making fun of such things, as well as highlighting them. I'm functionally focused on what works, as I test, and see over time. And is a process where I try to lighten things up when I can, to the extent I can. Helps me feel better at least.

I see myself currently as the biggest defender of the system, which is a responsibility I take very seriously. And that becomes more clear in time.

If you thought something else, why?

There is a LOT of hard evidence that I'm working hard to maintain certain things. Over and over again have gone to a lot of effort to support mathematics globally. Including taking the time to be careful as to how that is done.

There is no rush for me. Mathematics is important enough.

Feel like mostly done though, which is great! Heavy lifting? Over. Helping with mathematics is already mostly fond memories. Don't spend much time on these things now anyway. Just lately have been cleaning up a bit. Have other things to do too. Maybe have to do a little something else here or there if field of mathematics gets endangered again, but plan is soon NOTHING else, to fix. Can still talk things though. Is fun.

Often have been forced to face things objectively, accept facts, and realize I'm lucky to have a role to play, and not think too hard about how big it might be or how strange it is. As then can feel all kinds of weird, which is not fun.

Oh yeah, thankfully I haven't seen mathematicians trotting out mathematical arguments still under review as mathematical proofs that are delicate recently. If still happening, luckily not that I've noticed.

I find that behavior undermines public confidence in mathematics. And be certain I consider that to be unacceptable.

Mathematical proofs are NOT delicate. Never think otherwise.

Mathematical proofs are absolute in a sense few things human beings can ever find can ever be.

That will never change.

James Harris

Friday, March 17, 2017

Critiquing math people reality

Some tasks can be really difficult but really worth doing, even if you're not sure how it will all work out. Have learned to appreciate perspective and just working at things until done.

So yeah back in 2003 I wrote a paper which passed through anonymous peer review and was published, where luckily paper is still available hosted by EMIS as part of its archive. And that paper looks correct. No mistakes under established mathematical rules. But its conclusion contradicts with very well established number theory! So you have to use a separate argument to know the paper must have something wrong.

How is that possible?

Forget that, how can that NOT freak out mathematicians?

Since then I've noted, was demonstrating an esoteric and intriguing flaw with use of the ring of algebraic integers where that flaw is EASY to explain, but let's get back to that paper which has no errors under existing rules which can be shown to contradict with accepted number theory--how can that not just flabbergast mathematicians?

Some may know that there WERE hostiles to my paper, which is SUCH an ironic thing to me, as they attacked it for doing what it was supposed to do. As they pointed out that by a separate argument it had an incorrect conclusion. The paper itself appears flawless--unless you know conclusion must be wrong.

But um how is that possible? Can I ask that enough?

The very ability of a paper to do such a thing challenges so much. And such a paper actually does stand alone. It should not be possible.

The problem gives the potential of writing something that looks like it is correct, and checked against the established rules passes, which is nonetheless WRONG.

To me more than a decade since that paper can find a way to get a bit of dark humor in the situation, which is very serious.

Yes I critiqued the mathematical community--globally.

If YOU could, would you?

Did you pass? Or fail?

Why would I test mathematicians all over the planet in this way? Why wouldn't I?

It was a unique opportunity. Such an opportunity may never exist on this scale again in human history. Test mathematicians all over planet Earth with something THIS simple? Too cool.

Yes, have struggled with the weight of it though, at times. Mathematics is so important for our world. Wondered the fate of our world if I failed. Wondered if I could fail--if reality would let me. Such concerns seem distant now.

The test is perfect. Proper reaction is to understand that until fixed doubt can exist throughout the field of mathematics, though the problem does not exist for arguments that rely on fields. It only exists for number theory relying on the ring of algebraic integers.

And it IS checkable by any mathematician as well as uses elementary methods. So a person doesn't have to be a number theorist, and other mathematicians have no excuse. And yeah I'm NOT a mathematician I note again. Is important, you know? Helps my detachment.

Notice that the test remains until fixed as any math student can simply check the paper, and read through noting--no error with argument under existing rules. And then simply note that conclusion is false under existing number theory.

Until things are fixed of course and is simply a matter of historical record and example is part of the curriculum for future math students.

I have given the fix of course, which lead me to putting forward the object ring.

So yeah the conclusion is wrong in the ring of algebraic integers but IS correct in the object ring.

Oh, so how easy to explain? Imagine someone says they are only using evens and considering factors, so yeah 2, 4 and 8 are ok, and factor easily enough. But they trot out 6, and you say, NO! Because now with those rules 6 and 2 do not share factors because 3 is odd. And yeah evens are NOT a ring. And the ring of integers does NOT have this problem.

But the ring of algebraic integers DOES have an equivalent problem which I've shown multiple ways and even given a full explanation lately. as apparently I just LOVE explaining it. So the problem with the paper is the ring declaration with which it begins.

So why has the global mathematical community failed my simple test on the whole for so long?

I do have theories.

But not everyone really failed. My paper passed two anonymous reviewers and was published. Was chief editor who tried to pull later. And EMIS has kept it up. Am sure there are plenty of people out there who...I don't know. Maybe just figure there should be.

And it could still take awhile. What's good now is EXPLANATION. For those who have wondered you can now finally see the complete picture that took about 13 years to fully get outlined by me. So much work too! But learned so much has been worth it.

Now you can read not only the point of that paper that got published but see a full mathematical system explaining it all out in extreme detail. I think explanation is awesome. At least now people can know why.

Reading through seems so dramatic though. Yup. And we are in the 21st century with new ways.

To me am in a functional process where I see what works, and also it helps me to handle it all.

Figuring it all out is a challenge so worth the effort.

After all, our world's knowledge is what is important. And mathematics is important enough.

James Harris

Monday, March 13, 2017

Some abstract reductionism demonstrated

With Diophantine equations in two variables you can see a LOT of apparent complexity with:

c1x2 + c2xy + c3y2 = c4 + c5x + c6y

Here x and y are the unknowns to be figured out, of course. But it turns out you can reduce which was known, but my abstract reduction methods make that easier in general to get to the basic form:

u2 + Dv2 = C

And with more generality than with other methods derived by Gauss.

But on that form those same methods--iterate:

u2 + Dv2 = C

then it must also be true that

(u-Dv)2 + D(u+v)2 = C(D+1)


And I decided to call that a binary quadratic Diophantine iterator, or BQD Iterator for short, and I find it fascinating. It stands alone in that you CAN verify it simply by multiplying out. And conceivably could have been discovered just by someone playing around but apparently was not.

Where yeah there are implications then of course for integer factorization too, which I usually don't like to mention, as EVERY integer factorization can be connected to an infinity of others, trivially.

So now we know that the very complex looking expression, with integers, connects to very simple, and more formalized abstract reductionism makes that so much easier--was able to improve upon Gauss.

And how many people can say THAT?

I've played a lot with it, and you can see plenty of posts by clicking on the label below.

That is just my favorite demonstration example of SOME abstract reductionism. Have pondered it for years. Why can find something one way which had NEVER been found by another? Maybe does help a lot when is EASIER from a formalized process. 

The point important to make then, being abstracted by a process, which I have used in more than one area. Just really easy to see quickly with certain things.

James Harris

Monday, March 06, 2017

Abstract reductionism

Realized have developed expertise in certain mathematical areas because I invented a certain approach, though built on what came before. If you invent an approach then naturally you have expertise with it. And a consistent theme in much of my research actually is reducing to the most important elements which has been abstracted and standardized with modular algebra.

That I rely on modular algebra is significant as often I see discussion is on modular arithmetic.

And modular algebra offers stunning amounts of reduction as can be seen with my method for generally reducing binary quadratic Diophantine equations.

So feel approaches in that area can be quite simply called abstract reductionism.

Formalizing reductionism fascinates me will admit. And have done some additional work in that area to generalize what I call tautological spaces. The full system I call modular algebra symbology.

Remarkably tautological spaces can be used to get to answers where they are not needed to be mentioned, so results can stand alone.

My work in this area I believe flows naturally from foundations laid by Carl Gauss. In essence I just took modular in mathematics one step forward, and created a fully formed modular algebra.

Modular is coming into its own as a key concept in the 21st century. Feel privileged to have my own place in that development.

James Harris

Friday, March 03, 2017

Why went looking and found my prime counting function

Now I feel better as can fully explain a result which really floored me years ago, and turns out one of the first things that I'd harass myself with after I thought had something was: will that be it?

Which really irritated me, and I wonder why I do these things. And yeah was also at the time still into that Fermat's Last Theorem thing but knew that the big thing was staring at me, I think. Is fuzzy as was over a decade ago, but to not have ONE THING I started playing around with counting primes.

And few weeks or a couple as not so sure, later had figured out my prime counting function on which have discussed much on this blog, and that was August 2002.

And then was like, oh so you're going to just have two things?

It did relax me a BIT though as then I'd say to myself, well I have a backup.

But reality was already had the sphere packing thing--I think. I can't find anything wrong with it, but still even now I don't claim that one SOLID on emotion. And I don't really need it, and it bothers me. Is too easy. How do I figure out something that Sir Isaac Newton didn't catch when he worked on same problem?

And I don't need it. It'd be cool to have though, I guess. But that can work itself out over time.

Maybe should admit that while I don't need for bragging rights, the approach I pioneered should have practical usage, and I suspect if so is probably being used.

THANKFULLY, I finally stopped taunting myself with that, is that all you got?

Which gives me more time to explain things.

And laughing to myself, reading back through and I think is the weirdest thing, but is true! How do you go from thinking you have some great discovery to promptly taunting yourself to do more?

Seems mean to me. But now is funny. So no, that wasn't all I had to discover. Now I'm like, I'm good. And inner voice is finding other things to mess with me over. Of course. Like that wouldn't be true.

James Harris