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Saturday, August 12, 2017

Some number examples

For some reason really like when have examples with numbers in posts and some are actually my favorites and thought to collect a few linking back to post, in a post.

Like here is one:

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

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

Just like to stare at it for some reason.

Seem to like sums of lots of squares. Continuing, like here is two more sums of 5 squares to a square:

42 + 6+ 10+ 14862 = 882

and

862 + 129+ 215+ 301+ 8812 = 9682

And here went ahead and summed 7 to get a square:

349672 + 7522 + 1128+ 1880+ 26322 + 41362 + 48882 = 357212

Where am using BQD Iterator for all of these. But is mathematical tool I have which makes such things easy. Those are ones just from my own research.

An example from even earlier though, where rely on previous known result is here where was talking size of what I now call the unary form of the two conics equation:

60*2551100302 + 2551100292 = 19924730292

That is related to something, talk it here and for reference: 297182 - 61*38052 = -1

Liking that the easier solution, which is historically known, fits nicely there.

Putting in one place is useful to me for staring at them purposes.


James Harris

Friday, August 11, 2017

Some math discoveries listed

Sometime this month, as don't remember exact day, will be 15 years since found my prime counting function. And thought along with that special event would just list some of my mathematical discoveries. Will not link to anything as everything is on blog somewhere.

So first, yeah found my own prime counting function which fastest for its size in its compact sieve form, but more importantly leads to a difference equation when fully mathematized, which has to be constrained to get it to count primes. But the difference equations leads to a partial differential equation. That is so cool. And celebrate 15 years this month since first found.

Moving on.

Developed my own mathematical discipline using what I call tautological spaces, as rely on complex identities. Entire field call modular symbology, and it realizes the first true modular algebra. And is my best example of my use of abstract reductionism. Oh, before called modular algebra symbology, but like it shorter! So will switch to that now. There is so much related it dominates this blog.

Oh yeah, so in that area found my own way to solve for the modular inverse! That is finally for humanity a third way, with other two there is Euclid's name on one and Euler's name on the other. Is now barely over three months since found that so still absorbing the thrill.

And found an axiom related to primes, decided to call prime residue axiom.

Biggest thing though, is found my own numbers. That one takes SO much explaining will leave it like that, but big part was finding my own ring which call the object ring.

And that I think covers enough of the highlights to satisfy my mood.


James Harris

Saturday, August 05, 2017

Web rules and my Diophantine reducer

One of my more informative results can help elucidate how the web has changed things with sharing even highly refined information, like consider if wanted to reduce:

x2 + 2xy + 3y2 = 4 + 5x + 6y

Can use my method for reducing to get: [-4(x+y) + 10]2 + 2s2 = 166

Which is an example have used since 2011, where s = 9, and x+y = 2, is one solution, from which you can find: x = 4, y = -2 as one solution. And another gives: x = 5, y = - 2

Kind of cool, huh? Same y works for two different values for x. And really glad the equation had integer solutions! And easy ones too. Is obvious why I picked it am sure. And s has an explicit solution as a function of x and y, but I don't use it, as just helps by giving two linear equations to solve for x and y, but DO talk about it in this post.

And copied example from this post showing my way to reduce what are called binary quadratic Diophantine equations and have also seen called two variable quadratic Diophantine equations.

For me reducing is for show.

But there are people who need to reduce these types of equations for am sure lots of reasons, so linking to my method is about usefulness. However, why link to something worse than other techniques? To do so would be illogical, and against common sense.

In fact have noted my method improves upon methods for reducing my research shows are from Gauss. But with his techniques you also need to check something called a discriminant. With my approach that is worthless effort, and I make no mention of such a thing.

Turns out you don't need it.

But what do you think is in some math textbook, eh? And am a HUGE fan of Gauss, but if he were alive today doubt he'd be surprised that his authority is not taken lightly. Who knows when academics will update.

Innovation tends to lead in dramatic ways which is more fun and rapid. People who NEED will just go to the best thing available, when know it is. Getting established? Is more of a process which to me is tedious and depends on others, whose motivations can vary.

Web makes all that irrelevant from MY perspective.

Web can just connect information DIRECTLY to the people who need it.

With such older research of mine the web is very efficient in linking to it. And I can check search engines based on my own results to see if search results will get to them, and do it routinely.

Usually Google wins, and will check against Bing more than others. However, at times with more recent research have seen Bing win. And I think Google is making more effort to rely on established authority, rather than just on web authority, which is a hypothesis to explain that result.

Over time though, best results will win, and notice that is true regardless of the web.

And academics lagging best methods is not new I don't think. The web though can simply link to best, though I think often is being done on various views of authority! Which I think is interesting.

Picked one of my most dramatic examples but can help explain other areas too. Right now have several results could have used, but I like this one as is connected to some serious practical things.

World doesn't sit and wait on academics. And never has. But web has made things easier.

So yeah, one thing I do routinely is check search engines to see better how they operate based on what happens with my own research results, when can check those objectively against what is known, where usually can.

Figuring out web rules is of interest to me.


James Harris

Wednesday, August 02, 2017

Prime difference and marking fifteen years

Things have worked out really well for me from a discovery perspective. And it is important should note this month will mark 15 years since found my prime counting function, which in its most mathematized form counts using a difference equation. I think at times have used the word mathematicized. Did I make that up? Guess so, as spell check doesn't like it.

And don't know when this month though so may as well make a post now. Just remember was August 2002, and guess could try to check? But is some post on Usenet and do NOT wish to dig through there.

On here as checking published posts now, see this post as first one, which was posted June 2005. And first post on blog is March 2005, so took a few months to get to it. Of course this was second thing I had for my math? Before had something before Blogger and was paying for a website so was very happy when could switch to something free. That is, talking webpages. For a LONG time was just arguing on Usenet and eventually through Google Groups at some point. I say a long time but think was from 1996?

There are SO many posts on this blog talking my prime counting function where think have labeled most, so if curious can go through.

That prime counting function was first result which really was just deriving something kind of because felt like it. Specifically asked myself to figure something out and few weeks later had it, really just from scratch.

My prime counting function is also my most stand-alone result. It doesn't derive from any other research I'd done, and it is very compartmentalized though can get sort of broad. So there is a sieve form, and a fully mathematical form, where yeah guess that is supposed to be called fully mathematized. And there is the difference equation itself which has to be constrained for it to give the count of prime numbers. Then there is the partial differential that follows from the difference equation. And that is all really just one package all to itself.

Was also first result where there was just no doubt as wasn't using techniques I'd invented or wondering how something so simple could be missed. Oh, was a bit before I had the partial differential equation that follows, but had it by the time created this blog under its old name. And that is this post, which was also June 2005. Updated that post at some point to use delta symbol.

So cool. Has been SO amazing on the discovery side. Changed my life for sure.


James Harris

Tuesday, August 01, 2017

Sum two squares to power of 10

Number authority is from the knowledge embedded in numbers which is infinite and absolute. Like consider:

12 + 32 = 10

82 + 62 = 102

262 + 182 = 103

282 + 962 = 104

3162 + 362 = 105

So yeah, every power of 10 out to infinity is a sum of two squares, which doesn't seem useful to me, just curious. What can it tell me? Such things I enjoy pondering though yeah, also just like looking at numbers as you may have noticed.

Am just using a specific of a more general result as 10 is such a big deal to many, including me.

The numbers KNOW and we can find things out, if we want. So much depends though, on asking questions. The numbers know infinite information. We can find some things out.

If it doesn't interest you, ok.

For me such things DO interest, and this result follows from 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. Copied much from an earlier post where you can go there to read more.

And there are LOTS of posts on my BQD Iterator, and you can find a list of those I've labeled by clicking on label beneath this post.

Monday, July 24, 2017

When truth is your hammer

Readily admit I will turn to absolute proof and even number authority itself, when need reassurance. And when you wield truth in a certain way it can be necessary to hammer through against people who rely more on feeling than fact, as we humans can be recalcitrant at times. And there are people who think truth is a moving target or most amazing to me, some think truth is about human opinion.

So yeah I posted this thing recently:


349672 + 7522 + 1128+ 1880+ 26322 + 41362 + 48882 = 357212


_____________________________

An absolute truth it doesn't care what you think of it, or me.

And I'd summed five squares before and the math is easy, but it just felt good. And I like to stare at it when get philosophical or, yeah need that reassurance.

Human beings will come and go.

Here in this time, when some may think that they have a will that can overcome mathematical proof.

Let them try.

When you are someone like me who wields truth as a hammer when necessary, or has the infinity results, then you can look at others with curiosity, to see what they believe.

Try to break the hammer, see what happens.

I'm curious. I, at least, am human. But in the end?

The math is not human. The math knows you. But you do not control the math.

The math does not care what you believe.

Possibly some of you as well will feel that urge to wield truth.

Do you have what it takes? Few can handle the truth at certain levels, who can find the most powerful infinity results and present to their world.

I could. I have.

This post was SO much fun. Yeah am a fan of comic books. But I read them a little differently than others am sure, now. And the movies? Are so much more fun, for me.

Takes someone like me.

Can you stand with truth as your best protector?

Or would you bend in fear?

I know.

For me maybe is more fun this way anyway, as suits my flair for the dramatic.

Coming up on 15 years since I discovered my prime counting function as just one example. Can you imagine? Could you simply stand for truth?

Challenge demands a certain person. Reality? Knows.

The math knows. The math chose. Reality bends not for you.

The future demands the one who will get it done.

There are more results out there--an infinity of them. Reality will choose those who will get it done.

Can you stand for truth? Come what may?

Readily admit as much as I LOVE discovery of my own, more and more wonder, where can these ideas lead that others might find?

Truth is out there.


James Harris

Thursday, July 20, 2017

Progression, abstraction and two conics equation

Worth noting the progression to my latest result, which has a lot to do with reducing the general equation for what I like to call a binary quadratic Diophantine equation which is also called a two variable one:

c1x2 + c2xy + c3y2 = c4 + c5x + c6y

Here x and y are the two unknowns to be figured out. The base result comes from this post from September 2008. But checking published posts, was referenced talking a general method for reducing binary quadratic Diophantine equations on this blog in this post in May 2011.

And copying from this post, where also note that with my method for reducing can get to the general reduced form:

u2 - Dv2 = C

Where u and v are unknowns. And while I've talked about with C=1 as the two conics equation before, the more general also gives two so that is just the unary case. Letters don't matter of course and like to show as:

x2 - Dy2 = F

where all variables are non-zero integers. And yeah a LOT of abstraction, in that progression, where now you can solve for x and y modularly. Looks like I figured that out in September of 2012.

With a non-zero integer N for which a residue m exists where--m2 = D mod N, and r, any residue modulo N for which Fr-1 mod N exists then a solution is:

2x = r + Fr-1 mod N and 2my = Fr-1 - r mod N

And use that to solve for the modular inverse.

 r-1 = (n-1)(r + 2my0) - 2md mod N


Where y0 is chosen as is m, with m not equal to r, and n and d are to be determined. They are found from:

2mdF0 = [F0(n-1) - 1](r + 2my0) mod N

and

F0 = r(r+2my0) mod N

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

And felt an urge to put all together to kind of see that progression from the most general form, to abstraction to the more basic two conics form, and then to a solution for the modular inverse.

So you end up going from something to do with binary quadratic Diophantine equations to something more general than them.


James Harris

Wednesday, July 19, 2017

Correct matters most

When found mathematical results have an absolute aspect you can find difficult to find elsewhere, like maybe only in logic. When you have the correct mathematical result, it is absolutely true. And have given an example of absolute proof.

However emotion can lead us astray as human beings and have been lead astray in the past by my emotions and did not like it when found out! Where could be SO confident and certain, when thought had a correct mathematical argument only to finally have that wrong belief dramatically collapse when finally could see my error.

That elicits a terrible feeling and I do not like it. I try not to repeat such failure.

The joy in believing you have something important is not worth it, if it is not even correct, as such a thing is completely empty.

Correct matters most, as only when absolutely correct do you have the mathematical proof, as proof is perfect.

To check against emotion I now employ a process which I think helps protects others as well, as while not good to lead one's self astray, so much worse to lead others! Which means I try to focus objectively, consider results as facts only when well established, and refrain from emotional appeals.

So please do not be surprised at a steady process which does not involve trying to convince you, but is sharing of mathematical ideas and process as well, so that truth can be determined.

For those who appreciate truth, working for the truth should be a privilege.

Am lucky in that most of what I have requires only what are generally called elementary methods. I like that phrase. Elementary methods.

Numbers have fascinated me in special ways for as long as I can remember. Like friends with personalities who are anxious to tell you cool things. And they never lie. But can lie to myself if I'm not careful so yeah, focus on--correct matters most.

And explanation helps.

Labels below this post consider various areas around the social aspect of presenting mathematics from celebrity, to what I call the social problem, and also instructional. Click on a label for more posts in that area! And thank you for your interest.

I try not to try to convince you, but I do appreciate your time and attention.


James Harris

Monday, June 12, 2017

Number authority

Find that I DO turn to numbers routinely. And the authority you feel when the numbers behave as mathematics requires is like no other to me.

The math does not care. But the math is never wrong.

And there is a comfort in that which I think turns into a sense of protection. So the math may not care but you can find shelter in truth.

So yes, will turn to numbers in comfort and love the conversation with the math even knowing the math does not care. But the math can talk to you. And you can talk to the math, and ask questions!

And if you ask the right questions, the math can give you the truth.

Without a doubt to me is one of the greatest phrases. And in mathematics truth can be found--without a doubt.


James Harris

Summing seven squares to a square

Discovered a simple technique to build sums of as many squares as you want to a square. For example, here is a sum of seven squares to get a square:

349672 + 7522 + 1128+ 1880+ 26322 + 41362 + 48882 = 357212

_____________________________

My basic result used to get an arbitrary length of sums of squares is that in general there must always exist nonzero x and y, such that for an integer n equal to 1 or higher, and an integer m equal to 3 or higher:

x2 + (m-1)y2 = mn

Where n starts at 1. Often I like to start it at zero so it's a count of iterations but is prettier starting at 1. And you get values for x and y using what I like to call a BQD Iterator, which is short for binary quadratic Diophantine iterator. I've talked about it a lot.

For a sum of c+1 squares: m = s12 +...+sc2 + 1

x2 + (s12 +...+sc2)y2 = mn

The BQD Iterator is:

Given nonzero integers u and v with

u2 + (s12 +...+ sc2)v2 = F

then it must also be true that

(u - (s12 +..+sc2)v)2 + (s12 +...+sc2)(u + v)2 = (s12 +...+ sc2 + 1)*F

So for 7 squares, I'll need 6 s's and I'll use primes: 2, 3, 5, 7, 11 and 13

Then m = 4 + 9 + 25 + 49 + 121 + 169 + 1 = 378

12 + 377*12 = 378

First iteration: (-376)2 + 377*(2)2 = 3782

Second iteration: (-1130)2 + 377*(-374)2 = 3783

Third iteration: (139868)2 + 377*(-1504)2 = 3784

Which is: (139868)2 + 4*(-1504)2 + 9*(-1504)+ 25*(-1504)+ 49*(-1504)2 + 121*(-1504)2 + 169*(-1504)2 = 3784

Can divide both sides by 16, and get rid of negatives to get:

Which is: 349672 + 7522 + 9*(376)+ 25*(376)+ 49*(376)2 + 121*(376)2 + 169*(376)2 = 1894

And now get final result where will show as all squares:

349672 + 7522 + 1128+ 1880+ 26322 + 41362 + 48882 = 357212

Which is interesting to me, I think. Looks more impressive that way. Of course watch it get built maybe less impressive? But still is, a sum of seven squares to get a square.

Don't really see a practical use, so to me? Is just pure math.

Am curious, if you know a number theorist, why not ask that person to produce an example of sums of squares to a square? It's not like it's actually hard to do, if you know how.

Of course, highlighting a cool result with mathematical tools I pioneered. Do I really know or care if number theorists can match me here? Not really.

For the discoverer? It's all good.


James Harris

Friday, June 09, 2017

Talking my modular inverse discovery

The modular inverse is a rather simple thing from modular arithmetic, like consider:

2(3) ≡ 1 mod 5

Here 2 is the modular inverse of 3 and vice versa because they multiply to have a residue of 1 modulo 5, which is the modulus. That can be written as:

≡ 3-1 mod 5

Importantly in my research as I use SO much modular long ago tired of copying and pasting the modular congruence symbol so just use equals, so I have:

2(3) = 1 mod 5 and then: 2 = 3-1 mod 5

May seem small but in my experience you can face purists who will be dismissive on such small matters! When am someone who is NOT going to waste time copying and pasting something all over a vast amount of research just to appease such people as is human convention. The math does not care.

The modular inverse as a concept has been around for some time, but only a few basic approaches for finding it were previously known. One relies on something called the extended Euclidean algorithm, which is very simple but will just link. The other approach depends on something called Euler's theorem, where learned the above from article on Wikipedia on what they call the modular multiplicative inverse.

And now can add another basic approach which relies on a system of equations I discovered about a month ago. For some residue r modulo N, its modular inverse is:

 r-1 = (n-1)(r + 2my0) - 2md mod N


Where y0 is chosen as is m, with m not equal to r, and n and d are to be determined. They are found from:

2mdF0 = [F0(n-1) - 1](r + 2my0) mod N

and

F0 = r(r+2my0) mod N

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

Copied from my post: Modular Inverse Innovation

It is also derived there. And my system, is kind of more direct like that which follows from Euler's theorem, but is also iterative like what follows from extended Euclidean algorithm. But allows you to fiddle with things in a way that neither does. So you can pick two key variables: m and y0

And they are so named because of how I discovered the system, as was just kind of puzzling over some things.

It is one of my most direct discoveries which surprised me a bit, as it just flowed. And I'm beginning to accept that I have years of experience which has given me a certain level of expertise.

Research about the modular inverse on the web I've done since, when yeah got REALLY interested, has indicated is also a practical result as calculations of the modular inverse are part of modern techniques, according to that research. So it has applied and pure math aspects.

The result is definitive though in terms of evaluating social aspects of how discovery is actually treated versus how one might imagine. I've had lots of experience in this area with prior results.

And the social problem label below covers that topic.

Good news though, looks like first basic major result at this level in over a century. Probably MUCH longer. Notice prior methods go back to Euler and Euclid. Wow. Supposedly such a possibility of such a basic find no longer existed. Am thrilled to be the discoverer.

So yeah, humanity has a new mathematical tool.

Mathematics discovered belongs to the human race.


James Harris

Sunday, June 04, 2017

Iterative example with my modular inverse method

My way to calculate the modular inverse at least gives a smaller modulus with another inverse to calculate and realized might help to show an iterative example.

Will start with the system which I will copy from my reference post, and then will explain lots, where am not doing so much shown calculation as is tedious. So very wordy post I warn. Less with shown work. Interested readers are invited to work through themselves as I think greatly helps understanding to fill in what I'm leaving out.

Also lets me explain some things a bit differently. Here is the system:

 r-1 = (n-1)(r + 2my0) - 2md mod N


Where y0 is chosen as is m, with m not equal to r, and n and d are to be determined. They are found from:

2mdF0 = [F0(n-1) - 1](r + 2my0) mod N

and

F0 = r(r+2my0) mod N

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

Copied from my post: Modular Inverse Innovation

It is also derived there.

Importantly you have variables that follow from: x2 - m2y2 = F

And you're solving for n and d, with other variables picked. Where I've tended to pick m such that m2 is greater than N, though is NOT clear if that is necessary. Is a new area I just found.

I do know that m should not equal r though. So let's try it all with N = 851 = 23(37).

I tend to let y0 = 1, where that is just for easy. Later research may indicate other more useful values. And will let m = 31, as that squared is greater than N and I like primes. Not necessary for m to be prime though.

Oh, and let's let r = 97. I like using primes because things won't work if r shares factors with N, of course as then the modular inverse does not exist! So helps me just using primes often.

Now can calculate F0, which is: 97(97+2(31)) mod 851.

So F0 = 105 mod 851.

Plugging everything in and reducing things and simplifying, I get:

298d = 325n - 166 mod 851

So now to solve for n can just find: 325n - 166 = 0 mod 298

That is: 27n = 166 mod 298, so now I need the modular inverse...oh wait. I notice that 298 - 166 has 3 as a factor, so can get:

9n = -44 mod 298

Oh, then I got clever. If we let n = 2z, then: 9z = -22 mod 149, as can divide 2 across everything.

So now finally, can look for modular inverse of 9, modulo 149. It works. Not in the mood to put all that here though. But just use the system already given and this time it just gives the answer without further iteration needed. Turns out 9-1 = 116 mod 149. Where just get the answer that time without needing to iterate further.

Will leave as an exercise for the reader to get to: n = -38, d = -42

Where since is modular algebra those aren't the only ones that will work. But I liked them because they're small. Also shows you can use negative as well as positive, of course.

Then r-1 = -39(159) - 62(-42) mod 851

And r-1 = 658 mod 851. And 97(658) = 1 mod 851 as required.

I think that's all ok, as to how it works. Copying from notes so may be minor errors here or there.

Main point is: you definitely get to a smaller modulus as went from 851 to 149. And also can just iterate with THIS method, so it will solve for the modular inverse.

Lots of research room though for best techniques.

And I noted some open questions, like what is best choice for m? Or y0? And does it matter to work more to pick a smaller F0? I just went with whatever. You can fiddle with things to make coefficients of n and d as small as possible, I'd think!

So much room for further research.


James Harris

Thursday, June 01, 2017

Solid result and am surprised

Just a quick update that my modular inverse solving method is solid. Luckily is easy math, so not much to check. For those who wonder, when an error is in there it usually is something simple. I have my own definition of mathematical proof for checking for errors, but human reality can be you just miss things. But my modular inverse innovation is perfect.

It adds to my collection of infinity results. And that label is below the post for those curious about the others. Just click on labels below posts to get other posts where I've so labeled.

But so wild was available! Am so glad just kind of was wondering and have had that modular factorization for YEARS and talked it up on this blog too. Oh yeah, decided had a responsibility to at least try and sent some emails to some mathematicians and one US Government agency. Felt like the right thing to do. No replies from the mathematicians. And just an auto-reply so far from US Government agency.

That's ok. Just doing my due diligence. But for those who wonder? No, am not expecting mathematicians to reply to me any more as what can they say? Think about it. Any reply can force them to do more than just reply, like help champion a really massively cool discovery which greatly adds to human knowledge.

Maybe I should give them the benefit of the doubt. Hasn't been that long. Still feel there is clearly indication of a certain poetic justice in play. Situation can be distressing for others am sure who may worry about controversy! But good news is, is more me disappointed than anything else. I have math ideas I'd just as soon see picked up by established mathematicians. Mostly though they just seem to ignore me.

Regardless of any of that am SO excited with the find!

Am so happy with this thing. Not my biggest result by far, but one that tells a lot. Why was it available for me?

Because I went looking for it, I guess. I don't know. Am just babbling now. Then again is also would think another example of the analytical power of abstract reductionism.

Main thing: is a solid result. May have a massive impact in many areas of number theory, remarkably enough. I suspect is ALREADY being used by now as information travels fast in our times, but not that folks tell me. But already discussed the why there.

The math is there. People just have to go look for it. And mathematics IS an infinite subject.

For those who love math? That is a reason for so much joy. You will never exhaust possibility.

And a reality check too I think, as how can people lead the future of mathematics in our world, if they can't simply discover?

Discovery defines mathematics. It is the base from which all else must build.


James Harris

Tuesday, May 09, 2017

Modular inverse innovation

Figured out a way to calculate the modular inverse of some residue r modulo some integer N, by leveraging a modular factorization. Will give the working system, show an example then give the derivation.

System:

 r-1 = (n-1)(r + 2my0) - 2md mod N

Where y0 is chosen as is m, with m not equal to r, and n and d are to be determined. They are found from:

2mdF0 = [F0(n-1) - 1](r + 2my0) mod N

and

F0 = r(r+2my0) mod N

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

Example: Calculate the modular inverse of 11 modulo 137.

With N = 137, I will use m = 12, as: 7 = 144 mod 137

And looking at simple will use y0 = 1, so:

F0 = 11(11+2(12)(1)) = 11(35) = 111 mod 137

61d = 49n - 84 = 7(7n - 12) mod 137

If we now let d = 7d', get to 61d' = 7n - 12 mod 137, and n = -7 works with d' = -1.

So d = - 7, with n = -7, giving r-1 = (-7-1)(11+2(12)(1)) - 2(12)(-7) mod 137

So:  r-1 = -6 + 31 = 25 mod 137, and 11(25) = 1 mod 137 as required.

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

Derivation:

Given, some residue r modulo N whose modular inverse is to be determined, find D such that: m2 = D mod N, and m does not equal r.

Then for some  x2 - Dy2 = F, I have the modular factorization modulo N:

(x-my)(x+my) = x2 - Dy2 = F mod N, and can set:

x - my = r mod N, then x+my =Fr-1 mod N.

Solving for x, with x = my + r mod N, and substituting into x2 - Dy2 = F gets me:

r(r+2my) = F mod N

So I can look at a y0 which will give an F0, and a difference of d, such that:

y = y0 + d

And subtract that F from a multiple n, of my initial F, substituting for y:

nF0 - F = r[(n-1)(r+2my0) - 2md] mod N

Now consider, nF0 - F = 1 mod N, which gives:

r[(n-1)(r+2my0) - 2md] = 1 mod N

And can solve from there for the modular inverse:

 r-1 = (n-1)(r + 2my0) - 2md mod N

Now can substitute into x+my =Fr-1 mod N, using x = my + r mod N, and simplify somewhat to get the control equation:

2mdF0 = [F0(n-1) - 1](r + 2my0) mod N

and from before I have to show the full system as given:

F0 = r(r+2my0) mod N

Derivation complete.


James Harris
-------------------------------

Want a PDF? You can get one at my math group. There is a post there with a PDF attached.

Friday, May 05, 2017

Chasing the modular inverse

For years I've talked about the easy modular factorization with x2 - Dy2 = F.

Which is, when m2 = D mod C, where C is some integer:

(x-my)(x+my) = x2 - Dy2 = F mod C

And then I'd use some residue r.

Where one way is: x - my = r mod C, then x+my = Fr-1 mod C

Where for years have wondered about that modular inverse, and found myself pondering, can you use these equations to figure it out? So why now? I don't know. Just playing around.

Next thing I knew was looking at C = 111 = 3(37) where picked for easy. And noticed that a close square is 121, so got m = 11, and D = 10.

Then went for a prime for r, kind out of habit, and need one coprime to 111, wasn't going to use 2, so went with r = 5.

So had: x - 11(1) = 5 mod 111, so x = 16 mod 111

Then I calculated F, from x2 - 10y2 = F, and got F = 24 mod 111.

So had 16 + 11(1) = 24r-1 mod 111, so 27 = 24r-1 mod 111

Which didn't seem to get me anywhere. But I noted means: 5(27) = 24 mod 111

So I'm like, ok, why not use y = 2? And after same process got: 5(49) = 23 mod 111

And, I realized could subtract one from the other to get 5(27 - 49) = 5(-22) mod 111, so:

5(89) = 1 mod 111

So I got the modular inverse, but was that luck? And now finally decided to go more formal, so here is the full system:

 x2 - Dy2 = F, m2 = D mod C, where C is some integer.

(x-my)(x+my) = x2 - Dy2 = F mod C,  x - my = r mod C, then x+my =Fr-1 mod C

Solving for x, with x = my + r mod C, and substituting into x2 - Dy2 = F gets me:

r(r+2my) = F mod C

So I can look at a y0 which will give an F0, and a difference of d, such that:

y = y0 + d

And played around enough to know I needed a variable n, where will just give what I got without explaining everything in detail. Is easy math though, so not hard to retrace.

nF0 - F = r[(n-1)(r+2my0) - 2md] mod C

And chasing after modular inverse, nF0 - F = 1 mod C, would be nice, so have:

r[(n-1)(r+2my0) - 2md] = 1 mod C

And then:

 r-1 = (n-1)(r + 2my0) - 2md mod C

Finally now then I have control equations with:

2mdF0 = [F0(n-1) - 1](r + 2my0) mod C

and

F0 = r(r+2my0) mod C

So now I don't even see the original x2 - Dy2 = F or the other equations. Just see those three, but originals must be valid for it all to work! Must keep that in mind. First was thinking oh, they just went away. Nope. Math knows they are there.

Well now can see how n = 1, and d = 1 worked, with r = 5 mod 111.

So was just luck, huh? So you CAN just say, pick n, but then have potential of needing a modular inverse to calculate d, or vice versa. Oh yeah, m cannot equal r for this approach to work! Should note that as well.

Also may as well check my new equation since know n =1, works, and clips off the front too.

And: r-1 = -2(11) = 89 mod 111, as we got above.

Went ahead and tried r = 41 mod 111, just to use something bigger, so again m = 11 and y0 = 1. And got:

105d = 3n - 66 mod 111, where d = 1 will work with: n = 57

And that gives r-1 = 65 mod 111, which works: 41(65) = 1 mod 111

And just playing around. So yes, can calculate a modular inverse this way, potentially. Definitely can get to another modular inverse to solve for n or d, but may not be necessary if can just figure them out like with my simple experiments. But who knows how often is that easy? I don't.

Wonder if could be proven.... Possibly something for some other time. These simple experiments have been rewarding and extremely interesting.

Glad did some experiments and some derivation.

Can be SO much fun to play with the math.


James Harris

Saturday, April 29, 2017

Thoughts on simplicity rules

Underlying reality itself, in SO many ways mathematics rules the world.

Theoretically as a human discipline mathematics should be the most objective, straightforward WITHOUT politics or social influence because mathematical logic does not care. But functionally have concluded that difficulty in understanding mathematical topics can bring those things BACK into the discipline.

But difficulty should only be there out of necessity, and not for social purpose.

So will admit would be so cool to flush mathematical industry of unnecessary complexity. And we should cheer simplicity and push the discipline away from worship of complexity as if it's a power to be not understood by most.

For me simplicity improves access and am sure helps reach. Am not someone who has the assumption that people will pay attention for any other reason than my math discoveries WORK.

Useless complexity though am sure can give room for charlatans to hide within mathematical field! And any such should be cleared to speed up mathematical discovery and progress. Humanity deserves that on the constant. And am sure science will need more advanced mathematical tools on the constant as well, where direction for tools with modular as a concept has unique appeal for its efficiency, simplicity and conciseness. Luckily have done lots of the groundwork over last decade plus which has a modular focus.

In number theory have been fascinated with realization that much of my own research relies on what I decided to call abstract reductionism, and how that really is about simpler approaches, including modular but not only.

For example one of my favorite results is the fastest way to count prime numbers for its size and complexity.

For those who wish to code, with positive 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)}

The P(x,n) function will count primes 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 is the fastest algorithm for counting primes for its size.

Turns out I noticed a neat trick which lets you use old ideas with an innovation that makes for a concise algorithm, but also leads to a partial differential equation and an explanation that eluded mathematicians for over a hundred years about the prime distribution. Simplicity ruled.

And abstract reductionism is even more powerful with what I think may be the first true modular algebra, which I introduced with a tautological statement, also known as an identity:

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

I call that a tautological space and with that I found a way to generally reduce binary quadratic Diophantine equations, where suggest you use web search to find if don't know where is on this blog already.

Finding by web search versus me just giving a link may be more impressive AND informative as can then see competing websites. And my way improves upon techniques put forward by Gauss himself! And as he is a HUGE hero for me that's a massive big deal. Simplicity rules again.

These simplifications though to me help make number theory more fun--and accessible.

Consider the irony, if some long, difficult mathematical argument which only a few people in the world understood--is simply wrong. And I dramatically demonstrated how it could even look correct under established rules of rigor.

Have concerns about to what extent hero worship took over the math field. People chase the appearance of correctness, daring someone to catch them if they're wrong, but oh, how do you know? In a complex argument which few understand how dare you challenge?

Shredding the incentives for empty celebrity I think helps mathematicians care for truth.

There may be simple ways to remove such.

Simplicity should rule.

That shatters political maneuverings, dismisses hero worship as basis for belief, and ends worship of position in mathematical fields, as who cares who figures out some math thing? I know the math does not.

People can, and it can lead them astray.

We should care in SOME way though I think. But what way is appropriate? Main thing is: irrelevant to mathematical truth who digs up that truth for the world.

Simplicity rules.

And simplicity has other benefits too, as being able to readily play with actual numbers is so great. Like my methods lead to a thing I wonder about usefulness. Consider:

42 + 6+ 10+ 14862 = 882

Which I copied from  a post where I demonstrate a general way to sum whatever number of squares you wish to get a square.

What use is it? Um, for me is just a way...um is fun. I guess. I've debated if should make more examples with it, but will admit...so maybe not so much fun for me. Maybe someone, somewhere out there at some time in human history will find it useful.

Main thing in my opinion is that you can just get to math. And will admit do cherish that I escaped the label of mathematician. The math does not care. Discovery should matter more than a label.

But I think it DOES matter when people can easily check a person. Overly complex and abstruse mathematical discourse in my opinion can often be about making that difficult, to force relying on people who know what they're doing--even when it may not actually be mathematically correct.

It can seem expedient to make a career over advancing your species.

I have no career as a mathematician to advance as am NOT a mathematician. And for me simplicity helps me do what I like to do.

And people should cherish valid knowledge. When you start looking at the person who discovered some mathematical truth, as if that decides if something is true or not--then you have lost mathematics.

Mathematics does not care.

Ultimately simplicity is a basic principle for obvious reasons.


James Harris

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

Sunday, February 26, 2017

More explanation for the curious

One of the things I've noted often as is really important is that I found a way to show a mathematical result which follows established rules, so is in full mathematical rigor, which nonetheless leads to a conclusion which can be shown to be false. That is, the mathematical argument is perfect under established rules, but using those same rules you can use a separate mathematical argument to prove the conclusion is false.

Which to me is rather fascinating. And you can check if you have knowledge of basic number theory by downloading a paper I got published where here is link to post that has a link to it, and have talked story often on this blog.

What's cool is of course it stands on its own but for the curious can explain LOTS about how that's possible and implications for modern mathematics.

What allows that actually is that the paper starts by declaring the ring to be the ring of algebraic integers. And the Wikipedia has an article on the algebraic integer for those who need a reference.

While I'm NOT a mathematician, I stumbled across a problem with the ring of algebraic integers, where after YEARS can actually show very simply and won't give all details here, but is so short can enough.

So 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.

Of course a simple reducible quadratic with those requirements is:

x2 + 3x + 2 = (x+1)(x+2)

And I can force symmetry by introducing k, where k is a nonzero integer, and not 1 or -1, and new functions f1(x), and f2(x), where:

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)

If you want to see more, I found a really in-depth post where talk much more here.

The forcing symmetry is what does it really. As I put an asymmetrical form at the BOTTOM as the base, and then force a symmetrical one later, and that does it! So cool to me.

Notice you have NO problems with our simple example:

k(x2 + 3x + 2) = (kx + k)((x+2-k) + k)

But all kinds of awesome happens when you have non-reducible quadratics for P(x). That's when the fun starts.

Turns out you can set things up so that it's easy to solve for the f's as roots of a monic polynomial with integer coefficients so they are DEFINITELY algebraic integers, but for the non-rational case you run into a problem with g1(x) = f1(x)/k as it forces one of the f's to have k as a factor.

So you can force that factor into algebraic integers and then NOT be able to see it directly. Knowing is there from mathematical logic. So absolute proof tells you is there but no way to see directly.

For example I often would show that 3+sqrt(-26) must have 7 as a factor for one of its two solutions which can seem strange if you follow human convention. But for example 5 + sqrt(4) is 5 ± 2 and equals 7 or 3. If that really bugs you not much I can do. Had wild arguments years ago where people would say that the convention is to take the positive so MUST be 7. And I'm like, what can you say to that?

So you find that in such cases both g's can NOT be algebraic integers. And I used a neat trick, where the f's can be, but I constructed with a throw-away factor of k, which means if you remove it, and reduce, you're forced to have:

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

So you can't run from the problem. The algebra says you can do it.

Logically it works! Is fascinating to me though, how human beings can struggle with it, and may be one of the best pure examples of such.

There's no reason NOT to be able to do it, but our species didn't notice that over a hundred years ago in the late 1800's when coming up with algebraic integers, so it took me to show, in our time.

Yeah it is just NOT intuitive to multiply by that k, and math folks get taught to go in opposite direction, removing factors if you can! Not adding them.

So I went against the usual flow.

And I showed how not knowing these things can let you use full mathematical rigor and come up with a wrong conclusion--in the ring of algebraic integers.

So why would I get a paper published showing that?

Well that was a great way to demonstrate, and may be the one key example known in human history to demonstrate the failure, where planned on a follow-up paper to explain the problem, thinking would have help from mathematicians, but that didn't happen.

What can someone do with this error?

Can appear to prove things with full mathematical rigor, which are NOT true.

And yes, I talked the question on this blog: Did mathematical rigor fail?

Answer will go ahead and note here is, no.

So yeah I had things not happen the way I expected. And was disappointed for years and acted out a bit online as well. But as time went on, I realized I had responsibilities and besides I could discover more things!

Eventually I focused more on mathematical industry as healthy, which includes lots more people than mathematicians, as lots of people all over the world use mathematics. The problem I found is important but rather esoteric. It helps to be a number theory expert to fully understand it and its implications.

And I am NOT a mathematician.


James Harris

Wednesday, February 01, 2017

When connections can fascinate

Interest from others has often intrigued me and on this blog my find of a deep connection between hyperbolas and ellipses is an area that keeps trending.

Like I see a post where I discussed much in 2010, which stays high in list of popular posts. Back then called what I now call the two conics equation "Pell's equation" which is one of those weird to me things described as a historical mistake. Problem was some guy named Pell didn't deserve credit? Then if is known, why keep giving to him? I followed convention for a while then stopped.

For me my own results are too familiar so it helps to see what resonates with others. It's like, on any given day I can, if I wish, consider something I've discovered, but how do I choose? Looking at what interests others gives me a place to consider.

So I discovered the connection and gave some examples.

The connection is from x2 - Dy2 = 1, with D positive so you have hyperbolas to ellipses where it helps to see an example.

Like consider: 172 - 2(12)2 = 1

and I have this key variable j, which is explained in the referenced post:

j = ((17+2(12)-1)/2 = 20 is a solution giving:

202 + 212 = 292

And I notice that if D-1 is a perfect square relationship found connects to circles but otherwise gives non-circular ellipses.

Also proved you can always find integer solutions for af2 + g2 = h2 by using a rational solution to what I NOW call the two conics equation. In that post you see the other usage all over the place. And that is far as I can see one of the most consistently popular subjects for YEARS on this blog.

But would be curious of any other situation where a rational engine drives integer and therefore Diophantine solutions. So yeah lots to ponder there. If you know of any other cases feel free to comment.

Read referenced post for more, and also there are other posts with label tce, and that label can click below.


James Harris

Wednesday, January 25, 2017

When it seems impossible

For those who read this blog you may realize that a conclusion I reached is that I found some new numbers previously not cataloged which should be impossible. And reality is I realized that back in 2003, and yeah to me is good to be at a full explanation level so quickly, now only around 13 years later.

But that was the reason to figure out the object ring where a post talking it is the FIRST post on this blog.

My feeling is that the assertion is so unbelievable that there are numbers that were missed that the greatest skepticism possible is required.

So yeah, one dead math journal in this story? Not a big deal considering.

But is it true? The mathematical proof seems solid to me! But that's not enough.


James Harris

Sunday, January 15, 2017

Refreshing on old for me concepts

So yeah have one published paper, where also make sure to note that the mathematical journal tried to literally pull the paper AFTER it was published. In that it was an electronic journal and the chief editor tried to go with just deleting my paper and saying "Withdrawn" in the Table of Contents, though I never withdrew it. That of course isn't how it's done. And was relieved to find that as far as the mathematical establishment rules go, I have one published paper and am considered to be a published mathematical author, so am listed as one.

That paper is also kept available by EMIS, and have talked such things before, where here will refresh with what I feel like are old concepts for me, as was over a decade ago all that drama. Am so grateful for those who DO follow the rules. And am a person who works to follow the rules, as it actually matters most for me.

The paper relies for analysis on x+y+vz= 0(mod x+y+vz), which it does not state, which I later started calling a tautological space, one of my favorite made-up expressions, which is part of my own discipline of math innovation I decided to call modular algebra symbology.

For my paper I used a cubic but later simplified to quadratics which looks like it happened around February 2006, as realized same approach SHOULD work if the mathematical ideas were not flawed.

There was concern and then lots of relief as mathematics continued to behave as expected, and was so much easier working with quadratics. Learned the lesson: when you can in mathematics, always take opportunities to simplify your analysis.

Most posts talking quadratic non-polynomial factorization on this blog actually rely on the condition that:

x2 + xy + y2 = z2

And I talk through how I analyze it with post: Under the hood

With that conditional with the tautological space I would use v = -1 + mf. As v is free to be ANYTHING I want, and I figured out, which is the art to the mathematical science, that analysis went a certain way I liked with that choice. Further along in analysis, would bring back x, with m = x. Why? It occurs to me that I just like having x around. Since I can with this approach, I do.

And z would get cleared out with z=1, but would keep f visible but integer with f = 7, as liked getting to some numbers. It does help looking at the expressions. So end up with two variable quadratics which can be solved for functions I'd call 'a' where a = y. Which I guess may be because I liked the way that looked better. The functions end up being a1(x) and a2(x) and am sure that looks prettier to me for some reason.

If you stare at mathematical expressions for years? It matters how they look to you. I've studied in these areas now for well over a decade and like my foresight in picking variables that sit well with me. I came up with tautological spaces back December 1999, but took some time to call them that, so yeah, helps to pick well.

And in fact I came up with tautological spaces JUST so I could have a variable I could make anything I want in order to analyze expressions in a different way. Then I could deliberately probe.

Have another post which is even simpler talking tautological spaces with x2 + y2 = z2, where didn't find anything I thought mathematically interesting so use it as a teaching example of absolute proof. As explains carefully how you KNOW the proof is absolute.

And probing x2 + xy + y2 = z2 turned out to be very interesting which is good as I picked that equation for its simplicity while notice I had to get SOME complexity as can compare with the same analytical approach done on x2 + y2 = z2 where I didn't notice anything as interesting.

But there is an art to it, which is why I emphasize that I didn't notice as doesn't mean there isn't more there, as v is an infinity variable. And also I used the simplest tautological space, but there are an infinity of them, though my approach means there probably are a finite number that can sensibly be used with any particular equation, I guess.

And that appeals to me as the mathematical science, where for some questions you have to go exploring to get the answers.

Remarkable thing though is that while you can show a problem with some traditional mathematical thinking with analysis using tautological spaces, there is also a direct route which is simple and doesn't require mentioning them at ALL.

There I finally realized could just consider the generalization of a factorization of a quadratic polynomial P(x):

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.

From there I easily get:

k*P(x) =  (f1(x) + k)(f2(x) + k)

by multiplying both sides by k, and introducing new functions f1(x), and f2(x), where:

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

Where my innovation was to multiply both sides by some constant integer. It's so EASY. So I speculate that multiplying by some constant factor is so against the grain of trained math people this path was just waiting for me. Standard training of course is to remove extraneous factors.

If k = 1 or -1, then you can get algebraic integers for the g's and f's. Easy.

That was HUGE to realize, as yeah you CAN solve for the g's easily enough using my approach, if k = 1 or -1. The unary case is specially unique here.

But if k is 2 or 3, for instance, the f's can still be algebraic integers, but also you have:

g1(x) = f1(x)/2, or g1(x) = f1(x)/3

And usually that can't be an algebraic integer. But still exists, so what kind of number is it?

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

And that does NOT know if you multiply it by k = 1, -1, or 2 or 3, or whatever. So do not make the mistake of figuring that based on your human choice the g's shift dynamically. They know not nor care what you do or what you think! The math just is.

So what does shift? Depending on what YOU choose for k, you're probing the factorization in different directions. If k = 1 or -1, you're looking at algebraic integer solutions for the g's. When you shift k, the analysis approach used starts showing you the other numbers.

It's about my techniques for analysis. Like analysis method is like the microscope, and k is the magnification? Cycle it up from 1 and -1, and you start peering into another number theoretic world. I use a special function I call H(x) to get a handle on things.

It took years of pondering from a different direction though, as lots happened from that wild story with publication in 2003 until I came up with what I see now as a very simple algebraic argument with quadratics using elementary methods and most dramatic innovation being multiplying by a constant integer factor.

So it took only around 13 years to explain it all, which is ok.


James Harris

Sunday, January 08, 2017

My partial differential equation connected to primes

Sometimes answers can be kind of just there. Like for a LONG time there was a puzzle over why the count of prime numbers could have ANY connection with continuous functions. But I found an easy answer with a partial differential equation I found that follows from my way to count prime numbers.

My prime counting method leads to a difference equation from which I can get to a partial differential equation:

P'y(x,y) = -(P(x/y,y) - P(y, sqrt(y))) P'(y, sqrt(y))

I talk more about in an earlier post I like for reference, and will admit maybe should discuss it more, given its implications but feel like am primarily repeating myself.

Gist of it though is that there is a sieve form for a prime counting function that counts by calling itself. That allows for a difference equation, which does the same, but is a difference equation as it can tell itself when a number is prime. And that leads to the partial differential equation which is the continuous function.

Here's a post I made in 2009 which shows them all together. I rarely do that for some reason.

So past mathematicians were approximating the integration of the partial differential equation, with no clue. But it's a simple, succinct explanation for how the count of primes could be approximated by a continuous function, which isn't exciting. I like to say, it isn't math sexy.

It's like if Gauss were around I could explain things that he died not knowing, but did he miss much? I'm not sure. Or Euler or even Fermat maybe, oh and definitely Riemann. Getting THE answer, would it have been worth it to them? I like to think, yes. But they died with the prime count connection to continuous functions a big mystery, and not something easily explained.

But no, no dramatic, extraordinary and profound possibly mystical explanation needed for prime counts connected to x/ln x or Li(x) or any continuous function. Just a rather mundane, simple and straightforward mathematical explanation.

In our time the answer doesn't seem to get much excitement going, which is ok with me, though also puzzling. But I know that sometimes answers can be, well just not that exciting. I find that intriguing to some extent, but it's just one thing I have. So um yeah, I explained the primes connecting to continuous functions thing, years ago, and moved on to other things.

There is no other like it though.

And there are clues to how big the partial differential must be.

There are no other known partial differential equations that follow directly from a method to count primes. It comes from a difference equation which is ALSO unique in that there are no others that can count primes.

Yup, worth saying again, the difference equation counts prime numbers, when properly constrained. There is no other that does.

Finds them on its own which rattled me for years. Shifted how I looked at the math.

It is one of my favorite discoveries. Also one of the easiest of the BIG ONES to check I think. And one that gives a LOT of perspective.


James Harris