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Monday, April 16, 2018

Finding quadratic residues with modular inverse

Am curious about numbers often and find myself trying things. And realized had not yet used one of my innovative ideas in a way that looks possible. So figured may as well try a few things. And doing so public helps my process.

Have focused on x2 - Dy2 = F, as a fundamental equation which controls much of the behavior of integers.

Importantly, realized could solve modularly, as for some integer N, will be true that D is a quadratic residue, such that D = m2 mod N, for some m.

Which means have a useful factorization available:

x2 - Dy2 = x2 - m2y2 = (x-my)(x+my) = F mod N

Where then, for some residue r, x + my = r mod N, and x - my = Fr-1 mod N

And used this direction to find my own way to get the modular inverse, but am thinking could be used to find m, as well. Though why bother? Well, am curious.

Can solve for x, as: 2x = r + Fr-1 mod N, and 2my =  r - Fr-1 mod N

If I simply pick r, then get its modular inverse, and set F, then I have set x.

And with this path can simply set y = 1 mod N, and F = 1 mod N, and then find D, from D = x2 - 1 mod N.

For example, let N = 119, where I like to use composites for these types of experimental musings, as primes can be too particular. This approach does not care though if prime or not, so using a composite covers more territory.

And let's let r = 2 mod 119, for easy. As then also r-1 = 60 mod 119.

So: 2x = 2 + 60 = 62 mod 119, and x = 31 mod 119.

And 2m = 2 - 60 = -58 mod 119, and m = -29 = 90 mod 119

And D = 312 - 1 mod 119 = 8 mod 119

And yeah 902 = 8 mod 119, as required.

Well still ended up squaring x, and definitely does not seem practical to me. Will do one more easy.

So will go up one, and let r = 3 mod 119, again for easy. As then also r-1 = 40 mod 119.

So: 2x = 3 + 40 = 43 mod 119, and x = 43(60) = 81 mod 119.

And 2m = 3 - 40 = -37 mod 119, and m = 60(-37) = 41 mod 119

And D = 812 - 1 mod 119 = 15 mod 119

So of course: 412 = 15 mod 119

And my curiosity is satisfied, for now. Though does not look like did much more than look at examples of the modular factorization. Obviously you can simply square numbers and get residue modulo N, directly without taking this slightly convoluted path.

Of course I set F = 1 mod N, just playing around. But can set to minimize x, or who knows. Just playing around just a bit, so will not speculate much.

Oh, but you could use the BQD Iterator, to get another set of solutions for x, with the same D, if the same r holds. That of course shifts y and F. And then you can solve for m.

Wow that might work. Don't even have to do another square either, as have D. And could be a way to loop through all possible m's for a given D. BUT it could just give the same m, as before. And maybe r does shift when you iterate. Will not try it myself though. Just note it here, for now.

Of course would bring BQD Iterator up, as use it so much. But think is the key controller across integers. That thing is SO wild. Is like this super key to just about anything with integers. Where just have to use your imagination. And that I guess I have.

These kinds of things can be stubs, where later may notice something else which follows. So is worth it to have these initial musings public, as may lead nowhere interesting, but then again, could lead to something else really cool. And how would I know at this point?

Just musing on some math for me is so much of the fun.


James Harris

Tuesday, April 03, 2018

Knowledge recognition reality

Much has shifted in the search for knowledge because of the ease with which a person can share knowledge gained. Like on this blog have plenty of mathematical results. And can from objective sources get a feel for the global interest.

For this blog according to Google Analytics, which is a source have used for somewhere around a decade, there were visits from 66 countries so far this year. And that is from 746 cities, with people using 55 languages. Where just read from what the application told me as went to it for this post.

And thanks for the interest! Should acknowledge my appreciation to those visiting here from all over the world. My focus is on the primary reason being--useful information.

Of course there is also question of official recognition of some kind which with me has only happened once with a paper which was published, and then things got wild as a chief editor tried to just delete out after. But have told that story SO MUCH, as yeah worked at processing.

So how can there be such a disparity? Well the answer is that math actually moves best on USE and that was more difficult to track--in the past. Which meant establishment sources were the best bet for trying to guess at what mathematics was actually doing out there. But that was so much about a gathering of people most passionate--in the past. In our times? Is debatable what establishment represents.

But also in our times, if you have math ideas? And you put them on the web? Then conceivably you can get objective data from others who are trustworthy sources to get a feel for how your mathematics is moving around the planet. That does free a researcher from the establishment.

It has occurred to me the establishment may not like that reality.

Would rather have objective evidence than to me, empty accolades, as how do they know really? And who are they, often? Reality of attention to your mathematics from those who toss out awards or whatever are rivals, with grudging approval dragged out of people who should wonder why they didn't figure out instead. They can be forced to know YOU because of what you DID figure out that they did not. If they aren't rivals then maybe then they think are better than you, but with what mathematics?

Objective reality does not care about social things--at ALL. Either the math works, or it does not.

Either the math is an improvement or it is not.

Wonder if is still taught that reality. Maybe is too harsh to admit to students such honest rivalry or the brutal reality check that no matter what you do, if your math is not good enough it is like you did nothing at all?

The solidity potential of mathematics can make competition stark. Few winners with absolute truths, but so many trying.

In our times you can have some mathematical discoverer cheered by community, which is actually tiny if you consider a planet of billions, whose work just does not move outside some small circles. But his buddies, as is often a male, like him. How far will that make math move? If not actually useful? Not far at all.

Is just easier to check in our times.

In the past might have been enough for much more in mathematics--being well liked.

You cannot compete with someone like me on the accolades of people like you. Their praise is as worthless to me as yours, if your math discovery does not justify interest.

If you need praise but somehow convince yourself you are into discovery, then maybe you really should go into some other area, like politics? Or entertainment? Because the math does not care.

Does push mathematics as a discipline BACK to hard, objectivity without delusions of importance of celebrity.

Mathematics is about logic, consistency, and absolute truth at best.

Humans can make into something else for a time, and call flawed ideas mathematics, but what works keeps pulling back the body of accepted mathematical ideas to better. Math is just SO useful.

More and more I think people are learning to trust objective data and my story helps! We need facts.

Web helps there as well. Readily admit started checking more to be aware of what happens--after the news fades. Can be a pressure to just chase after the latest without watching people over time. Web is really great there, as long after news organizations have moved on, you can consider what a person is doing, or not. If they do anything on the web. But also you can usually especially with academics at least keep up with where are working, or not, if is at a university.

Reality is, the rush of attention can fade so rapidly. While for me? Have been pondering global attention levels for over a decade now. Is just SO steady which puzzled me for years. Realized had been trained wrong. Human interest builds over time with math. People keep building more things in actuality. So yeah, is the actual way.

The math becomes more in demand as is proven in the minds of more as they use, and the need continues to grow, as humanity continues to progress.

And you can have someone like me, who is global on a scale few humans can actually comprehend, which I say as have worked for YEARS at trying.

But yeah, I invented a new math discipline, where the math does algebra demonstrably better than any human. Discovered for the world a new primary way to calculate the modular inverse. And even figured out the 'why' of the connection between counts of prime numbers and continuous functions like x/ln x. Among other big things actually but just grabbed for a few quick cool ones.

Try to compare with someone else's discovery in the 21st century. I know I don't even bother.

So yeah, the Google Analytics data is probably an underestimate, which is ok. I like to use as a baseline.

Math moves when useful. Human interest is so much about what works. And with the web, in our times, you can watch human interest pull that needed knowledge.

Is a great thing really. Your discovery? Can MOVE, and YOU can watch.


James Harris

Monday, April 02, 2018

Where identity is key

One of those things did years ago, had me wondering why I bothered. But again shows how tautological spaces which are what I call complex identities can do MUCH when it comes to algebra.

Copying from my post September, 28, 2011:

...result that in general x2 - y2 = (x+y)(x-y), which can be constructed from within the tautological space as an intrinsic property:

x+y+vz = x+y+vz, x= -y - vz + (x+y+vz), squaring both sides, gives:

x2 = y2 + 2yvz + v2 z2 - 2(y+vz)(x+y+vz) + (x+y+vz)2

x2 - y2 = 2yvz + v2 z2 - 2(y+vz)(x+y+vz) + (x+y+vz)2, and setting v = 0, I have:

x2 - y2 = - 2y(x+y) + (x+y)2, which is: x2 - y2 = (x+y)(-2y + x +y),

so: x2 - y2 = (x+y)(x-y).

Source: Under the hood

Was like the coolest thing to me when realized was possible. Have pondered and wouldn't be surprised if you can build every part of algebra using tautological spaces, and in so doing relate everything back to identity.

So, given: x+y+vz = x+y+vz, you have x+y, and x-y, and a standard result. Follows logically from the tautological space itself. Oh yeah, also need some basic operations of addition, multiplication and need the distributive property. And the integers, should not forget. But realized also just need -1, 0, and 1, actually as of course from there can get the rest.

Reducing mathematics to identity? Resolves questions of truth.

That basis in truth then conceivably is the basis for mathematics itself: identity.

If so, then you just need identity and mathematical operations, and can build ALL of mathematics.

I like that idea. May as well make a post that goes ahead and states something else have been pondering for years now.


James Harris

Friday, March 30, 2018

BQD Iterator is a very surprising tool

Realized that one of the more profound results for me both practically and philosophically is a mathematical tool I call the Binary Quadratic Diophantine iterator, or BQD Iterator for short.

Where have this post as my established reference, posted November 28, 2014, which talks in-depth. For here will just give my favorite form again.

Given: u2 + Dv2 = F

then it must also be true that

(u-Dv)2 + D(u+v)2 = F(D+1)

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

Where do minor variations on that theme here and there as of course can shift with symbols used and signs, and what is weird about that thing is, seems to connect to just about every property of integers.

Wouldn't that be one of the weirdest things ever? Like, ONE thing being key to so much where for thousands of years so many searched and found pieces of the puzzle, when the biggest thing was SO simple.

And yeah I know, may not look like much. Have stared at it often for years now, pondering.

The BQD Iterator quite simply may control, or in so way be related to the controls for all integers. It is the one tool that the math may use, just about everywhere with integers. I find that remarkable, and wonder if is true.

Like MAY lead to THE template for ALL integer factorizations: But does it factor?

Where is one of those areas where I do NOT check thoroughly, as delves into too much scary. (And will collect here another post where talked scary areas with this post with regard to implications from other things.)

The AMAZING thing about the BQD Iterator is that no human found it directly, which includes me, as I used tautological spaces. What is it about it that not even just playing around no human just stumbled across it? Or maybe someone did, but is just not big in the historical record? Have wondered.

And I LOVE to play with it now. Though took a few years, where helped out much when named it for some reason. Like sqrt(3) approximately equals xn+1/yn+1, where:

xn+1 = 1351xn + 2340yn

yn+1 = 780xn + 1351yn

and x0 = 1, and y0 = 0;

So get x1 = 1351, and y1 = 780. Which is a decent approximation, and next is:

x2 = 3650401 and y2 = 2107560

And 3650401/2107560 is approximately:  1.73205080757

Have used to explain result where Euler and Ramanujan played around, along with lots of other things. And emphasized for me just how much simple we may never know, as how do we know what we do not know?

So much of being human is about aspects of our design with which we are simply born, which must impact much what we can discover. Which is kind of depressing actually.

And yes, talked in this post where mused about my recent find of a third primary way to calculate the modular inverse. Where have been trumpeting that one for a bit now.

For me it is interesting then to have a result where yeah, was me there. And the math didn't hand me THAT result, but then again took me years to notice something that followed from something clever. Where kind of wonder, why did I think of it?

But then again, have been as I say, talking to the math now for quite some time. And has dawned on me that I DO think differently now. Maybe there does happen that better ability to look outside the box as the saying goes, when have watched a process give answers WAY outside of what humanity found on its own.

Yeah that does make sense. Watching the math DO math, could conceivably have shifted how I look at mathematical problems. Like if had watched some human teacher working on math, but am watching an infinite intelligence instead. Wow. These are the kind of posts where am glad just start typing and find out where will go.

Like, what greater teacher of mathematics could there be to watch? Than the math? As can watch the math DOING mathematics, and just be like a student, learning?

And later, me, the student could figure out something cool entirely on my own. Yeah. Maybe.

So yeah, watching the math, do math, can help teach a human, me, how to do math better? Wow. What a concept. Can explain much if true. And I do so like to hedge, do notice. If true.

Yeah but did have the packing of spheres thing, before. Though still bothers me for some reason, how simple that modular approach is. And of course, did figure out myself how to use tautological spaces. Still guess do love the idea of the math itself as one of my greatest teachers. And why wouldn't I?

Makes sense to me.


James Harris

Wednesday, March 28, 2018

More playing with square root of three

Number theory is focused on integers. And mainly am focused on number theory. But there is math that I find myself just doing for the fun of it, which goes into approximation. But really only because is just another demonstration of some of my math discovery. Also though is kind of a reminder of why x2 - Dy2 = 1 had historical importance. As solutions could be used to approximate square roots.

With my research noticed special cases with my BQD Iterator for two and three, where could use to get easy approximations. And is just kind of fun for me and even relaxing to advance. Last post was this one where ended up with sqrt(3) approximately equals xn+1/yn+1, where:

xn+1 = 362xn + 627yn

yn+1 = 209xn + 362yn

and x0 = 1, and y0 = 0;

And my first post was this one for my reference.

So need the BQD Iterator, with D = -3, to advance some more:

u2 - 3v2 = 1

then it must also be true that

(u+3v)2 - 3(u+v)2 = -2

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

(362x + 627y + 3(209x + 362y))2 - 3(362x + 627y + 209x + 362y)2= -2

Which is: (989x + 1713y)2 - 3(571x + 989y)2= -2

And one more iteration to be able to divide off some factors:

(989x + 1713y + 3(571x + 989y))2 - 3(989x + 1713y + 571x + 989y)2= 4

Which is: (989x + 1713y + 3(571x + 989y))2 - 3(989x + 1713y + 571x + 989y)2= 4

Which is: (2702x + 4680y)2 - 3(1560x + 2702y)2= 4

So now can divide off that 4, where happens because 2 is the only prime factor of D+1, where is every two iterations, which is why is a special case. Wonder if will always get a cycle if D+1 only has 2 as a prime factor? Sounds like something to check with a computer. But might be mathematically provable also. Checking with computer first could test the hypothesis. Ok, so dividing off the 4.

And finally: (1351x + 2340y)2 - 3(780x + 1351y)2= 1

From which I have, sqrt(3) approximately equals xn+1/yn+1, where:

xn+1 = 1351xn + 2340yn

yn+1 = 780xn + 1351yn

and x0 = 1, and y0 = 0;

So get x1 = 1351, and y1 = 780. Which is a decent approximation, and next is:

x2 = 3650401 and y2 = 2107560

And 3650401/2107560 is approximately:  1.73205080757

Just using how much a web calculation showed. Another trick is to square, and look at what that gives. Where looking at prior post got more digits before.

And 36504012/21075602 approximately equals 3 is what my pc things are saying, so is too close for more with precision of the defaults. Have had LOTS more digits before but just playing around so not interested in fiddling with settings to find out why so few now.

Well guess at limits then of just being able to check with usual pc things. Oh well.

Well that was simple fun. Find it kind of relaxing. Just playing with numbers.


James Harris