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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