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Thursday, May 24, 2018

Modular simplicity is fascinating

Have a much better handle now on a lot modular where recognize much of my own success with discovery has been because of modular algebra tools. Where discussed much recently as yet again explained tautological spaces, but now with a more instructional tone. And finally emphasized how represents a new discipline where the math does algebra for you. Which is really cool.

It is worth noting how odd math history becomes then, when you realize how much EASY was waiting for anyone who could discover. Like have gone on much about:

x2 - Dy2 = F mod N

where N can be prime or composite, and nonzero. Where I don't think it gets interesting at all until you reach N = 3 or higher, where I stick with positives. And just with THAT expression was able to figure out how to calculate the modular inverse, so have my own modular inverse method! Which means I get to mention myself with Euler and Euclid. Where extended Euclidean algorithm is how Euclid gets in there. While Euler and Fermat are the others who actually came up with their own ways to figure out modular inverses. And Euler simply extended Fermat who had an idea that only worked with a prime modulus.

Was cool idea anyway. I discuss simply with post: More overview on modular inverse

Given other things from my research is not surprising am also rethinking some notions that more are from others, like the idea that integer factorization is a hard problem. I fear it is NOT actually, with a proper modular approach.

Like have seen naive mathematical discussion looking at, say xy = C mod N, where C is some composite, ironically in my opinion noting that x and y are easily calculated with the modular inverse.

Um, yeah but you have to not know what you're doing to focus on that expression. And I guess they did not as to folks who put such things in mathematical texts. But how would they know what they did NOT know, which we know now? I like that sentence.

Advances in human thinking fascinate me now, so have ranged more widely. Having success in mathematics gives massive benefits. And simplicity with modular means much can be easily checked by others, which I think is a great thing.

Mathematics should be about what works.

Human beings can think all kinds of things.

With mathematics is possible to gain absolute certainty with absolute proof.


James Harris

Thursday, May 17, 2018

More on finding quadratic residues with modular inverse

Last month started pondering if one could find quadratic residues using the modular inverse.

Copying much from my post last month will update also as answer a question raised then.

Focused on x2 - Dy2 = F, where 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

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.

Get to set some variables, like simply set F = 1 mod N and will set y, and then find D, from:

D = (x2 - 1)y-2 mod N.

Noticed before that WILL work, but also wondered if could use BQD Iterator, to possibly get different values for m, and realized that MIGHT change r, but how to know?

From the start: 2x = r + Fr-1 mod N

The BQD Iterator will give next: 2(x+Dy) = r + F(-D+1)r-1 mod N

So can just solve. Then r = 2(x+Dy) + F(D - 1)r-1 mod N, so:

2x = 2(x+Dy) + F(D - 1)r-1 + Fr-1 mod N = 2x + 2Dy + FDr-1 - Fr-1 + Fr-1 mod N, so:

0 = 2Dy + FDr-1 mod N

Assuming D is coprime to N, which is odd, have then: y = -F(2r)-1 mod N

Which surprised me. But y is free here so can be easily set. Guess you get one iteration then per try?

For example, again let N = 119, and again start with easy with let r = 2 mod 119. Then also r-1 = 60 mod 119.

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

Now there is a change from before as y = -4-1 mod 119 = 89 mod 119

And 2m(89) = 2 - 60 = -58 mod 119, so: 59m = 61 mod 119, so: m = 116 mod 119

And D = (312 - 1)(16) mod 119 =  9 mod 119

And yeah 1162 = 9 mod 119, as required.

Now though can iterate with the BQD Iterator, knowing have same r, which then is way to handle that particular thing.

So next value for x is: 31 + 2(89) mod 119 = 90, and next y is: 31+89 mod 119 = 1 mod 119.

Then 2m' = 2 - 60(-9 +1) mod 119 = 2 - 60(-8) mod 119 = 6 mod 119, so: m' = 3 mod 119

Which is just the negative of the first one. But at least it worked! Cool. There may be potential here. But now I wonder: will it always simply negate?


James Harris

Monday, May 07, 2018

Quadratics easier with more degrees of freedom

One thing that DID surprise me was when found solutions were actually easier when I focused on more variables with a quadratic case:

c1x2 + c2xy + c3y2 = c4z2 + c5zx + c6zy

where the c's are constants. Years ago, actually shied away from and went simpler by setting z=1, to get my method for generally reducing binary quadratic Diophantines. And FINALLY got curious enough to look at the more general case, where handled in two key posts:

Three variable quadratic reduction

and

Trinary Quadratic Iterator

Where yeah my BQD Iterator is my primary workhorse still, where who knows really what the TQ Iterator adds and for the moment am not working to find out.

What is really cool to me though is how you can end up with quadratics easier to solve for integers. Like copying from my post giving the reduction method, let c1 = 1, c2 = 1, c3 = 1, c4 = 1, c5 = 1, c6 = 1, so:

x2 + xy + y2 = z2 + zx + zy

Which gives:

A = -3, B = -2, and C = 4

[-3(x+y) + 2z]2  + 3m  = 16z2

Where now have two unknowns m and z determining existence of a rational solution. But by inspection has an infinite number of rational solutions with m = 0. Still shouldn't just assume that then x and y always will be integers. Easy enough to check:

-3(x+y) + 2z = +/- 4z, so: -3(x+y) =  2z or -6z

So there will be a set of integer solutions for every nonzero integer z.

Where a trivial set is: x = y = z

But you can also find others.

Here's one a little more complicated. Let c1 = 1, c2 = 2, c3 = 3, c4 = 4, c5 = 5, c6 = 6, so:

x2 + 2xy + 3y2 = 4z2 + 5xz + 6yz

Where similarly can show that you can always find solutions with: x+y = 2z or 3z

Which means can always have integer solutions with an integer z, with one reduced constraint.


James Harris

Saturday, May 05, 2018

Modular inverse anniversary and reality check on modern discovery

Today marks a year from time I found out there was a direct way to calculate the modular inverse. Where yeah for people who wonder you now have a reality check on how hard it can be to get official recognition in modern world for discovery.

Yet has been HUGE in terms of impact on blog, where I have access to the web metrics where contrarily as usual the total numbers for visits don't shift much. So I end up focusing on things like linking behavior as reported by Google Webmaster Tools as one of the more important indicators.

For me? Is just more of the same as have been witnessing now for over a decade.

Have several major results at this level, and some well beyond. So I can cross compare.

For others? Yeah can notice how much you might have to do, if you ever wish to get official recognition for a mathematical result.

Consider, I found the first direct route to calculation of the modular inverse, which is the third primary way to calculate. My competition in this arena? Euler who figured out his totient function on top of a result known by Fermat, and ideas shared by Euclid related to calculation of the greatest common divisor.

So we are the three for the modular inverse now--me, Euler and Euclid.

Primary results at this level were supposedly all found, and apparently certain people who do not understand the web believe if they refuse to acknowledge they can discredit useful mathematics.

But we know better. The web has changed so much! Thankfully is the knowledge that is important and yeah, from multiple indicators that information rocked our world. And possibly changed much across the entire web itself. Useful mathematics gets used. Believe that.

My modular inverse is a 21st century result for a computer age, where information rules.

So cool. So marking first anniversary of the discovery of the PRIMARY way to calculate the modular inverse as human mathematics seems poised to shift into fully modular, for the first time in its existence.

Modular algebra rules numbers we now know.


James Harris