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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.

And also, 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

Friday, April 27, 2018

Considering modular impact

Once focus is on modular methods it does become clear that you get a HUGE difference in terms of mathematics. And does seem pertinent that Gauss had started much. Also Euler had done some things where usually am thinking more about Gauss myself. And then we get this remarkable pause, until the 21st century.

And I found out there was another primary way to figure out a modular inverse, where Euler had one, and another goes back to ideas shared by Euclid, where tried to explain those simply. And have talked my modular inverse discovery much! As was a telling result for me, nearly one year ago back May 5th, and shifted my thinking on other things as well.

Now can show that x2 - Dy2 = F mod N, is the key equation controlling behavior of integers along with my BQD Iterator. But modular goes much deeper as my first major result back in 1996 relied on for packing of spheres. And even what I say is an axiom reveals lots.

Like p1 mod p2 where just consider primes modulo each other makes no sense to claim they have a preference. And THAT by itself resolves the Twin Primes Conjecture and refutes Goldbach's, if accepted. Where did an overview recently and also one can web search: prime residue axiom

When consider fights that have emerged in this area, they aren't even interesting to me.

Human beings can be weird in predictable ways with thousands of years of known history. There is nothing to learn there.

Yet is worth noting that there are recent mathematicians who tried to erase Euler from his zeta function and hand to Riemann, who am sure would have been horrified at the attempt. While I noted a simple explanation for that question he was asking--which I found back in 2002.

Modular methods lead to a true modular algebra, which can manipulate algebra itself, and in so doing give VAST analytical ability to anyone who learns. The math does better algebraic manipulation than any human can manage. Who knew? That is actually kind of freaky. And I found out when improved on Gauss with reducing binary quadratic Diophantine equations, which I did years ago.

Have also talked some of my history with modular with a Google Group posting on my MyMathGroup, where checked and found out yeah, can link to it.

Good news is that the math apparently has been picked up rapidly. Explaining how THAT is clear to me feels kind of complicated and also kind of awkward. I think has to do with emergent reality which is in process.

That web enabled reality have discussed before. Is kind of interesting I guess. So social problem is not that big of a deal really in one way. However is still troubling in many ways.

How does it impact me? Um, I don't talk that either, any more. Have discussed in the past.

So modular was the next big thing in mathematics, and waited for quite a bit of time, until the 21st century to reveal just how big. Makes sense though.

Human thinking keeps advancing in certain ways, for some reason there does seem to be a logic to timing often, as modular?

Well modular is perfect for an age with computers. Even my story actually involves web much.

Interesting.


James Harris

Monday, April 23, 2018

Knowledge reality in age of the web

Knowledge does move fast in our times. And readily admit rely on that much with my own research. So yes, can now say definitively that I had one of the greatest mathematical discoveries, when figured out there was a simple explanation for why count of primes connects with continuous functions like x/ ln x and that ROARED, through math circles.

Then other things didn't happen and our world learned some things, while most humans did not.

My assumption was to assume I was missing something, which is pursuing simplest explanation which actually worked out great.

Then in 2004, when demonstrated with a published paper that with existing rules a person could create an apparent contradiction from a coverage problem apparently that roared too, through math circles. That coverage problem DOES allow mathematicians to fake mathematical discovery should note. And world learned more, but most humans did not.

And I assumed maybe I was missing something, or who knows, maybe was even WRONG. Which was the smart thing to do, so I pondered: How do you know mathematical truth?

And I came up with a functional definition for mathematical proof.

Is is SO cool. And is it unintuitive that I assume that I'm wrong and get MORE? Is one of the best things ever. In math it is better to work with the possibility you are wrong than to ever rest on certainty you are right. As math can be SO subtle.

If you ARE right, try as you will, your math will handle every attack. Math does not care.

So yeah, web spread the information widely and I learned could simply use blogs, as less and less relied on other means like Usenet. Though DID go on Usenet through Google Groups deliberately in the past to maximize wide distribution and lessen censorship ability.

When one looks at how others reacted to my discoveries is not telling about me, but is about them.

And there is some naive I think. Mathematicians who do not realize they were accurately judged from then on by their behavior, even if also judging their peers.

World apparently judged as well but assessing is harder. Nations can be so cagey. There is a use for academics even they may not fully understand. Do they really need to be correct?

But we DO understand emotion with truth. Is so telling really. Which confused me greatly for years as I got this JOY from answers. I really did want to know. Took quite some time for me to more fully comprehend how could there be these people who did not.

Where lately I noted I increasingly was worried less about correctness when proven as solidified tools for determining, than with trying to handle my global attention reality.

But so much faith maybe from some in a world they never understood.

Like you think recognition is from awards? That's funny. Or maybe you think the press coverage is what makes things happen?

There was a time that I did, will admit, which is embarrassing to me now. But is better to be honest I think. But how was I supposed to know better?

Took experience for me to learn.

And yeah, if you think brilliant mathematics and great discovery is primarily about winning awards or getting covered by press people then you have no clue why and how math discovery works.

And should those of us who do know inform such people who do not? In the past I thought yes, with persistence. Now am like just share well as one can--and trust, as now am more interested in talking to others who know.

We know.


James Harris