Tuesday, March 25, 2014

Mainstream concerns

One of the issues that I worry about at times is the concern of possibly leading math students astray especially younger ones, so it's worth noting how I've acted with that concern in mind, which can explain some things which may have puzzled readers of this blog.

For instance it was originally titled "My Math" which I thought was an emphasis on these being original ideas of a particular individual, which at the time seemed like a way of stressing that I'm NOT part of the mainstream mathematical order. However, noting web traffic I noticed that possibly I had created a situation where kids could get directed here when they were trying to get to a site built for them:

At the time I was still trying to process web stats from Google Analytics telling me this blog was getting annual readers from over 120 countries. It just seemed strange and didn't connect back into my regular life.

I was also dominating search results with that title which to some extent I now dominate search results, at least on Google, with the current one. So it DID seem to be an issue I should try to handle and I broke the possible mis-direction connection with the shift from "My Math" to "Some Math" and problem was solved.

The other benefit was that after years of pondering how I could possibly be read in over 120 countries with it not seeming to matter much and with domination of search results in math areas I had the distinct pleasure and sense of disquiet when with the new name I looked at blog stats of 0.

No visitors at all, which was of course expected.

At that time there were no links to this blog except from my other two blogs (since removed), so there was no reason for there to be traffic. People didn't know where it was. It was in that sense new with the new name.

And then I could watch as traffic slowly began to arrive and watch as the country counts built up but they never returned to the previous level. For instance last year Google Analytics just told me as I just checked that there were visits from only 56 countries. So my name change cut the country count in half, whatever that means.

Also ALL my domination in web search for the site was reset, so all went away and I could watch to see what of my research returned to search domination. For instance my research on reducing binary quadratic Diophantine equations did so nicely.

In my latest posts I've made a point of emphasizing I'm not a mathematician. This site is not meant to replace mainstream mathematics sites and it's not meant to direct math people away from them. I think it can augment mathematical understanding for people with a thorough foundation in mathematics who should be able to test what they read most effectively.

In my view I am doing what's necessary to be a good citizen on the web with ideas that some might find worrying or disruptive.

For those who wondered about the name change and have kept with this blog through it, maybe the above will finally explain something that might have puzzled, and can give confidence that it won't happen again.

I'm actually giddily happy with the name change and was lucky it was available for me. I think it fits better, and is more appropriately focused less on me than on abstraction as a math site with some math on it.

James Harris

Monday, March 24, 2014

My top three results

When I worry I think about 3 mathematical results I find most compelling, which can maybe help visitors to this site understand. And I'll go in reverse order from number three to number one:

3. Reducing binary quadratic Diophantine equations

Turns out that equations of the form:

c1x2 + c2xy + c3y2 = c4 + c5x + c6y

where the c's are known and x and y are unknown are called binary quadratic equations, and Diophantine means you're looking for integer solutions. An example of such an equation is:

x2 + 2xy + 3y2 = 4 + 5x + 6y

It has solutions x = 4, y = -2, or x = 5, y = -2, which I found from a simpler form:

(-4(x+y) + 10)2 + 2s2 = 166

Found using my own research. 

Few people can claim to have improved on Gauss but my method is more straightforward than prior techniques where you need to use a discriminant. Simpler, easier--it is then the best in the world:

That seems good enough for it come in at #3 on my personal scale of most compelling results.

2. Modular solution to binary quadratic Diophantine equations

For me one of my most surprising results because of its simplicity came from just noticing something rather obvious with equations of the form: x2 - Dy2 = F

I realized that in modular arithmetic you could always factor and in so doing solve for x and y modulo some N:

x2 - Dy2 = F = (x - my)(x + my) mod N

Where: m2 = D mod N

Given any nonzero integer D, there exists an N for which it is a quadratic residue.

Find r, any residue modulo N for which Fr-1 mod N exists then:

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

One of my easiest results to derive it's also one that could have been known in the time of Gauss, so why am I the guy talking about it in the 21st century?

That's just weird. I keep wondering about this one, highly suspicious that it's already out there.

But then I can do some things with it, like in this blog post also a paper:

For those reasons it easily comes in at #2.

1. Non-polynomial factorization

Who knew that breaking from the mold with factoring polynomials would turn into a wild adventure. This result of mine is easily the most controversial as well as most tested, well-worked with the wackiest wildest story. Everything about it, including that the very first post on this blog is related to it, make it by far the #1 most compelling result.

You see, I got bored with polynomial factorization and figured out a way to factor a polynomial into non-polynomials:

7(175x2 - 15x + 2) = (5a1(x) + 7)(5a2(x)+ 7)

where the a's are roots of

a2 - (7x-1)a + (49x2 - 14x) = 0

The techniques used to derive my method for reducing binary quadratic Diophantine equations were developed deriving that non-polynomial factorization. Reality is, I played with binary quadratic Diophantine equations to build confidence in those methods. Things felt that weird. I had to find something else to build confidence. Those techniques are my most ridiculed and criticized of all. And represent possibly the area where people come out of the woodwork with the greatest effort to discredit anything I have. So there is no doubt about that position at a solid #1.

And those are my personal top three. Others might have a different set, but for me, these are the ones.

James Harris