Wednesday, January 25, 2017

When it seems impossible

For those who read this blog you may realize that a conclusion I reached is that I found some new numbers previously not cataloged which should be impossible. And reality is I realized that back in 2003, and yeah to me is good to be at a full explanation level so quickly, now only around 13 years later.

But that was the reason to figure out the object ring where a post talking it is the FIRST post on this blog.

My feeling is that the assertion is so unbelievable that there are numbers that were missed that the greatest skepticism possible is required.

So yeah, one dead math journal in this story? Not a big deal considering.

But is it true? The mathematical proof seems solid to me! But that's not enough.

James Harris

Sunday, January 15, 2017

Refreshing on old for me concepts

So yeah have one published paper, where also make sure to note that the mathematical journal tried to literally pull the paper AFTER it was published. In that it was an electronic journal and the chief editor tried to go with just deleting my paper and saying "Withdrawn" in the Table of Contents, though I never withdrew it. That of course isn't how it's done. And was relieved to find that as far as the mathematical establishment rules go, I have one published paper and am considered to be a published mathematical author, so am listed as one.

That paper is also kept available by EMIS, and have talked such things before, where here will refresh with what I feel like are old concepts for me, as was over a decade ago all that drama. Am so grateful for those who DO follow the rules. And am a person who works to follow the rules, as it actually matters most for me.

The paper relies for analysis on x+y+vz= 0(mod x+y+vz), which it does not state, which I later started calling a tautological space, one of my favorite made-up expressions, which is part of my own discipline of math innovation I decided to call modular algebra symbology.

For my paper I used a cubic but later simplified to quadratics which looks like it happened around February 2006, as realized same approach SHOULD work if the mathematical ideas were not flawed.

There was concern and then lots of relief as mathematics continued to behave as expected, and was so much easier working with quadratics. Learned the lesson: when you can in mathematics, always take opportunities to simplify your analysis.

Most posts talking quadratic non-polynomial factorization on this blog actually rely on the condition that:

x2 + xy + y2 = z2

And I talk through how I analyze it with post: Under the hood

With that conditional with the tautological space I would use v = -1 + mf. As v is free to be ANYTHING I want, and I figured out, which is the art to the mathematical science, that analysis went a certain way I liked with that choice. Further along in analysis, would bring back x, with m = x. Why? It occurs to me that I just like having x around. Since I can with this approach, I do.

And z would get cleared out with z=1, but would keep f visible but integer with f = 7, as liked getting to some numbers. It does help looking at the expressions. So end up with two variable quadratics which can be solved for functions I'd call 'a' where a = y. Which I guess may be because I liked the way that looked better. The functions end up being a1(x) and a2(x) and am sure that looks prettier to me for some reason.

If you stare at mathematical expressions for years? It matters how they look to you. I've studied in these areas now for well over a decade and like my foresight in picking variables that sit well with me. I came up with tautological spaces back December 1999, but took some time to call them that, so yeah, helps to pick well.

And in fact I came up with tautological spaces JUST so I could have a variable I could make anything I want in order to analyze expressions in a different way. Then I could deliberately probe.

Have another post which is even simpler talking tautological spaces with x2 + y2 = z2, where didn't find anything I thought mathematically interesting so use it as a teaching example of absolute proof. As explains carefully how you KNOW the proof is absolute.

And probing x2 + xy + y2 = z2 turned out to be very interesting which is good as I picked that equation for its simplicity while notice I had to get SOME complexity as can compare with the same analytical approach done on x2 + y2 = z2 where I didn't notice anything as interesting.

But there is an art to it, which is why I emphasize that I didn't notice as doesn't mean there isn't more there, as v is an infinity variable. And also I used the simplest tautological space, but there are an infinity of them, though my approach means there probably are a finite number that can sensibly be used with any particular equation, I guess.

And that appeals to me as the mathematical science, where for some questions you have to go exploring to get the answers.

Remarkable thing though is that while you can show a problem with some traditional mathematical thinking with analysis using tautological spaces, there is also a direct route which is simple and doesn't require mentioning them at ALL.

There I finally realized could just consider the generalization of a factorization of a quadratic polynomial P(x):

P(x) = (g1(x) + 1)(g2(x) + 2)

where P(x) is a primitive quadratic with integer coefficients, g1(0) = g2(0) = 0, but g1(x) does not equal 0 for all x.

From there I easily get:

k*P(x) =  (f1(x) + k)(f2(x) + k)

by multiplying both sides by k, and introducing new functions f1(x), and f2(x), where:

g2(x) = f2(x) + k-2 and g1(x) = f1(x)/k,

Where my innovation was to multiply both sides by some constant integer. It's so EASY. So I speculate that multiplying by some constant factor is so against the grain of trained math people this path was just waiting for me. Standard training of course is to remove extraneous factors.

If k = 1 or -1, then you can get algebraic integers for the g's and f's. Easy.

That was HUGE to realize, as yeah you CAN solve for the g's easily enough using my approach, if k = 1 or -1. The unary case is specially unique here.

But if k is 2 or 3, for instance, the f's can still be algebraic integers, but also you have:

g1(x) = f1(x)/2, or g1(x) = f1(x)/3

And usually that can't be an algebraic integer. But still exists, so what kind of number is it?

Remember: P(x) = (g1(x) + 1)(g2(x) + 2)

And that does NOT know if you multiply it by k = 1, -1, or 2 or 3, or whatever. So do not make the mistake of figuring that based on your human choice the g's shift dynamically. They know not nor care what you do or what you think! The math just is.

So what does shift? Depending on what YOU choose for k, you're probing the factorization in different directions. If k = 1 or -1, you're looking at algebraic integer solutions for the g's. When you shift k, the analysis approach used starts showing you the other numbers.

It's about my techniques for analysis. Like analysis method is like the microscope, and k is the magnification? Cycle it up from 1 and -1, and you start peering into another number theoretic world. I use a special function I call H(x) to get a handle on things.

It took years of pondering from a different direction though, as lots happened from that wild story with publication in 2003 until I came up with what I see now as a very simple algebraic argument with quadratics using elementary methods and most dramatic innovation being multiplying by a constant integer factor.

So it took only around 13 years to explain it all, which is ok.

James Harris

Sunday, January 08, 2017

My partial differential equation connected to primes

Sometimes answers can be kind of just there. Like for a LONG time there was a puzzle over why the count of prime numbers could have ANY connection with continuous functions. But I found an easy answer with a partial differential equation I found that follows from my way to count prime numbers.

My prime counting method leads to a difference equation from which I can get to a partial differential equation:

P'y(x,y) = -(P(x/y,y) - P(y, sqrt(y))) P'(y, sqrt(y))

I talk more about in an earlier post I like for reference, and will admit maybe should discuss it more, given its implications but feel like am primarily repeating myself.

Gist of it though is that there is a sieve form for a prime counting function that counts by calling itself. That allows for a difference equation, which does the same, but is a difference equation as it can tell itself when a number is prime. And that leads to the partial differential equation which is the continuous function.

Here's a post I made in 2009 which shows them all together. I rarely do that for some reason.

So past mathematicians were approximating the integration of the partial differential equation, with no clue. But it's a simple, succinct explanation for how the count of primes could be approximated by a continuous function, which isn't exciting. I like to say, it isn't math sexy.

It's like if Gauss were around I could explain things that he died not knowing, but did he miss much? I'm not sure. Or Euler or even Fermat maybe, oh and definitely Riemann. Getting THE answer, would it have been worth it to them? I like to think, yes. But they died with the prime count connection to continuous functions a big mystery, and not something easily explained.

But no, no dramatic, extraordinary and profound possibly mystical explanation needed for prime counts connected to x/ln x or Li(x) or any continuous function. Just a rather mundane, simple and straightforward mathematical explanation.

In our time the answer doesn't seem to get much excitement going, which is ok with me, though also puzzling. But I know that sometimes answers can be, well just not that exciting. I find that intriguing to some extent, but it's just one thing I have. So um yeah, I explained the primes connecting to continuous functions thing, years ago, and moved on to other things.

There is no other like it though.

And there are clues to how big the partial differential must be.

There are no other known partial differential equations that follow directly from a method to count primes. It comes from a difference equation which is ALSO unique in that there are no others that can count primes.

Yup, worth saying again, the difference equation counts prime numbers, when properly constrained. There is no other that does.

Finds them on its own which rattled me for years. Shifted how I looked at the math.

It is one of my favorite discoveries. Also one of the easiest of the BIG ONES to check I think. And one that gives a LOT of perspective.

James Harris

Some assessments on math status

Figured may as well talk some things I now consider boring, as way back had vague notions about how things might go if found interesting and important mathematics. As firmly believe found some, hence the name of this blog, is worth talking reality versus that fantasy?

Most important shift that happened years ago was I stopped trying to write math papers for mathematical journals. Turns out a LOT of what mathematicians learn in school is how to write papers, um am thinking is a lot as didn't take math courses beyond needed for my physics degree.

Regardless, sent some papers and even had one published, so can say, yeah there are certain expected formats and I am NOT a mathematician and do NOT want to become one.

Of course a reason to bother would be to get my own math known if the web weren't around. But the web is here.

So how do you know my math works if you don't have some mathematicians to tell you?

If you need someone to tell you if math is correct or not, then you probably are at the wrong blog.

Of course there is no choice with MOST things in human life. You are forced to trust that people are telling you the truth in so many areas. But not in mathematics! Woo hoo!

In mathematics, you can check.

And remember, you can take anything here to a mathematician if you wish.

Mathematics at its best, when it is true mathematics, is perfection.

Gave me lots of leeway though will admit. Some things I did just to see if I could.

I do puzzle on such things sometimes, but less and less as the years go by.

Reality now, is I have greater reach with just this blog which is not my only one, than I expected to ever have, just on my own. There really isn't anything to add there.

James Harris