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:

x

^{2}+ xy + y

^{2}= z

^{2}

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 a

_{1}(x) and a

_{2}(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 x

^{2}+ y

^{2}= z

^{2}, 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 x

^{2}+ xy + y

^{2}= z

^{2}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 x

^{2}+ y

^{2}= z

^{2}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) = (g

_{1}(x) + 1)(g

_{2}(x) + 2)

where P(x) is a primitive quadratic with integer coefficients, g

_{1}(0) = g

_{2}(0) = 0, but g

_{1}(x) does not equal 0 for all x.

From there I easily get:

k*P(x) = (f

_{1}(x) + k)(f

_{2}(x) + k)

by multiplying both sides by k, and introducing new functions f

_{1}(x), and f

_{2}(x), where:

g

_{2}(x) = f

_{2}(x) + k-2 and g

_{1}(x) = f

_{1}(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:

g

_{1}(x) = f

_{1}(x)/2, or g

_{1}(x) = f

_{1}(x)/3

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

Remember: P(x) = (g

_{1}(x) + 1)(g

_{2}(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

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