Back December 1999 found myself excited about math again. Earlier had felt horrible after months with yet another failed attempt at solving an intriguing problem from Fermat. The mere mention of it seems to be a harsh negative, which puzzles me. Regardless having decided all elementary method paths available to me were checked, had wondered, how might I make more?
By December had my answer, would use identities, where could deliberately add another variable, which would call 'v' for victory...well also was available. But could have used 'w' for win. I think I considered but liked simpler. After all, w really is double v. And had:
x+y+vz = 0(mod x+y+vz)
What's interesting to me is that I had it that way from the start, once had something that actually worked. Took about two weeks if I remember correctly to figure out something that did. Like I discovered I needed an asymmetric form. So yeah there was research AFTER the basic idea as asked myself: how exactly do I add another variable? And figured it out.
The specific trend that has made the difference in my research am thinking, is that I look modular, first.
Which realized recently. So am usually looking for a modular angle, I guess. As yes, thinking back when I finally realized an asymmetric form wrote it like I do now. So from FIRST useful version was the current version.
And then after talking out on math groups, when faced criticisms would switch to the explicit:
x+y+vz = x+y+vz
That is telling I realize. It is interesting the things that can make such a huge difference just based on which way people tend to go. By focusing modular, I simplified and ended up creating a true modular algebra.
Is SUCH a simple idea. You take an identity, and subtract an equation THROUGH the identity, where with my use of modular algebra, you end up with a residue, which has all the properties of the original expression, as it must. And made an absolute proof demo post where prove that:
If: x2 + y2 = z2 then: (v2 - 1)z2 - 2xy = 0(mod x+y+vz), where v can be any value.
That is compact as well. Which is all about going modular. Without that modular approach would not be practical as the expressions are then too complex to be useful. And of course would play with values. Like had not realized before that 2xy then has x+y+z as a factor, and also x+y-z, but then could easily prove with simple algebra. Just not the kind of thing playing around had noticed otherwise.
That v can be any number keeps being fascinating to me. Like let v = i.
Then: -2z2 - 2xy = 0(mod x+y+iz)
Which is then easily verified, by algebra.
Still would like looking at numbers, like: -74 = 0(mod 3+4+5i) = 0(mod 7+5i)
And of course: 74 = (7+5i)(7-5i)
And back over 18 years ago it all seemed kind of odd. Would do example after example where would do with modular algebra and also would go back through and show everything explicit.
Now I simply note: invented my own math discipline.
So along with geometry, algebra, calculus and other math disciplines there is also math of tautological spaces. And yeah study of tautological spaces relies on modular algebra, but is like, calculus relies on algebra. And how did that happen really, do ponder. Guess...just is hard to process.
Should I admit that ALL attempts with that basic tautological space to probe the key equation from that famous proposition by Fermat failed? There is more to that story, as eventually expanded BEYOND the basic form, in only situation where felt so forced. Later developed a full methodology, and even considered how might work with calculus:
Tautological spaces and calculus
But yeah was a challenge for me. Worked at it for a couple of YEARS too.
Is interesting, when you keep with something even when it doesn't do what you want.
Somehow kept faith in the basic idea, maybe because it just seemed so cool. But for a bit had nothing to show for the effort. Luckily later found other more interesting things anyway. Oh yeah, with the Fermat thing eventually thought had something with a more complicated tautological space, but couldn't nail down the ending, and now agree with Gauss that the proposition is not really of much interest in and of itself.
However, yeah got me motivated to figure out my tautological spaces. Can get philosophical in this area. You do wonder about the ways you get to some place.
And have had 18 years to use as I wished against some problems, where got most of my cool results to talk about by analyzing a general quadratic, first with two variables, and more recently with three variables. Does make it easier when you are the one person you know of doing such research.
Was in complete control with no pressure. Still am.
Is weird though, but hey, is a huge thing and readily have admitted have been lucky. And introduced first true modular algebra, where is so intrinsic was no way would be using congruence sign all over.
Oh yeah technically though: x+y+vz = 0(mod x+y+vz) is an equivalence. One of my first early results which kind of feels profound.
The things that change your life. And for me a natural path, which reached for, without even thinking about much, completely changed mine. That modular reach was itself profound I now know. And thanks to it would find so much.
Having your own mathematical discipline does change you too, am sure. There is just a different way I now realize I began to look at so many other things.