Harristotelian Logic

The start of my major mathematical successes came in December 1999, when in pursuit of more variables to chase after a short proof of Fermat's Last Theorem I first wrote:

x+y+vz = 0 (mod x+y+vz)

where I fiddled with various forms, before deciding to stick my new variable v next to z.

Introducing the concept on math newsgroups to howls of derision, one of the more remarkable objections was that it was an invalid expression despite my explaining that is equivalent to

x+y+vz = x+y+vz

so it's an identity. So my big idea was to start with an identity, but use it with congruences.

So how do you use it? Well you subtract out some expression from an expression derived from it, and analyze the residue, which I now call the algebraic residue.

So for instance, say you want to analyze x^2 + y^2 = z^2, to keep things familiar.

x+y+vz = 0(mod x+y+vz), so

x+y=-vz(mod x+y+vz)

and squaring both sides gives

x^2 + 2xy + y^2 = v^2 z^2(mod x+y+vz)

and now you can finally subtract out what I call the conditional expression as

x^2 + 2xy + y^2 = v^2 z^2(mod x+y+vz)

is still an identity so it is always true, and you get

2xy + (v^2-1)z^2(mod x+y+vz), which is

(v^2 - 1)z^2 - 2xy = 0(mod x+y+vz).

And that is the algebraic residue. It is only true if x^2 + y^2 = z^2, so for instance, with

x=3, y=4, and z=5

(v^2 - 1)(25) - 2(3)(4) = 0(mod 7+5v)

and you can trivially prove that is always true without regard to the value of v.

But that is a trivial example while you can do more with more complex examples and in fact all the controversy generated by my paper that went to the now defunct journal SWJPAM, was over an argument using

x^2 + y^2 + vz^2 = 0(mod x^2 + y^2 + vz^2)

with a variant of the FLT equation with p=3, as I think I used something like x^3 - xy + y^2 = z^3, and subtracted to get a residue on which I did some analysis and the rest is history.

Now then at the fundamental level of developing core mathematics what I've done is extend Gauss's idea of using congruence relations in a rather natural way, as of course, if you put in numbers for the variables you just get regular congruence relations. Where to me one of the more remarkable things is that I've introduced using identities that are in ways more complex than the other expressions used with
them.

Using identities of course is common as I showed the familiar completing of the square with my post on factoring where with

x^2 + 2xk = y^2 + k^2 mod T

a more traditional congruence where I put things slightly differently as I put the () around my more complex congruence relations as I think it looks neater—you can of course use k^2 = k^2 to complete the square:

x^2 + 2xk + k^2 = y^2 + k^2 + k^2 mod T

and that is a traditional way to use identities where the identity is LESS complex than the expression it is used with as you have one variable versus four—x, y, k and T.

In contrast with x+y+vz = 0(mod x+y+vz) you have MORE complexity with the identity as you have 4 variables, x, y, z, and v, while the expressions to be subtracted out that I use have only 3, x, y and z.

I say it like that as I don't see why you couldn't subtract out expressions with more variables or whatever as it's new research that I've pioneered and I just use it a certain way. More brilliant minds may find other ways.

The more astute of you who know your math history, know about Gauss's research on congruences, and what he introduced, as well as what is done in modern math in this area may realize that I have taken the position as a successor by extending the concept of using congruences in this way, and there really is nowhere else to go with congruences. So it is finishing research completing what Gauss started.

Now the algebraic residue is interesting as it is only true when what I call the conditional is true, and it must be true if the conditional is true.

And what happened was years ago feeling a bit down about my failed attempts at proving Fermat's Last Theorem, I thought to myself that what I really needed were more variables!

I'd been playing with x, y and z for years and had done every approach I could think of, and in desperation deliberately set out to introduce more variables, so I invented what I now call non-polynomial factorization as a direct result of necessity.

Oh yeah, I call x+y+vz=0(mod x+y+vz) a tautological space. It is an identity so it is always true. And I call x^2 + y^2 = z^2, a conditional, as it is not always true but is conditionally true depending on the values of x, y and z.

So I linked logic, calling identities tautological, to mathematics, and that is a still unexplored branch of this approach.

Now then I have research that covers a lot of territory, but this area is the biggest as any equation or expression available by traditional mathematics can be analyzed by subtracting it from a tautological space and poking at the algebraic residue.

So it encompasses all prior known mathematical research, and extends it.

More than enough for massive resistance. More than enough for a dead math journal.

More than enough to mean that human progress is stalled while the resistance continues no matter what anyone else says or how much they disagree.

Quite simply, while the impasse continues, humanity is not progressing in basic number theory research at all.