Harristotelian Logic

It is remarkable to me that a technique I first introduced to attack Fermat's Last Theorem has now been key to solving the factoring problem, which is a technique of adding extra variables.

So with FLT I ended up adding a new variable v, and to solve the factoring problem I needed two new variables, k and p, where p is an odd prime.

What you see now in my talking about the solution is an extensive refinement on techniques I've figured out over years where the odd thing now is that there is so much room to simplify so I can give the solution to the factoring problem very succinctly so I will do so again now.

Prior attempts in the math field had the start right:

z^2 = y^2 + nT, or z^2 = y^2 mod T

but failed to realize that adding a few variables blew the problem away as in solved it very quickly:

x^2 = y^2 mod p

2x = k, and z = x+k

I actually came across this approach by deliberately looking for ways to factor one number with another, an approach I christened surrogate factoring, and I've deliberately looked for methods that had completion of the square for some time but did not realize how simple it all could be.

So you go from 2x = k, to 2xk = k^2, and add it to x^2 = y^2 mod p to get

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

and then complete the square by adding k^2 to both sides to get

(x+k)^2 = y^2 + 2k^2 mod p

and now simply enough with z = x+k, you have that

nT = 2k^2 mod p

and doing the substitutions you also have that z = 2k/3, so everything is connected trivially and you have a way to find z modulo p just like that.

For years I've worked on the question of how to find k, or how to pick k as in previous research I'd often set it, but now realize that k is determined and you look for it using

k = 2^{-1}(nT) mod p

where next the issue came up of how to pick p, which I now know is an odd prime.

It took some time and basic research involving theory plus experiments to finally figure out that there was a limit on the size of p, where I erroneously for some time thought that the size of factors f_1 and f_2 where f_1*f_2 = nT determined the size of p, but finally realized that additional congruence relations gave the answer as

f_1 = k mod p

and

f_2 = 2k mod p

so easily enough, k must be greater than p—I keep k a positive integer—as otherwise there would be a contradiction as k would be a factor of nT and of z when z^2 = y^2 + nT.

Given those rules it's then trivial to come up with what is probably the best factoring method possible, which involves picking a large prime p near the limit in size which is at the minimum k possible which would only happen with nT a perfect square, which means a prime near 2sqrt(nT)/3.

Since you need a quadratic residue of that prime, there is a 50% probability that any given odd prime p will work.

Oh yeah, about the variable n, it is one other addition I forgot to mention which is just there to make z divisible by 3, so if T mod 3 = 2 then n=1, but it can, equal, say 5 if T mod 3 = 1, so its size impact is nominal.

Remarkably as you search for k, which would mean picking the first even k near the minimum that has the residue modulo p required by your prime and then iterating by 2p, you can, of course, pick additional primes as well as you increase the size of your test k's, since the rule is that p be less than k.

And THEN of course you can loop modulo more primes as if you have

k mod p_1 and k mod p_2

then of course you can get k mod p_1*p_2

which means that with theory alone you the reader now know this method will be faster than anything else previously known.

Since I know that I previously found a proof of Fermat's Last Theorem using an advanced technique which math society ignored, though I managed briefly to get a key technique published, and I have my prime counting function discovery with its remarkably unique features which math society ignored, and my prime gap equation which math society ignored, plus my work on logic which math society ignored I am
comfortable in saying that the modern math field is completely corrupted.

For instance, supposed great research works are proven by my research to be flawed where I'm still amazed by the many ways I could refute the work of Andrew Wiles where his supposed proof of FLT fails many ways including the use of a simple logical fallacy called cum hoc, ergo propter hoc.

The best explanation for what has happened to the math field is that in esoteric "pure math" areas, people have simply done fake research and claimed it was otherwise, feeling safe and secure in that because the research had no practical value. They clearly believed that the charade could be maintained indefinitely.

But they had two problems to handle:

1. Computer science advances could long ago have introduced computerized checking of most or all claims of mathematical proof.


2. The emergence of a major researcher who could point out numerous errors.


3. So they handled each problem in turn in various way—attacking the notion of computers being able to check math proofs and building a system that would allow them to just ignore major research findings.

In addressing the issue of how to handle the blocks they raised I determined that I had to move from "pure math" areas to a practical arena where they could not simply deny a result so I turned to the
factoring problem.

That problem is now solved as explained in this post.

The current delay in acceptance is all about the corruption of the modern math field.

My take on the situation is that most modern number theorists rarely if ever tell the truth, especially about math.