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 *** 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.
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.
James Harris
|
