Re-cap, why factoring is done

I'm going to ****ft variables to give you all something more familiar
with factoring and explain the significance of a remarkable and simple
congruence result:

Given

x^2 = y^2 + cN

where N is a target odd composite to factor, not divisible by 3, and c
is a control variable that equals 1 if N mod 3 = 2 and 5 if N mod 3 =
1, it can be trivially proven that

x^2 = 8^{-1] (9cN) mod p

when 8^{-1] (9cN) is a quadratic residue for an odd prime p, where p <
2x/3.

For some of you it may be an extremely counter-intuitive result or
terribly hard to understand, so I'll give an example and then talk
about why it's significant.

Let N = 299 = 13(23), and since 299 mod 3 = 2, c=1.  The only primes
available (I'll leave the proof to the reader) are 3, 5, 7, 11 and 13
but since I know 13 is a factor I'll try, 11.

With p = 11, I check to see if the quadratic residue exists:

x^2 = 8^{-1} (9(299)) mod 11 = 7(7) mod 11, so that's easy and x = 7
mod 11 or -7 mod 11.

For those who wonder about the modular inverse 8^{-1} mod 11 = 7
because 7(8) = 1 mod 11.

Just knowing that x needs to be greater than p, you could try 11+7 =
18, and

y = sqrt(18^2 - 299) = sqrt(25) = 5, as I only care about the positive
solution.

So I have (18+5)(18-5) = (23)(13) = 299, and a factorization with this
approach.

The result is significant for two reasons:

1.  No one knew in mathematical history that given x^2 = y^2 + cT, you
could get x mod p, where p is an odd prime less than 2x/3, when x is
forced to be divisible by 3, and it is a "pure math" result with
amazing significance there because you have these prime numbers coming
in out of the blue in this remarkable way.

And prime numbers are cool.

2.  Given x mod p with successive primes, you can get x exactly, which
means you can factor T.

That is, if you have something like:  x mod p_1 and x mod p_2 and x
mod p_3, you can find

x mod p_1*p_2*p_3

and if p_1*p_2*p_3 > x then, of course, you have x exactly.

So the short of it is, if it had been known early on that you could
get x mod p then factoring would NOT have been thought to be a hard
problem, so it's that significant.

From a socio-political aspect the result is significant because it's
easy to derive, so easy to prove and has extraordinary social impact
when developed into a fast factoring method so there is no way that a
healthy mathematical society would not inform the world about it (and
start celebrating, cheering and being generally happy that such a
remarkable and beautiful result exists).

That isn't happening yet and my guess is that part of it is that
mathematicians hate me and they're political animals so they are
playing politics with the result.  Ha ha.

I laugh because they can only get away with that for so long and then
their funding can be taken away---all of it--in "pure math" areas so
these people will be looking for new jobs soon enough.  Ha ha.

Oh, so how do you derive the result?  Easy.  I've given the derivation
in other posts with different variables but wanted to use more
familiar variables to help some of you understand.

I have other mathematical research having proven Fermat's Last
Theorem, found the prime counting function, delivered a prime gap
equation, and if you Google "definition of mathematical proof" you can
find my definition in the top 10.

Mathematicians hate me because I say many of them routinely lie
because they do lie and they do it for money and prestige and because
doing real mathematical research is harder than lying about your
research and playing pretend.  Ha ha.

Part of the purpose of postings like this one is to remove their naive
student sup****t, which will hurt their feelings very much because they
are actors, playing at being mathematicians so it will bug them for
you to look at them like they're crappy human beings, which they are.

And to me that actually is such a fun thing to contemplate that I have
a sense of satisfaction already.

After all, Ribet or Andrew Wiles really just want butt-kissers which
so many of you have been for so many years so the shock of that tragic
look in your eyes will devastate the con artists, which they deserve.

Merry day!


James Harris