> Oddly enough to me the most fascinating find from surrogate factoring
> which has created the means to end the impasse is a remarkably simple
> result that follows from a relatively simple equation:
>
> x^2 = y^2 + nT - (1 + á^2)(k_0^2 + 2pjk_0 + p^2*j^2) - (k_0 + pj)pr_2
>
> That is the equation that comes from letting 2áx = k + pr_2, and z = x
> + ák, when
>
> z^2 = y^2 + nT
>
> and considering k = k_0 + pj, to see how
>
> nT - (1 + á^2)(k_0^2 + 2pjk_0 + p^2*j^2)
>
> behaves as you increment or decrement k with j.
>
> So actually I just kind of expanded out the traditional difference of
> squares.
>
> Um, that's what they call thinking out of the box.
>
> And you have trivially that as j increments OR decrements, r_2 will
> tend to be negative to compensate, so
>
> nT - (1 + á^2)(k_0^2 + 2pjk_0 + p^2*j^2)
>
> will have a minima and change around that value as j increases which
> is just an incredibly powerful result as it allows you to to get an
> idea of the value of z.
>
> So the approach to the factoring problem is really tackling finding
> how to get z, when
>
> z^2 = y^2 + nT
>
> and all those variables are just helpers in that task.
>
> You're just trying to get in the neighborhood.
>
> And it can be shown that if x, y and z are rational
>
> k^2 = (1 + á^2)^{-1}(nT) mod p
>
> so you can go looking for z by looking for k, where you can get k's
> residue modulo p, an odd prime.
>
> Qualifications are few.  Yes, for a given choice of p, x, y and z
> rational may not exist such that all equations are satisfied but you
> can try different primes.  Um, there are, after all, a LOT of primes.
>
> So you have prime numbers as helpers that disappear after helping you
> to factor, and you have a surprisingly simple result with a parabolic
> minima, and you get quadratic residues, and it's the factoring problem
> and I've been talking about this latest research for days, and still
> math society waits...
>
> Um, could REAL mathematicians wait?
>
> Would Gauss or Euler or Fermat?  Archimedes?
>
> My place in history is secure even though I know a lot more than most
> of you clearly know so I also know that there may not be much history
> left!  Not human history at least.
>
> But not understanding is what this situation is all about, as some
> people didn't understand that lying about math would invite the
> retribution of the math because they didn't believe in mathematics
> itself.
>
> The poor field was overrun by people who hate math but found a way to
> work the system by lying.  That's all.  Nothing more.
>
> But without advancement in mathematics, humanity has no future, so the
> Universe will just kill off the species as no longer of further use.
>
> By stopping mathematical progress, these people removed a key element
> in the purpose of the continued existence of the entire species so the
> clock is ticking down faster than any of you can imagine because
> you're too dumb to realize that if YOU lied and got things wrong, why
> couldn't others have?
>
> Guess at how many years are left, and I'm sure you'll be wrong.
>
> Yup.  The test of humanity was a subtle one but it was very fair.
>
> It was all about mathematical absolutes.
>
> James Harris