Re: JSH: SF: Finally, surrogate factoring



Tim Peters wrote:
[Rick Decker]

...
So completing the square w.r.t y first yields

(2*y + 10*z + 5*f_1 - f_2)^2 = (4*z + 3*f_1 - f_2)^2 + 4*T

Completing the square w.r.t. z first yields

(42*z + 10*y - 3*f_2 + 19*f_1)^2 = (4*y + 3*f_2 - 5*f_1)^2 + 84*T

and rewriting both these as differences of squares yields the same
(useless) factorization of T:

T = (y + 3*z + f_1)(y + 7*z + 4*f_1 - f_2)


Macsyma agrees ;-)


and it's not hard to verify that

g_1 = y + 3*z + f_1

g_2 = y + 7*z + 4*f_1 - f_2


Given that James started with (although it was obscured by the presentation):

f_1 = w + x - 2*z
f_2 = w + 3*x + 2*y + 2*z
g_1 = w + x + y + z
g_2 = 3*w + x - y - 3*z

that's immediate. So, after small mountains of tedious manipulation, we get back a minor respelling of the initial assumptions.

Of course. My observation was nothing more than a verification
that all of this completing-the-square taradiddle was indeed correct
(and, as you indicate below, trivial). Nice typesetting, BTW.

What I don't understand is how anything other than that outcome could be _hoped_ for here. No amount of rearranging and cross-substituting the initial equations (whatever they may be) is going to yield new information, and there's never a step that even requires the quantities to be integers (as opposed to, e.g., arbitrary complex numbers). How can someone imagine that insight into integer factorization could result from this insight-less symbol-pushing?

I think that what we interpret as obfuscation on James' part is actually
a consequence of the fact that his understanding is extremely shallow.
This is, I think, the reason that he thinks his "prime counting
function" is truly new and innovative--he really is incapable of even
the slightest bit of abstraction that to all mathematicians is as
natural as breathing.

As usual, I couldn't make sense of his original writeup before you showed the correct result of completing the square wrt y first, at which point I could work backward from that to deduce what you thought James was trying to say. Also as usual, you got that right. Therefore :-) you must also know why he thinks this kind of approach _could_ yield something useful.

See above. The kind of self-editing we're accustomed to by inclination
and training is something he simply doesn't get. For example, a tiny bit
of thinking makes it obvious that no matter what collection of linear
equations one starts with, as long as they have a unique solution
the end result of the "small mountain of tedious manipulation" will
be the completely unsurprising T = g_1 * g_1, which we knew from the
start.

Or is this another case where you know he's right, and are keeping silent about which initial equations actually do work just to protect your career? Clever, if so ;-)
Clever you for deducing that. True, I've found that a simple
modification of James' argument will allow one to factor N in
log^2 (log N) steps, but I'm witholding it (a) to protect my career
and (b) to crack all those RSA codes out there and make a bazillion bucks by theft, deceit, and blackmail.

Be afraid, be very afraid.


Regards,

Rick

.

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