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.