Re: JSH: SF: Finally, surrogate factoring

[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

[jstevh@xxxxxxx]
The only problem I have with that is that you have too many solutions
for y.

There is one explicit solution for y given by solving the 4 linear
equations.

That solution is

y = (7g_1 - 3g_2 + 5f_1 - 3f_2)/4

Woo hoo! Correct!

which what you give covers, but you could have gotten that by working
backwards FROM the explicit solution,

He could have, but he didn't. Neither did I.

and there is one problem.

If T has only two prime factors p_1 and p_2, there are 8 possible
values for y for a given f_1 and f_2, which represent g_1 = T, p_1,
p_2, or 1, and the negatives, and g_2 = 1, p_2, p_1, or 1, and the
negatives.

But your solution has more than that because it gives

y = (5f_1 - 3f_2 + 21g_1 - g_2)/4

as a solution as well.

[Rick Decker]
No. However, it would be interesting to see how you got this.
Ah. Perhaps you were working from h_1 * h_2 = 21 * g_1 * g_2,
like this:

Are you psychic or what? I had no idea how he came up with the thing
containing 21g_1, and never would have guessed he was just pulling it out of
his *** :-)

Let h_1 and h_2 be chosen so that h_1 * h_2 = 21 * T

h_1 + h_2 = 10*y + 42*z + 19*f_1 - 3*f_2 [1]
h_2 - h_1 = 4*y - 5*f_1 + 3*f_2 [2]

I think you meant to write h_1 - h_2 on the LHS of [2].

Then we can write

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

in the form

(h_1 + h_2)^2 = (h_1 - h_2)^2 + 4 * h_1 * h_2

This part would be clearer with the correction above.

Then, from [1] and [2] we solve for y to get

y = (5*f_1 - 3*f_2 + h_1 - h_2) / 4 [3]

While this conclusion _needs_ the correction above.

Then, since h_1 * h_2 = 21 * T = 21 * g_1 * g_2 we may as well
pick h_1 = 21 * g_1 and h_2 = g_2 so [3] becomes

y = (5*f_1 - 3*f_2 + 21*g_1 - g_2) / 4

Right?

Yes, you are psychic!

If that was your reasoning, it's wrong. You can't pick any old
values for h_1 and h_2. Watch:

Solving [1] and [2] for h_1 and h_2 we get

h_1 = 7(y + 3 * z + f_1)
h_2 = 3(y + 7 * z + 4 * f_1 - f_2)

That also needs the correction above ;-)

But from your original four linear equations we can derive

g_1 = y + 3 * z + f_1
g_2 = y + 7 * z + 4 * f_1 - f_2

in other words, we are forced to choose

h_1 = 7 * g_1
h_2 = 3 * g_2

and not your h_1 = 21 * g_1 and h_2 = g_2.

And to force the conclusion, in that case [3] becomes

y = (5*f_1 - 3*f_2 + 7*g_1 - 3*g_2)/4


But let's give James something else to worry about :-) Take

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

expand it, use the quadratic equation to solve for y, and then substitute to
get rid of z and T:

z = -(3*f_1 - f_2 + g_1 - g_2)/4
T = g_1*g_2

The result is:

y = (5*f_1 - 3*f_2 + 5*g_1 - 5*g_2 +/- 2*(g_1 + g_2))/4

Pick "+" and you get the result James wants:

y = (5*f_1 - 3*f_2 + 7*g_1 - 3*g_2)/4

Pick "-" and it's different:

y = (5*f_1 - 3*f_2 + 3*g_1 - 7*g_2)/4

Woo hoo! Centuries of mathematics down the tubes again, or can James spot
the bogosity? Hint #1: this isn't an algebraic error; you really do get
that result for y. Hint #2: you get the same two results for y if you do
the same thing but starting from

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

instead.

<snip>

It's why I decided after thinking that this method MUST lead to a
surrogate factoring solution as the only way the algebra can avoid the
contradiction is to use a surrogate factorization.

You claim otherwise with your post.

No. That was your (incorrect) deduction, as I show above.

The suspense is killing me.

Your claims mean that 4 linear equations can be wrong.

I wonder if you just lied.

You just can't resist, can you? Are you naturally boorish, or do
you have to work at it?

I strongly suspect that bit of gratuitous assholishness was deliberate. God
only knows why, but James got it into his head that he needs to _provoke_
people into replying when he thinks they know something he wants to find
out. That's just his despicable way of trying to goad you into doing his
work for him. It's especially idiotic in this case, since if he had any
memory he'd recall that you typically respond much better to polite requests
than to his stupid baiting tactics.

But, in this case, I'm afraid what he'll take away is "ha! it worked again",
without a shadow of a clue that it was neither necessary nor helpful to
behave like an ass.


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