Re: Surrogate factoring approach, analysis

From: John Roberts-Jones (johnrj_at_globalnet.co.uk)
Date: 01/01/05 

Date: Sat, 01 Jan 2005 18:29:21 +0000

In <1104591454.431335.128080@z14g2000cwz.googlegroups.com>,
jstevh@msn.com wrote:

<snip>

>
>I've found I can use some simple expressions to consider my approach
>to the factoring problem:
>
>a_1 x + b_1 = f_1
>
>a_2 x + b_2 = f_1
>
>And A = a_1 b_2 + a_2 b_2
>
>where all are rational.

So you are working in the field of rationals.

>
>And multiplying the first equation times the second gives
>
>a_1 a_2 x^2 + Ax + b_1 b_2 = f_1 f_2
>
>and if you solve for x, you can get to
>
>x = (b_2 f_1 + b_1 f_2 - 2b_1 b_2)/A
>
>and initially I attempted an approach where f_1 f_2 would equal the
>integer you were trying to factor, and not surprisingly, found a
>dependency on that factorization, which can be seen in the solution
>for x here.
>
>But I moved on, basically just checking possibilities, and considered
>the case where f_1 f_2 - b_1 b_2 equals the number you're trying to
>factor, and that gave what at first appears to be a surprising
>solution.
>
>But it's mathematics, right? So there has to be a logical explanation.
>
>And looking at the solution for x, it makes sense, as x is dependent
>on b_1, b_2, f_1, f_2 and A, and as outlined in the paper on this
>site, b_1 b_2, f_1 f_2 and A are all chosen arbitrarily (in the paper
>b_1 b_2 = k and f_1 f_2 = T).
>
>See http://groups.yahoo.com/group/simplefact/files/
>
>So the dependencies are all on the factorizations of numbers that are
>picked, and the mathematics, when given the products of those
>numbers, cycles through the factorization for you--for one of them--if
>you power the engine, you might say, with the factorizations of
>another.
>
>The logic is straightforward, and it just turns out there is a way to
>do this, which I guess others didn't notice. But just because
>something isn't noticed doesn't mean it doesn't exist.
>
>So I focus on the factorization f_1 f_2, where you cycle through all
>possible integer factors of that product, which pushes you through the
>factorizations of b_1 b_2, and down the line should give those of your
>target (unless I'm missing something).

In the rationals, each nonzero integer is a factor of f_1 f_2.
How do you propose to iterate through all of them? 

-- 
John Roberts-Jones

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  • Re: Surrogate factoring approach, analysis
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