Puzzling over latest surrogate factoring
JSH
surrogate factoring and have been puzzling over it. It seems that
there are factorizations with small test numbers that indicate some
mathematical requirement but I haven't been able to prove it!
In any event, here's the approach I am puzzling over.
With a target composite S, where
(x+k)^2 = y^2 + S
consider a non-trivial factorization, x+k-y = f_1 and x+k+y = f_2,
where S = f_1*f_2, so k is some chosen non-zero integer, so x and y
are then determined by the factorization.
Now expanding gives
x^2 + 2xk + k^2 = y^2 + S
and if you assume ahead of time some non-zero integer T, such that x^2
= y^2 + n_1*T, then
2xk + k^2 = S + n_1*T
and if you subtract 2k^2 from both sides then you have
2xk - k^2 = S - 2k^2 + n_1*T
and NOW if you assume that S - 2k^2 = n_2*T, you have
2xk - k^2 = (n_2 + n_1)* T, so
n_1 + n_2 = (2xk - k^2)/T
where now you have that T needs to be a factor of 2xk - k^2, and if
you look at factors coprime to k, the real focus is on 2x - k, and
I've checked with some test factorizations and it always has been such
that n_1 and n_2 are integers.
If that were a rigid requirement then there would always exist some T
a factor of S - 2k^2, such that
x = (k+T)/2 if k is odd, or
x = k/2 + T
if k is even, assuming that S is odd, as why have an even S to factor.
The weird thing to me is that it has worked with dinky test numbers
and I don't know why, as couldn't it be true that no prime factors
were in common between S - 2k^2, x^2 - y^2, and 2x - k, given
x+k-y = f_1 and x+k+y = f_2
where S = f_1*f_2?
After all, k is then the only choice as x and y are then forced by a
non-trivial factorization, for instance,
2(x+k) = f_1 + f_2.
Oh, there is one remarkable thing as well in that trivial
factorizations are blocked!!!
It's not possible for this technique to pull S itself as f_1 or f_2,
where there is an easy little proof of that fact.
James Harris
Mas Plak
"JSH" <jst...@gmail.com> wrote in message
news:1190865621.5...@19g2000hsx.googlegroups.com...
> Recently I started considering a reverse approach with what I call
> surrogate factoring and have been puzzling over it. It seems that
> there are factorizations with small test numbers that indicate some
> mathematical requirement but I haven't been able to prove it!
I was going to suggest it, => Factoring Surrogate
Just reverse the current approch.
>
> In any event, here's the approach I am puzzling over.
>
> With a target composite S, where
>
> (x+k)^2 = y^2 + S
or
S = x^2 +2*x*k + k^2 - y^2
> consider a non-trivial factorization, x+k-y = f_1 and x+k+y = f_2,
> where S = f_1*f_2, so k is some chosen non-zero integer, so x and y
> are then determined by the factorization.
> Now expanding gives
>
> x^2 + 2xk + k^2 = y^2 + S
why introduce f_1 and f_2 if your not going to use them?
> and if you assume ahead of time some non-zero integer T, such that x^2
> = y^2 + n_1*T, then
Where is n_1 defined?
x^2 = y^2 + n_1*T
but
x^2 = -2*x*k - k^2 + y^2 + S
or
y^2 + n_1*T = -2xk - k^2 + y^2 + S
and therfore;
n_1*T + 2*x*k + k^2 = S
>
> 2xk + k^2 = S + n_1*T
this is wrong => you make sign mistake Should be
2xk + k^2 = S - n_1*T
all the rest is wrong. Next time check your work.
you get an F.
gjed...@gmail.com
Why are you 'puzzling' over a dead idea?
I thought you were due to post some RSA factors this week? What's the
excuse this time? Protecting the economy again?
local host
> surrogate factoring and have been puzzling over it. It seems that
> there are factorizations with small test numbers that indicate some
> mathematical requirement but I haven't been able to prove it!
But you have proven over an 8 year period, that using simple algebra it is
imposable to do anything with surrogate factoring.
you have exhausted this subject.