Re: JSH: Factoring and residues


<jstevh@xxxxxxx> wrote in message
news:1153022857.985835.215450@xxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
Tim Peters wrote:
[Tim Peters]
...
And these things are generally false, although James wishes they were
true:

x = x_res modulo T
x^2 = y^2 modulo T
1 < gcd(x+y, T) < T
1 < gcd(x-y, T) < T


[jstevh@xxxxxxx]
The correct mathematical view is that given S and k, you have an
infinite range of possible x_res's and T's that will satisfy

S = 2x_res*k mod T

and

x^2 - y^2 = 0 mod T

so you can find an S and k that will work with a particular x_res you
choose, your desired residue for x, but there are an infinity of other
x_res's and T's possible for which S and k will also work.

It's as simple as that and there is no reason for anyone to act like
there's a big mathematical mystery here.

I agree that there isn't, but stop pretending that was your _original_
view
of this method. At its start, this was all delusional nonsense like:

I was wrong in thinking that picking x_res and getting k with a chosen
S would guarantee T.

However, while I have been reasonable, you have not, as you keep
fighting a political battle to convince other people that there is
nothing to these ideas.


[JSH]
Kind of one of those odd things, but this idea just did come to me
Saturday in that I just wrote down some equations and hey! I solved
the factoring problem.

Etc, etc.

You didn't solve it, and the primary purpose of my post was to show that,
and explain _why_ the method usually fails to factor the T you give it.
A
secondary purpose was to help others understand what your method _is_,
since
almost nobody else replying to you appeared to have any real idea what
you
were trying to say (your own writeups were unclear on several key
points).


But you haven't shown that the method usually fails.

You've stated it.

I've stepped in to show what's mathematically shown, having to revise
my own views as there were things I missed or was wrong about, as I
learn more, myself, about this approach.

No, you do not learn. You are a Crackpot

Now you've backtracked to something that's true: while there's no reason
to
hope this method can efficiently factor the T you give it, it does
efficiently factor /other/ integers, vaguely related to T via chains of
information-losing congruences. That's incredibly weaker than the
original
claims, but at least it's a true claim.


I haven't back-tracked as I've learned more given more information, and
had some mistaken ideas cleared up.

It's basic research. Some expansion of knowledge along the way is
understandable.

You don't care about that.



For example, for those who didn't follow the full story on other
newsgroups,
given T=91 x_res = S = 1 as inputs, it can end up deducing that 459 is
divisible by 17 and 27. The doesn't look useful to me, but if you can
make
something useful of it, more power to you.

That opens the real question of how often when you find S and k for
your x_res, will you get some other answer coming in,

More often the larger T's smallest prime factor, and you're already very
unlikely to factor incoming T with 6 digits.


But you just make statements without proof as if the idea of
mathematics has escaped you.

Mathematics is not about just making statements, as people who are
mathematicians believe that you can actually prove things to be true,
or not true, using the tools of mathematics.

instead as the math shifts to some T other than the one you want, and
the good news is that other T's will tend to be integers of absolute
value bigger than yours,

Because the largest T' satisfying the congruences is |S - 2*x*k|, and
when
you pick S=1 x_res=1 (as you've always done so far), k = (T+1)/2.
Therefore
the largest T' satisfying the congruences is |1 - (T+1)*x|, which has a
_minimum_ value of T at x=1 (x=0 isn't useful).


Well, it seems to me that there must be some pressure against larger
T's, as there are an infinity of possible ones, correct?

But as you go up you get larger and larger numbers, so the likelihood
of getting them drops rapidly.

Sounds like a strategy to me for maximizing your chances of getting the
T you want could be lurking in there--or in plain sight.

which means there are probably strategies for limiting that happening.

See above. Pick larger S, and, more importantly, since

x = z-k = (f_1 + f_2)/2 - k

pick f_1 and f_2 to minimize the absolute value of that expression.


Which is not a mathematical statement.

Now I think you're playing politics in an effort to convince others to
ignore this research.

People who look over my posting of a roadmap of my research may wonder
how it's possible I could have so much research and it not be
recognized.

Well, people like Tim Peters are part of the problem.

I LIKE mathematics. And I LIKE talking about mathematics.

And there are people who LOVE to reply to me just to tell people that
I'm wrong or that what I found is useless.

I'm just one voice.

I'm tempted to delete off the rest of Tim Peters reply, but it might
help to leave it in, especially so that some of you can understand the
other weapon of people like him--tedium.

They make long posts which are a pain to reply to where they say
nothing of mathematical importance.

Eventually I get tired of bothering with them and abandon threads,
which they take as an opportunity to have the last word.

There are LOTS of them and one of me.

Now the mathematics here is simple and I think neat, but the politics
are high, so rather than admit that future development is possible, or
even bother giving mathematical reasons for how certain things can
happen Tim Peters is just pushing the position that it can't work well.

I explained earlier that, given any inputs and any specific way of
factoring
S+k^2, your method ends up computing specific integers x and y. The set
of
T for which the congruences hold is then the set of integer divisors of
x^2-y^2, no more, no less. Just like in the example above, given T=91
x_res
= S = 1, your method computes x=5 and y=22 (given one way of factoring
S+k^2), so that T then satisfies the congruences if and only if T divides
22^2-5^2 = 459.

There's no mystery about that either.


<SNIP CRAP>.

Well, maybe, maybe not, but why one way or the other?

See above. You have no way here to find useful solutions to:

x^2 = y^2 (modulo T)

for a /fixed/ T. Instead you're effectively picking x and y, then
solving
for T. But that's trivial to solve -- none of the machinery in your
method
is needed for that, and it's of no use for factoring a /fixed/ T
regardless.


<SNIP CRAP>

That's why I hate how political this damn field has become as you run
into operatives like Peters who are fighting a war--what I call the
math wars--and the mathematics is just a pawn in a game about
convincing people one way or another.

I have no stake in whether you "solve the factoring problem" or not.
It's
your ego that's hugely involved in that, not mine. When you post claims
that aren't true, sometimes I take time to refute them; when you're going
down a road that I clearly see is a dead end, sometimes I try to tell you
why. You haven't noticed that I haven't been wrong about one of those
yet?
I have ;-)


<SNIP CRAP>


.

Relevant Pages

  • Re: JSH: Factoring and residues
    ... If some integer in his colon needs factoring, ... Pick some integer S coprime to T. ... Now the mathematics here is simple and I think neat, ... happen Tim Peters is just pushing the position that it can't work well. ...
    (sci.math)
  • Re: JSH: Factoring and residues
    ... information-losing congruences. ... Mathematics is not about just making statements, ... Now I think you're playing politics in an effort to convince others to ... people like Tim Peters are part of the problem. ...
    (sci.math)
  • Re: JSH: Factoring and residues
    ... x = x_res modulo T ... information-losing congruences. ... happen Tim Peters is just pushing the position that it can't work well. ... I have no stake in whether you "solve the factoring problem" or not. ...
    (sci.math)
  • Re: JSH: Mystery increases
    ... solution to the factoring problem. ... I am not qualified to speak about most of your results, but the one I feel most qualified to speak on, your attempt to prove P = NP, I had a strong feeling from you: your proofs were based on "it feels right," as opposed to fully rigorous derivations. ... It appears to be mostly intuitions that you claim proved you correct; intuitions about your algorithms led me to suspect gaping holes. ... An outsider who claims to be able to refute 100-year old cornerstones of mathematics is more likely to be wrong than right. ...
    (sci.math)
  • Re: JSH: Of course theyre lying
    ... The reason for me picking the factoring problem is that the lies ... Your group claims to care about mathematics, ... "Understanding Godel isn't about following his formal proof. ...
    (sci.math)