JSH: Generalizing quadratic residues solution idea
JSH
quadratic residues I have and realized I should look at the
generalization, so I wrote down: f_1 = ak mod p, and f_2 = bk mod p,
and quickly realized I had:
k = (a+b)^{-1}(f_1 + f_2) mod p
with T = f_1*f_2 = abk^2 = abq mod p
to solve for k, when k^2 = q mod p.
But it's so weirdly simple and worse ab and a+b are unique!
So to the math when you have k = (a+b)^{-1}(f_1 + f_2) mod p, where
f_1*f_2 = abq mod p, then you are TELLING it what 'a' and 'b' are, as
that's forced. So the math knows what you're playing with as it's
unique!!!
But then again, with p an odd prime, you can have a BUNCH of potential
'a' and 'b' that will work, so conceivably what will happen is that
they'll keep kicking out the same k, unless a given choice is blocked
for a particular T.
So how can a choice be "blocked" for T? Well, like if ak < p and bk <
p, but T does not have k as a factor, then that T is blocked.
But if there are no blockings then this method should give k.
That is so weird. But even weirder if it works well and was missed!!!
One reason it could be missed is that math people learn that
congruences are USELESS with factoring as, given say,
product = mn mod p
where m and n are residues modulo p, there are p-1 factorizations:
that is, there is a factorization for EVERY nonzero residue!
Math people have it banged into their heads that for that reason
residues and factoring don't mix.
But I've found that looking at just the product is wrong!!! You can
also get a SUM and that SUM AND PRODUCT ARE UNIQUE.
Such a simple difference and so much history can change. Maybe if
Gauss had noticed this thing we'd have never had public key
encryption, eh?
IMAGINE how the history of our world would have changed! History
turned on a simple miss.
James Harris
Mark Murray
> Earlier today I was pondering this weirdly simple idea for solving for
<skip a bit>
:
> So to the math when you have k = (a+b)^{-1}(f_1 + f_2) mod p, where
> f_1*f_2 = abq mod p, then you are TELLING it what 'a' and 'b' are, as
> that's forced. So the math knows what you're playing with as it's
> unique!!!
Suddenly maths has a personality and a will.
> That is so weird. But even weirder if it works well and was missed!!!
Oops. The grandiose claims are coming out.
> One reason it could be missed is that math people learn that
> congruences are USELESS with factoring as, given say,
>
> product = mn mod p
>
> where m and n are residues modulo p, there are p-1 factorizations:
> that is, there is a factorization for EVERY nonzero residue!
The shrillness really kicks in here:
> Math people have it banged into their heads that for that reason
> residues and factoring don't mix.
>
> But I've found that looking at just the product is wrong!!! You can
> also get a SUM and that SUM AND PRODUCT ARE UNIQUE.
>
> Such a simple difference and so much history can change. Maybe if
> Gauss had noticed this thing we'd have never had public key
> encryption, eh?
>
> IMAGINE how the history of our world would have changed! History
> turned on a simple miss.
... and James comes off the rails by forgetting the maths and letting
his overactive imagination run wild.
James - earlier you asked about NPD. What you wrote in the last few
paragraphs is exactly the sort of stuff that fuels the NPD theorising.
It does your maths no good at all and makes you appear as a prize prat.
M
JSH
> Earlier today I was pondering this weirdly simple idea for solving for
> quadratic residues I have and realized I should look at the
> generalization, so I wrote down: f_1 = ak mod p, and f_2 = bk mod p,
> and quickly realized I had:
>
> k = (a+b)^{-1}(f_1 + f_2) mod p
>
> with T = f_1*f_2 = abk^2 = abq mod p
>
> to solve for k, when k^2 = q mod p.
Just replying to note there are, of course, Goldbach's Conjecture
implications.
With f_1 = p_1, and f_2 = p_2, you have a composite C = (a+b)k mod
helper_prime_p.
Where also now because of the result you get T = f_1*f_2 = p_1*p_2 =
abq mod helper_prime_p.
And, of course, k^2 = q mod helper_prime_p.
So the result may offer an attack on Goldbach's Conjecture through the
use of quadratic residues, which if that pans out could be a truly
beautiful result.
Of course I needed to note that for credit purposes to make sure I DID
note it early on.
I should have first credit I think.
James Harris
amzoti
> turned on a simple miss.
>
> James Harris
People would still wake up in the morning and take a piss and a shit!
It is a good day!
Mark Murray
> So the result may offer an attack on Goldbach's Conjecture through the
> use of quadratic residues, which if that pans out could be a truly
> beautiful result.
>
> Of course I needed to note that for credit purposes to make sure I DID
> note it early on.
>
> I should have first credit I think.
From "... may offer ..." to claiming first credit on solving Goldbach's
conjecture.
THIS is evidence of crankiness, and is no more than fuel for the NPD
theory.
Do you really want to look like a complete idiot? If so, carry right
on.
M
MichaelW
>
> Math people have it banged into their heads that for that reason
> residues and factoring don't mix.
>
> But I've found that looking at just the product is wrong!!! You can
> also get a SUM and that SUM AND PRODUCT ARE UNIQUE.
>
> Such a simple difference and so much history can change. Maybe if
> Gauss had noticed this thing we'd have never had public key
> encryption, eh?
>
> IMAGINE how the history of our world would have changed! History
> turned on a simple miss.
>
> James Harris
Call for reference. Show me anywhere that a technical maths paper
"bangs into their heads" that modular arithmetic is not connected to
factoring.
I would say the reverse is true and MA is a standard part of modern
encryption. This is why (for example) the RSA standard includes an
appendix on MA:
http://www.rsa.com/rsalabs/node.asp?id=2367
Regards, Michael W.
JSH
Ok, I wouldn't be surprised if I over-reached in certain statements or
was just wrong.
IN any event, not looking to do anything further any time soon. Could
even drop this for some months as I consider other things. But don't
know.
It's a simple result that seems too simple to me. But it does seem to
work in cases I've considered, though I have reasons to wonder how
well it'd do as you get to bigger numbers.
It's just too simple for me to not wonder how useful it is, and right
now, not terribly interested in playing much more with it. So, on to
other things.
James Harris
Mark Murray
> Ok, I wouldn't be surprised if I over-reached in certain statements or
> was just wrong.
As many people have said, both of those cases are very common. So much so,
that if either is not so, it is remarkable (to use a word that you over-
use).
> IN any event, not looking to do anything further any time soon. Could
> even drop this for some months as I consider other things. But don't
> know.
So full retreat with no apologies for the insults left behind?
That would also be somewhat commonplace.
> It's a simple result that seems too simple to me. But it does seem to
> work in cases I've considered, though I have reasons to wonder how
> well it'd do as you get to bigger numbers.
It is a pity you have a policy of not following up on such musings.
> It's just too simple for me to not wonder how useful it is, and right
> now, not terribly interested in playing much more with it. So, on to
> other things.
Translation: "I'm busted, and I'm going to take my ball and go home."
M
rossum
wrote:
>It's a simple result that seems too simple to me.
A good thought James. It is simple enough that Gauss could have
worked it out. That is probably enough to indicate that this result
will probably not be as significant as you thought it might be.
>But it does seem to
>work in cases I've considered, though I have reasons to wonder how
>well it'd do as you get to bigger numbers.
Exactly James, That is the crux of the problem. Factoring small
numbers quickly is easy. Factoring large numbers quickly is
difficult. Remember that the essence of the factoring problem is
speed. Unless you test you method on large numbers and can show that
it factors them quickly then you will not have solved the factoring
problem.
rossum
Junoexpress
> Earlier today I was pondering this weirdly simple idea for solving for
>
> But it's so weirdly simple ...
>
> That is so weird. But even weirder if it works well and was missed!!!
>
>
> turned on a simple miss.
>
> James Harris
Isn't it amazing, all of the "simple" ideas you've discovered that
somehow missed detection by the greatest mathematical minds over the
last 300 years?
M
Gordon Burditt
>was just wrong.
Can you give an example where you didn't do one or both of those
things? Ever?
Junoexpress
The question you've set up is an interesting logic puzzle.
Question 1:
If James always over-reaches or is wrong (inclusive or), then what can
his responses to your question be?
Question 2:
Suppose we don't know if the hypothesis in question 1 is true or
false. What are the possible responses James can make that will allow
you to decide if the hypothesis is true or not?
M