Wednesday, June 20, 2007

 

Composite factorization coupling

For quite some time mathematicians have focused on factoring composites as a global problem, where they would focus efforts on a single number, which is following in the footsteps of past mathematicians, like good little soldiers.

But past mathematicians didn't have computers.

Being trained as a problem solver since I was a little kid, I thought to re-think the problem, and wonder if it wouldn't be possible to shift factoring one number into factoring a lot of other numbers and found that hey, it's easy to do with trivial algebra.

That's modern problem solving which it takes YEARS to master. And here is why it's so important as often a seemingly hard problem becomes simple with a small shift in perspective.

Given a target composite T, you start with the traditional—see no throwing out the old—difference of squares:

x^2 = y^2 + aT

Then you add in a very small extra which is the new, where yes, b must have k as a factor:

2xk = k^2 - bT

And bringing together the old with the new, where you already have the classical difference of squares to start, you now get ANOTHER difference of squares to finish:

(x+k)^2 = y^2 + 2k^2 + (a-b)T

and you have coupled one factorization to another, so that the factorization of aT is connected directly to the factorization of 2k^2 + (a-b)T.

So mathematically EVERY composite factorization comes with another factorization connected to it, so no composite factorization stands alone.

And that's over infinity.

Now to Fermat, or Euler, Newton or Gauss there might have been a, so what?

As for them, with their pencils and paper or pen and paper, figuring stuff out was hard enough on its own, without showing that one factorization could lead to another.

But we are children of the modern age and we have computers.

Computers can whir through millions of calculations of alternate factorizations without trouble, but seem to bog down if you do two things:
  1. Get two very large primes, and multiply them together.

  2. Try to factor them.
But the composite coupling equations show that EVERY factorization—even that of the product of two large primes—is connected to ANOTHER factorization, which is VERY UNLIKELY to also be the product of just two large primes.

And so what if it is? In the blink of your slow eye, a modern computer can whir through a tremendous number of alternates until it finds one connected to your target and thereby gets the factorization of the product of two large primes.

That is modern problem solving—where you think out of the box.

The trouble is that MOST MATHEMATICIANS DO NOT DO MODERN PROBLEM SOLVING.

Instead they waste time learning a lot of abstruse stuff and a lot of techniques and styles, and they also learn how to ignore brilliant ideas that make them look stupid.

Now I know this from my other mathematical research where mathematicians can just blissfully continue no matter what I prove, but my emphasis on factoring is about stupid people pretending to be smart who will ignore ANYTHING because their entire lives are invested in the act.

Their entire lives are invested in the act.

They are wannabe actors pretending to be great mathematical minds.

I feel pity for some of you because of the years you've spent trying to learn dumb crap that doesn't actually work because you trust groups of people who tell you that it's brilliant and YOU are stupid if you find it hard to understand or realize that hey, a lot of this stuff just doesn't make sense!

But you see, a lot of modern math is hard because it's wrong, and modern math people are invested in keeping things the way they are so that they can make easy money.

I call it white collar welfare.

[A reply to someone who wrote that all his previous tests of James' ideas have shown that they can factor numbers but are slower than trial factorisation.]

Remember I have said recently that you need a number that is roughly as hard to factor as your target.

The method is a booster method that is an adjunct to other methods and there are more than enough headlines coming out to scare real mathematicians.

But you people are not real mathematicians, now are you?

United has a mysterious shut-down of its computers for two hours.

How would you demonstrate to a buyer that you could crack systems if you were a hacker looking to make millions?

The Pentagon(?) has to shut down its email because of some thing or other. Or was it the Department of Defense?

You people want the world to turn on its head, end the domination of the United States and usher in an era of anarchy, based on your ability to hold, hold, hold people who might do something just long enough until it's too late.

(Is that it? You are enemies of the United States? You WANT the global economy to crash to embarrass and remove the US from power?)

And people may die as a result, but you parasites just can't see beyond your limited needs.

It's a sad world because too many people trust too many others who are, parasitic humans who are not that smart, and care not a whit for the rest of the world.





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