Friday, January 18, 2008

 

Factors mod p

Possibly the form I've given has made this result difficult for many of you to understand, so I'm shifting to directly considering composite factors modulo p.

Given a target composite T and integer factors f_1 and f_2, such that f_1*f_2 = nT, and any prime p, the following relations must be true:

f_1 = ak mod p

and

f_2 = a^{-1}(1 + a^2)k mod p

where

k^2 = (a^2+1)^{-1}(nT) mod p.

Those represent the fundamental factoring relations that underpin ALL composite factorizations.

Example: n=1, T=119, p=11, and a=2 gives

k^2 = (5)^{-1}(119) mod 11 = 9(9) mod 11 = 4 mod 11.

And k = 2 mod 11 is a solution, so f_1 = 4 mod 11, and

f_2 = 2^{-1}(1+4)(2) mod 11 = 5 mod 11.

In this case, subtracting 11 gives one prime factor as f_1 = -7 mod 11, and f_2 = -6 mod 11. Subtracting 11 again from f_2 gives -17.

I think many of you fail to understand why this result is so important because you don't realize that the congruences relations are not just sometimes true, but ALWAYS TRUE and they determine what is possible for integer factorizations.

There is no composite factorization that escapes those rules. None. There has never been any either. It's just mathematicians did not know the rules existed.

It is trivial to prove with them that powerful factoring algorithms can be developed.

And their derivation is backed by easy mathematical proof.

So easy mathematical proof shows my case. If you doubt that I challenge any of you to focus on the specifics here in a mathematical discussion asking me to expand on any particular point, from the derivation of the relations to how they can be used to factor efficiently.

Replies simply requesting that I factor a large number are just heckling when I say my position is firmly established by mathematical proof and I am willing to elaborate on any and every detail that proves my case.





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