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Topic: Factoring dead zones in large semi-prime composites.
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dan73

Posts: 465
From: ct
Registered: 6/14/08
Factoring dead zones in large semi-prime composites.
Posted: Feb 2, 2013 4:49 PM
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By dead zones I mean very large sequential blocks of
integers that could not possibly contain a factor of the
composite.
These could also be polynomials within the composite but
in this case these are multiple super cycles that can
eliminate may individual sequential blocks of integers.
Depending on the density or tightness of these super
cycles is what can collectively be a large elimination of
these sequential blocks of integers.
Completely eliminating them and then factor the remaining
gaps can help in factoring large semi-prime composites.

As an example below I show starting points of each of
these super cycles that need a mathematical process
to complete to the largest remaining zeros then
transitioning too largest remaining 9's in the largest
divisor.
This process can take a short time considering the size
of the block of integer it will eliminate.

Using rsa2048 as an example composite where its smallest
divisor is influenced by +/- e+153 too e+154.

These blocks are about e+154 in length which vary slightly
between super cycles e+152 too e+154.
A very large addition can possibly be made to each of
these because of the nature of the ratio between terms of
the divisors at the starting point(beginning point of the
super-cycle at point 1 zero remainder in the largest
divisor too 152-154 zeros and then a transition to 152-154
nines in the remainder).

The addition could be on the order of +/- e+302 to the
smallest divisor.
This still does not address the remaining 6 high order
digits of the divisor.
Leaving that much to be factored is significant I know.
Then again this can depend on the density possibility
of the super-cycles which I am sure there is a limit
to this density.

The new group of super-cycles I calculated below using the
smallest divisor for each where each has (2) distinct sets
of 0 and 9 remainders in the largest divisor.


The smallest divisors =

sqrt(rsa2048)/sqrt(2)
" " " " /sqrt(1.9)
" " " "/sqrt(1.8)
" " " "/sqrt(1.7)
" " " "/sqrt(1.6)
" " " "/sqrt(1.5)
" " " "/sqrt(1.4)
" " " "/sqrt(1.3)
" " " "/sqrt(1.2)
" " " "/sqrt(1.1)

Each above have (2) distinct blocks of sequential integers
that are dead zones with digit lengths e+153 ---e+154

Some new additions are ratios between divisors
1.19 and 1.21 just to test the tightening up
of these (3) super-cycles (1.19)--(1.2)--(1.21)
Each has (2) distinct dead zones of zeros and nines.
(Zero and nine) as one distinct dead zone.
There are two for each ratio.

The start of these super cycles usually have no zero
or nine remainders in the largest divisors but have to
be started with a certain odd smallest divisor.
Then taken from there +/- to 152 to 154 zeros and
then transition to 152 to 154 9's in the remainder.

Any thoughts on these dead zones and how they can be
incorporated into a special factoring algorithm that
eliminates them completely for possible factors. Then just
factor the gaps between these super-cycles.

I did not use actual numerical examples because that
can sometimes be confusing but will add if requested.

Dan



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