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dan73
Posts:
465
From:
ct
Registered:
6/14/08


Factoring dead zones in large semiprime composites.
Posted:
Feb 2, 2013 4:49 PM


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 semiprime 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 supercycle at point 1 zero remainder in the largest divisor too 152154 zeros and then a transition to 152154 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 supercycles which I am sure there is a limit to this density.
The new group of supercycles 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) supercycles (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 supercycles.
I did not use actual numerical examples because that can sometimes be confusing but will add if requested. Dan



