option for finding multiple factors during trial factoring - Page 3

chris2be8 · 2014-06-07, 15:35

Quote:

Originally Posted by R.D. Silverman

Do you understand that factoring (prime base) Cunninghams reveals the structure of finite fields?

Does this also apply to (prime base) Brent tables entries?

Chris

R.D. Silverman · 2014-06-07, 21:32

Quote:

Originally Posted by chris2be8

Does this also apply to (prime base) Brent tables entries?

Chris

Try Google for "finite field"

tha · 2014-06-10, 23:31

I got this reply from James, it clarifies how the graph is made and things were just the opposite of what I (we) thought.

Those "weird" sections are were someone (no idea who) has done factoring differently (read: more thoroughly) than ranges around it. Normally TF will go until the first factor is found and then break, but someone has gone through some ranges and factored beyond finding the first factor. Pulling up some numbers, you can see that the number of factored exponents is as expected across all 4 ranges, but the number of known factors is much higher in the 130M-131M range (the one range that shows up in the weird section of the graph):

110-111M: 31678 factored exponents, 39107 known factors (1.234 factors/exponent)
120-121M: 31071 factored exponents, 38239 known factors (1.230 factors/exponent)
130-131M: 30646 factored exponents, 43852 known factors (1.430 factors/exponent)
140-141M: 30242 factored exponents, 37265 known factors (1.232 factors/exponent)

Note that on average there's 1.23 factors per factored exponent, but there's 16% more known factors in the weird zone.

Looking at the distribution, it seems that the extra TF was done up to 2^64 rather than stopping around 2^52.

So, bringing it back to my mersenne.ca graphs -- the height of the colored bars is the number of factored exponents (compared to the background grey which is the number of candidate exponents) -- not the number of known factors. The color of the bar is the bit size of the factor (averaged over the 0.1M or whatever a pixel represents on the graph scale). So for the weird ranges, the same number of exponents have at least one known factor as the surrounding ranges, but on average more larger factors are known, so the colors are biased towards darker blue. Make sense?

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2014-06-10, 23:31	#25
tha Dec 2002 5×163 Posts	I got this reply from James, it clarifies how the graph is made and things were just the opposite of what I (we) thought. Those "weird" sections are were someone (no idea who) has done factoring differently (read: more thoroughly) than ranges around it. Normally TF will go until the first factor is found and then break, but someone has gone through some ranges and factored beyond finding the first factor. Pulling up some numbers, you can see that the number of factored exponents is as expected across all 4 ranges, but the number of known factors is much higher in the 130M-131M range (the one range that shows up in the weird section of the graph): 110-111M: 31678 factored exponents, 39107 known factors (1.234 factors/exponent) 120-121M: 31071 factored exponents, 38239 known factors (1.230 factors/exponent) 130-131M: 30646 factored exponents, 43852 known factors (1.430 factors/exponent) 140-141M: 30242 factored exponents, 37265 known factors (1.232 factors/exponent) Note that on average there's 1.23 factors per factored exponent, but there's 16% more known factors in the weird zone. Looking at the distribution, it seems that the extra TF was done up to 2^64 rather than stopping around 2^52. So, bringing it back to my mersenne.ca graphs -- the height of the colored bars is the number of factored exponents (compared to the background grey which is the number of candidate exponents) -- not the number of known factors. The color of the bar is the bit size of the factor (averaged over the 0.1M or whatever a pixel represents on the graph scale). So for the weird ranges, the same number of exponents have at least one known factor as the surrounding ranges, but on average more larger factors are known, so the colors are biased towards darker blue. Make sense?