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Old 2020-09-17, 00:18   #1
Ensigm
 
Aug 2020

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Default Is it realistic to calculate and store the following info

For every (composite) mersenne number, if it had a factor of x bits/digits, how likely would have the factor been already discovered.


If we store by digits from 21 to 70 then there will be 50 floating point numbers for each exponent. Storing by bits for the same range will yield about 166. Either way it's not a huge demand for storage, so the only problem is to calculate it correctly.

Last fiddled with by Ensigm on 2020-09-17 at 00:21
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Old 2020-09-17, 02:08   #2
firejuggler
 
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Yes it is possible but it is useless info. there is 3 method of factoring Mersenne.
PM1, Trial factoring, and ECM.
PM1 is probalistic. 53853801 has a factor. When I used it , the algo calculated that I had a 7.04% chance of finding a factor. It doesn't say anything about the size.
TF has a 1/bit level chance of finding a factor. again unrelated to size.
while ECM (eliptic curve method) work by size of factor. (called T level). If you complete T35, there is no chance of any factor below 35 digit on that particular M exist.




Edit : Sorry for the multiple edit. I wanted to make myself clear. And sorry for anyone quoting me in between edit

Last fiddled with by firejuggler on 2020-09-17 at 02:26
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Old 2020-09-17, 02:14   #3
VBCurtis
 
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Quote:
Originally Posted by firejuggler View Post
]If you complete a T35 on a certain exponent and found no factor, then the chance of a sub 35 digit factor is all but nill.
This isn't quite true- when a T35 is finished, there is a 1/e chance of missing a 35 digit factor (if one exists).

It's more accurate to say that after a T35 is done, chances of a 30 or smaller digit factor is nil (1/e^-6 or worse). A 33 digit factor has about a 10% chance (1/e^2, roughly) to be missed by a T35.
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Old 2020-09-17, 07:38   #4
Ensigm
 
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Quote:
Originally Posted by firejuggler View Post
while ECM (eliptic curve method) work by size of factor. (called T level). If you complete T35, there is no chance of any factor below 35 digit on that particular M exist.
I thought T35 only exclude a large majority of factors below 35 digits so that it makes more sense to search for factors with 35-40 digits? Someone knows more about ECM can correct me.



Quote:
Originally Posted by firejuggler View Post
TF has a 1/bit level chance of finding a factor. again unrelated to size.
If a bit level has been TF'ed, then if a factor is in that level, the probability of discovering that factor would be 100%.
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Old 2020-09-17, 16:01   #5
VBCurtis
 
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Quote:
Originally Posted by Ensigm View Post
If a bit level has been TF'ed, then if a factor is in that level, the probability of discovering that factor would be 100%.
This is correct- he meant the odds of finding a factor have nothing to do with the size of the input number. That's why the data you want to collect would not be useful- none of the 3 methods he mentioned have odds changed by the size of the input.
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Old 2020-09-17, 19:54   #6
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Quote:
Originally Posted by VBCurtis View Post
This is correct- he meant the odds of finding a factor have nothing to do with the size of the input number. That's why the data you want to collect would not be useful- none of the 3 methods he mentioned have odds changed by the size of the input.

I never mentioned the size of the input number. What I was talking about can be paraphrased as, in your words, "the chance of missing an x-digit factor (if one exists)". For each exponent, this will produce a line chart with bit or digit as the x-axis and probability as the Y axis.



This can be useful when people manually do curves with bounds that are not the typical values of T25, T30, etc. Under current rules I assume they are rounded down to the closest typical value? (Such as if I do 100 curves with B1=1e5, B2=1e7, they will only be counted as 100 T25 curves.)
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Old 2020-09-17, 20:05   #7
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Quote:
Originally Posted by Ensigm View Post
I never mentioned the size of the input number. What I was talking about can be paraphrased as, in your words, "the chance of missing an x-digit factor (if one exists)". For each exponent, this will produce a line chart with bit or digit as the x-axis and probability as the Y axis.



This can be useful when people manually do curves with bounds that are not the typical values of T25, T30, etc. Under current rules I assume they are rounded down to the closest typical value? (Such as if I do 100 curves with B1=1e5, B2=1e7, they will only be counted as 100 T25 curves.)
Good question.

What I do with the ECMNET server running at 83.217.167.17:8194 is first to multiply the number of curves by the B1 reported by a client and use that as a measure of work done. Each candidate in the server has its own record of the amount of work done by that measure. That total is used to set an appropriate B1 for a client when the candidate is allocated to it.
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Old 2020-09-17, 21:09   #8
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mersennne.org does something similar- I run ECM with B1=80k, B2=100*B1, and every 5 curves I run earns credit of 8 standard B1=50k curves. It's not a perfect conversion, but it works well enough for B1 near the expected T-level.

Where it breaks down is for B1 choices far smaller than useful- e.g. M1277; 1000 curves at B1 =1e6 are useless there, but get credited at just over 1 curve @850M. We hope people don't do that...
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