20150601, 15:25  #12  
"Bo Chen"
Oct 2005
Wuhan,China
2×5×17 Posts 
Quote:
then you have 0.1 resource. the second .1 is come from mersenne.org, http://www.mersenne.org/various/math.php which says, Code:
Looking at past factoring data we see that the chance of finding a factor between 2^X and 2^{X+1} is about 1/x. So find a factor between 10^55 to 10^60 is about ln(60)ln(55) = 0.087 I had read your paper that you sent me many years ago, but I could not understant it, it seems like you only use B2 = k * B1 to judge how many curves and which B1 should select. But you do not said nfs in that paper, If ECM use more time than nfs, the direct thinking is switch to nfs. I do not have profound knowledge about ecm and snfs, I read some papers about nfs, the ecm is more than a tool to me to use it. I do not know where the 2/9 come from, but it seems like all persons from this forum use it. 

20150601, 16:02  #13  
Feb 2012
Paris, France
7·23 Posts 
Quote:
between y and y^(1+e) is about e/(1+e). If I replace y by 10^55 and e by 1/11, I get probability of having a factor p such that 10^55 < p < 10^60 is about 1/12 = 0.0833 which is pretty close to 0.1. Now, why 55 digits? Because a t55 has been run and it produced no factor (so there is still an 1/e = 0.37 probability of missing a 55 digit factor). I read your paper but it looks like I won't be able to derive the result I'm looking for from that reading :). Last fiddled with by YuL on 20150601 at 16:41 Reason: Typo 

20150601, 16:35  #14  
Nov 2003
2^{2}×5×373 Posts 
Quote:
GIVEN that the probability of missing a p55 was 1/e, what is the conditional probability of finding a factor between 55 and 60 digits? 60 and 65? 65 and 70? etc. This is what Bayes thm does for us. One can compute the likelihood of suceeding with ECM even though the factor size is unknown. This tells whether more ECM is worthwhile when you compare the ECM time against the SNFS time.... 

20150604, 21:13  #15  
"Serge"
Mar 2008
Phi(4,2^7658614+1)/2
17·563 Posts 
Quote:
This is not a miss, just normal luck. Having it found by ECM would have been lucky; not having found  normal. A job well done! Congrats! 

20150605, 10:17  #16  
Feb 2012
Paris, France
7×23 Posts 
Quote:
yesterday, of course p135*p136 would have been wonderful, but a p65 on that one is certainly not an ECM miss. It's just the way it is, I may be luckier with the next one who knows.. Two years ago I thought a SNFS(240) was out of reach and now I just achieved a SNFS(271). Quote:
(which is a good thing cause it makes us learn things). I've been thinking about your above post lately and here is what I can come up with: Let N be the number we want to factor, we have Code:
factor size (digits) 5055 5560 60log10(sqrt(N)) probability 1/11 1/12 11/111/12 = 109/132 Now suppose we do a t55 and it fails to produce a factor it means that if there is a 5055 digit factor probability that we missed it is 1/e. Using Bayes formula we find that probability of having a 55 digits factor is now (e^1/11)/(e^1/11+10/11) = 1/(10e+1) = 0.035. But I don't see how this influences probability of N having a 5560 digit factor. One thing I thought about is using the fact that t55 is equivalent to 0.2t60 hence probability of missing a 5560 digit factor is e^0.2 (is it right?) thus probability of having a 5560 digits factors becomes (e^0.2/12)/(e^0.2/12+11/12) = 1/(11e^0.2+1) = 0.069 Does that mean that the amount of ECM I should do before switching to NFS is 7% of 45000 thread.hour = 3150 thread.hour ~ 3150 curves @260e6 ? Last fiddled with by YuL on 20150605 at 10:19 

20150605, 11:13  #17  
Nov 2003
2^{2}×5×373 Posts 
Quote:
the probability of having a factor of a given size from Dickmans's fxn (or an approximation such as CanfieldErdosPomerance, or the approximation I give in my paper) absent any information from ECM. You use my approximation in your discussion above. But: ECM gives us additional information. The data can be turned into a sample density function. Now apply Bayes' Thm. Convolve the prior with the sample density to derive a posterior density function. Now use the expected value of that density function as the best guess for the size of the unknown factor. 

20150605, 16:53  #18  
Jun 2005
lehigh.edu
2^{10} Posts 
Quote:
to be somewhat beyond amateur analysis/stat. Of course, I do recall years of rereading the SilvermanWagstaff tables in the hardcopy you sent me in the early 80's; and trying to understand the parameter revisions in Peter's thesis (hardcopy he sent on receiving his degree) using more efficient step 2 (ecm/fft, before Paul's ecmnet project; with ever better step 2s). But I'm reasonably sure that you'd agree we oughtn't to attempt extrapolation from reflexes established while searching for p40s too far past searching for p45's; much less in to the p65's of the OP's questions (cf. "pentiump90years for ever!"). So in RE your earlier post Quote:
p65searches (rather than p40/p45)? As long as we're here, I was interested to see Sean's ecm count Quote:
optimal p75 parameters (skipping 2.9e9/p70). Seems to fit a new definition of a Most Wanted number as getting t65+. Bruce (somewhere past SNFS(27X)) 

20150605, 17:16  #19  
Nov 2003
2^{2}·5·373 Posts 
Quote:
when I was a grad student there and taught a course) If one uses the idea (associated with what RaiffaSchleiffer calls postposterior analysis IIRC) of applying LOSS functions to the cost of making a wrong estimate (for a parameter), it is a theorem that using the expected value of the posterior minimizes the unitlinear loss function. Howard went through a proof of this theorem in class. IIRC it used a transformation to the characteristic function of a density function(s), went through a little Fourier analysis on the convolution, then did a reverse transform. I will see if I can find my old notes. I don't remember seeing a proof of this in print. 

20150605, 17:18  #20  
Nov 2003
2^{2}·5·373 Posts 
Quote:
Also, the 2/9 rule of thumb is only an approximation. The value "2/9" should slowly decrease as the numbers get bigger (because the likelihood of having an outofECMreach factor increases as the composites get larger) Last fiddled with by R.D. Silverman on 20150605 at 17:20 Reason: added sentence 

20150605, 23:25  #21 
"William"
May 2003
New Haven
2^{6}×37 Posts 
I've got a draft analysis that says it would take four times as much work by ECM as to do the SNFS sieving. Even if correct, I don't think that answers the question. How should you include the post processing work? At the very least we should include the thread hours, but those aren't fungible with sieving thread hours.

20150606, 08:22  #22 
"Victor de Hollander"
Aug 2011
the Netherlands
10010011000_{2} Posts 
If anybody has a script which calculates the required ECM effort based on the N, (G/S)NFS size and ECM done, then I hold myself recommended :). Yes, I'm lazy!
Last fiddled with by VictordeHolland on 20150606 at 08:22 
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