20040420, 00:08  #1 
Mar 2004
101111101_{2} Posts 
ECM on Mersenne numbers with prime exponents
An elliptic curve is successful, if the group order (p  1 + sigma) of a given prime factor is smooth.
When looking for p1 on mersenne numbers, we take advantage of the property, that a factor has the form p = (2k*exp) + 1. so exp can be taken into the group order of the p1 factor. Is it possible to choose the sigma of the ECM group order as multiple of 2*exp? Sigma = 2*k*exp. In that case that will give some extra digits and increase the efficiency. Does that theoretically work, and is that optimisation already included in prime95? 
20040420, 08:28  #2  
Bamboozled!
"๐บ๐๐ท๐ท๐ญ"
May 2003
Down not across
2C35_{16} Posts 
Quote:
Unfortunately, because the value "sigma" is used for two quite different quantities and the value used to choose the term is not the same one as appears in the formula for the group order. Paul 

20040420, 18:59  #3 
Mar 2004
575_{8} Posts 
I know that there are many different definitions of sigma.
I try to change my question then: Is it possible to choose the coeffizients of the ECM group order as multiple of 2*exp? That we choose sigma, that the group order is p  1 + 2*exp. which has a higher probability of being smooth? I don't know how the relation between the group order and sigma (in prime95 or gmpecm) is, but if jus needs to be chosen that the group order has that form. 
20040420, 19:55  #4  
Nov 2003
2^{2}·5·373 Posts 
Quote:
You want to choose coefficients so that the group order over Z/pZ is a priori divisible by 2*exp. It is not possible unless exp = 2,3,4,6, or 8. We can construct curves that explicity have points of order 4,6,8,12, or 16. Read P. Montgomery's 1987 paper for a full explanation. 

20040421, 04:03  #5 
Sep 2002
2×331 Posts 
I noticed when entering the values for F14 using M32768 with ECM on Prime95,
there is a checkbox with the label Factor 2^N + 1. My understanding is the M32768 as 2^327681 which covers F14's 2^16384+1 so it was left unchecked. Is this correct ( sure hope so ) ? Is there any use for the checkbox with ECM and Fermat numbers ? 
20040421, 04:44  #6 
Sep 2002
2×331 Posts 
My previous post belongs in the other ECM thread, ignore it, I'll repost.

20211101, 19:28  #7  
Aug 2020
79*6581e4;3*2539e3
1F7_{16} Posts 
I know this thread is old, but I couldn't find any other explanation more detailed than
Quote:
I guess that's as expected, but I couldn't figure out how to use that to calculate the correct group order. I used this script. I assume it has to be rewritten to be used for Mersenne factors? 

20211107, 09:41  #8 
"Alexander"
Nov 2008
The Alamo City
2·401 Posts 
I usually have success adding my GIMPS ECM factors to FactorDB using the standard group order formula. I used to dump my aliquot ECM factors into FactorDB regularly, and I ran into similar errors as you on rare occasions, so I'm not sure it's Mersennerelated. I never understood what exactly happened in those instances.

20211117, 23:18  #10 
Dec 2016
2^{3}·3·5 Posts 
I originally found this factor using mprime, not gmpecm. I'm not sure if the Sigma values are "compatible", i.e. if the same Sigma yields the same result in both programs. I have no idea about Bur's question about group order, btw.

20211128, 04:05  #11  
"Alexander"
Nov 2008
The Alamo City
2×401 Posts 
Quote:
Last fiddled with by Happy5214 on 20211128 at 04:06 Reason: Trimming quote 

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