From Phil Moore, who cannot join the Mersenne mailing list right now because Luke Welsh's scruznet.com ISP went under. Luke will host his mailing list sign up page when he has some free time.
>Date: Fri, 26 Oct 2001 14:43:17 -0700 >From: "Phil Moore" <[EMAIL PROTECTED]> >Subject: ECM testing > > From time to time, I see letters on this list asking if there is still > valuable work to be done with older, slower computers. If you would like > an alternative to factoring and double-checking assignments, I suggest > ECM (Elliptic Curve Method) testing. For one thing, you already have the > Prime95 (or mprime) software in your computer. For another, ECM curves > on small numbers run relatively quickly. Two projects on the web-pages at: >www.mersenne.org/ecm.htm really could benefit from additional computers. > >1) The smallest numbers on the Cunningham lists 2^N+1 and 2^N-1 will >become future condidates for factoring by the SNFS (Special Number Field >Sieve) method. Already, M673 and P647 are on the Cunningham list of the >"10 most wanted numbers"; see: >http://www.cerias.purdue.edu/homes/ssw/cun/third/index.html >(Also, P653, P659, and P661 are on the list of "more wanted numbers".) >SNFS requires enormous computer resources, and the people doing this don't >want to waste those resources if these numbers might have 45 or 50 digit >factors which could be more easily found by ECM. So usually, some major >ECM effort precedes SNFS work. In addition, there is a small, but >non-zero chance (maybe one in a million?) that running one curve at >B1=44,000,000 on one of these numbers could find a record-sized ECM factor >of >55 digits! > >2) The second project I would like to recommend is ECM factoring of Fermat >numbers. I have been working on this for over 2 years without success, >but I still think that the average computer's chance of finding a Fermat >factor is considerably higher than finding a Mersenne prime. The last >Fermat factor found by ECM was the factor of F18 discovered in April 1999, >but I would estimate that there are probably 2 or 3 additional Fermat >factors within reach of Prime95. The work involved scales roughly as (# >of digits)^0.2, so that finding a 40-digit factor takes roughly 10 times >the amount of work needed to find a 35-digit factor. So running 19,300 >curves at B1=44,000,000 to look for 50-digit factors of the 14th Fermat >number P16384 would take around 175 years on a 400 MHz Pentium II, while >running the 10,300 curves at B1=11,000,000 for 45-digit factors would only >take 20 or so years. More time has been spent on the numbers F14, F20, >and F22 which have no known factors, but I notice that the Fermat numbers >F17, F19, and F21 have had relatively less searching. > >So if you have a computer which will work for electricity, download the >lowm.txt and lowp.txt files into your Prime95 folder and start >searching! You can always run one curve just to see how long it >takes. Small exponents should only take a couple of hours. I have run >curves which took about a week each on F23 on 400 MHz Pentium II's running >Windows 98 with 128M of memory, but if you decide to work on these large >numbers, depending on what else you are running, you may have to >experiment with the memory allocation in the main menu to find an optimum >setting. Good luck! > >Phil Moore >Lane Community College >Eugene, Oregon _________________________________________________________________________ Unsubscribe & list info -- http://www.scruz.net/~luke/signup.htm Mersenne Prime FAQ -- http://www.tasam.com/~lrwiman/FAQ-mers
