I have been running ECM curves on Fermat numbers lately on numbers of the form 2^(2^n) - 1 (rather than + 1, as in the definition of Fermat numbers.) The number 2^(2^n) - 1 is the product of all the Fermat number F0 through F(n-1), so by running a curve on M32768 = M(2^15), I can search for factors of all the Fermat numbers up through F14. Why would I want to do that, you may wonder? Well, on both a Pentium 3 and an Athlon 1400MHz machine, I'm finding that the curve on M32768 runs about 3% or 4% faster than running one curve each on the 12th, 13th, and 14th Fermat numbers P4096, P8192, and P16384. For larger values of n, the increased efficiency is even more, in the range of 8% to 15%. George Woltman tells me that this is probably because Prime95 can choose from a larger set of FFT sizes for M numbers than P numbers. At any rate, since there may only be 2 or 3 undiscovered Fermat factors anyway of a size that may reasonably be found by the elliptic curve method, I find ! this method of searching more psychologically satisfying than concentrating on one Fermat number at a time. I am running curves on M32768 with a stage one limit of B1=44,000,000, also curves on M131072 with B1=11,000,000, curves on M524288 with B1=3,000,000, on M2097152 with B1=1,000,000, and on M8388608 with B1=250,000. The program seems to be working just fine, at least I am able to recover most of the smaller known factors of Fermat numbers, and I would like to invite anyone interested to try it. You need to edit the lowm.txt file with the known factors of all the Fermat numbers up to the largest one you are trying to factor. For example, for M32768, I have listed below the lines I inserted in lowm.txt, which include all known factors of F0 through F14 up to 50 digits. (I'm assuming that the probability of finding the 62 digit factor of F8, for example, is extremely small, and if it did turn up, it would be of interest as a new ECM record of sorts.) The lines need to ! be inserted in order of increasing order of exponent, i.e., just i not work. Also, on a number as large as M8388608, I would recommend allocating 64Mbytes of memory, or even more, as more memory will definitely speed up stage 2 processing. It has been over 3 years since a Fermat factor has been discovered by ECM, so it would be nice to end this drought!
Phil Moore M( 32768 )C: 3 M( 32768 )C: 5 M( 32768 )C: 17 M( 32768 )C: 257 M( 32768 )C: 641 M( 32768 )C: 65537 M( 32768 )C: 114689 M( 32768 )C: 274177 M( 32768 )C: 319489 M( 32768 )C: 974849 M( 32768 )C: 2424833 M( 32768 )C: 6700417 M( 32768 )C: 26017793 M( 32768 )C: 45592577 M( 32768 )C: 63766529 M( 32768 )C: 6487031809 M( 32768 )C: 190274191361 M( 32768 )C: 2710954639361 M( 32768 )C: 67280421310721 M( 32768 )C: 1238926361552897 M( 32768 )C: 1256132134125569 M( 32768 )C: 59649589127497217 M( 32768 )C: 2663848877152141313 M( 32768 )C: 3603109844542291969 M( 32768 )C: 167988556341760475137 M( 32768 )C: 3560841906445833920513 M( 32768 )C: 5704689200685129054721 M( 32768 )C: 319546020820551643220672513 M( 32768 )C: 4659775785220018543264560743076778192897 M( 32768 )C: 7455602825647884208337395736200454918783366342657 (To create lines for, say M(131072), you would change the exponent from 32768 to 131072 and add additional lines for the known factors of F15 and F16 which can be found in the lowp.txt file.) Good luck! _________________________________________________________________________ Unsubscribe & list info -- http://www.ndatech.com/mersenne/signup.htm Mersenne Prime FAQ -- http://www.tasam.com/~lrwiman/FAQ-mers
