Re: Mersenne: Request for help (especially ECM)

[Alan Powell]

Brian

FYI I have directly computed the group order #E for each of your factors/sigmas
below based on the Richard Crandall EC parameterization in Prime95.  They are:

M1123:   #E = 777288435636813540 = 2^2*3*5*11*31*1823*38917*535489
 
M1621:   #E = 649412561966194102824 = 2^3*3^2*11*857*3967*94727*2546119
 
M1787:   #E = 838124596917111720588 = 2^2*3*23*29*613*12713*35543*378041
 
M1867:   #E = 597702354112546800 = 2^4*3*5^2*13*17*2843*3061*258983
 
P721:    #E = 8142767081751608532 = 2^2*3^3*2579*15937*33487*54779
 
M111409: #E = 17304916346160435696 = 2^4*3*11*31*601*1051*4861*344327

Note:

1. The parameterization used always results in a #E divisible by 12 thus one
   gains about 1.2 digits of smoothness.

2. AFAIK the implementation is deterministic in that if in stage 1 we carry out
   EC operations up to B1=penultimate factor of #E and then in stage 2 we carry
   out EC operations on blocks of primes up to a B2 such that the last block of
   primes contains the highest factor of #E then we will always detect a factor
   or possibly a composite containing that factor.

3. This determinism should not be affected by any post-processing based on the
   contents of lowM.txt or lowP.txt ! 

Regards

Alan Powell



At various times, you wrote:
> With sigma=1459848859275459, the factor 777288435261989969 of M1123 is found
> - in stage 1 with B1 >= 535489
> - in stage 2 with B1 >= 38917 & B2 >= 534241
> ............................................
> With sigma=7324432520873427, the factor 649412561933038085071 of M1621 is found
> - in stage 1 with B1 >= 2546119
> - in stage 2 with B1 >= 94727, B2 >= 2543311
>
> With sigma=5643809308022499, the factor 838124596866091911697 of M1787 is found
> - in stage 1 with B1 >= 378041
> - in stage 2 with B1 >= 35543, B2 >= 378001
>
> With sigma=6305161669623833, the factor 597702354293770769 of M1867 is found
> - in stage 1 with B1 >= 258983
> - in stage 2 with B1 >= 3061, B2 >= 258301
>
> With sigma=5956836392428930, the factor 8142767081771726171 of P721 is found
> - in stage 1 with B1 >= 54779
> - in stage 2 with B1 >= 33487, B2 >= 54390
> ..........................................
> With sigma=8866098559252914 the factor 17304916353938823097 of M111409 is found
> - in stage 2 with B1 >= 4861 & B2 >= 343351.