27-digit factor of (M+2)6972593 by ECM by mprime287-OSX

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bearnol

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Jul 27, 2016, 1:15:40 AM7/27/16
to Mersenneplustwo
[Tue Jul 26 20:57:27 2016]
ECM found a factor in curve #11, stage #2
Sigma=8282816111310800, B1=250000, B2=25000000.
2^6972593+1 has a factor: 142921867730820791335455211 (ECM curve 11, B1=250000, B2=25000000)

Desmond:~ james$ gp
                  GP/PARI CALCULATOR Version 2.7.4 (released)
           i386 running darwin (ix86/GMP-6.1.0 kernel) 32-bit version
        compiled: Jan  8 2016, gcc version 4.2.1 (Apple Inc. build 5566)
                            threading engine: single
                 (readline v6.3 enabled, extended help enabled)

                     Copyright (C) 2000-2015 The PARI Group

PARI/GP is free software, covered by the GNU General Public License, and comes 
WITHOUT ANY WARRANTY WHATSOEVER.

Type ? for help, \q to quit.
Type ?12 for how to get moral (and possibly technical) support.

parisize = 4000000, primelimit = 500000
? (2^6972593+1)%142921867730820791335455211
%1 = 0

Desmond:~ james$ time (echo '142921867730820791335455211' | bc | tr -d '\\\n';echo) | math/superfac13.gmp-5.0.1.intelOSX.64.static -e
random seed = 1469654310
base = 317388680447007
number to be tested:
142921867730820791335455211

real 0m0.564s
user 0m0.003s
sys 0m0.012s

Desmond:~ james$ time (echo '142921867730820791335455211' | bc | tr -d '\\\n';echo) | math/gmp-ecpp/atkin249.gmp-5.0.1.intelOSX.64.static -q
random seed = 1469672677
error_shift = 1000
precision = 10000
Bmax = 2000
Dmax = 20
N[0] = 142921867730820791335455211
a = 0
b = 78204068999952157777412276
m = 142921867730844669126398571
q = 1082175739430484588559
P = (3407908693, 532418724915329385542950)
P1 = (0, 1)
P2 = (84171469144138423728736142, 45268612727935761206023740)
Bmax = 2000
Dmax = 20
N[1] = 1082175739430484588559
a = 0
b = 220082507323440004042
m = 1082175739496276881377
q = 10599707410333
P = (2141215857, 69896538264253247575)
P1 = (0, 1)
P2 = (935454441880619554901, 647765648841261245242)
Bmax = 2000
Dmax = 20
N[2] = 10599707410333
a = 0
b = 6728517948383
m = 10599701166801
q = 102119533
P = (1509888801, 5413324155857)
P1 = (0, 1)
P2 = (6988843835299, 1886999262635)
proven prime

real 0m4.043s
user 0m2.007s
sys 0m0.015s

Desmond:~ james$ time (echo '142921867730820791335455210' | bc | tr -d '\\\n';echo) | math/superfac13.gmp-5.0.1.intelOSX.64.static -e
random seed = 1470346257
base = 278199544019489
number to be tested:
2
3
5
29
61
B=1000, curve#1, a=826142455357139                    
12618511
B=1000, curve#2, a=741282647051094                    
6972593
30608861

real 0m0.101s
user 0m0.049s
sys 0m0.006s

Desmond:~ james$ time (echo '142921867730820791335455212' | bc | tr -d '\\\n';echo) | math/superfac13.gmp-5.0.1.intelOSX.64.static -e
random seed = 1470384209
base = 4847038301466
number to be tested:
2
2
B=1000, curve#1, a=1098948183274188                    
150323
B=1000, curve#2, a=433327493894013                    
1851973
128344897209157

real 0m0.093s
user 0m0.063s
sys 0m0.006s

sage: def FindGroupOrder(p,s):
...      K = GF(p)
...      v = K(4*s)
...      u = K(s^2-5)
...      x = u^3
...      b = 4*x*v
...      a = (v-u)^3*(3*u+v)
...      A = a/b-2
...      x = x/v^3
...      b = x^3 + A*x^2 + x
...      E = EllipticCurve(K,[0,b*A,0,b^2,0])
...      return factor(E.cardinality())
sage: FindGroupOrder(142921867730820791335455211,8282816111310800)
2^3 * 3 * 19^2 * 251 * 317 * 1327 * 5839 * 53959 * 495877

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