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

[bearnol]

[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