23-digit factor of (M+2)32582657 by ECM by mprime255

[bearnol]

[Mon Dec 6 01:56:07 2010]
ECM found a factor in curve #51, stage #2
Sigma=1483035008190041, B1=11000, B2=1100000.
2^32582657+1 has a factor: 13526662966442476828963

Desmond:math james$ gp
GP/PARI CALCULATOR Version 2.3.5 (released)
i386 running darwin (ix86 kernel) 32-bit version
compiled: Mar 1 2010, gcc-4.0.1 (Apple Inc. build 5490)
(readline v6.1 enabled, extended help available)

Copyright (C) 2000-2006 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
? (2^32582657+1)%13526662966442476828963
*** the PARI stack overflows !
current stack size: 4000000 (3.815 Mbytes)
[hint] you can increase GP stack with allocatemem()

? allocatemem()
*** allocatemem: Warning: doubling stack size; new stack = 8000000
(7.629 Mbytes).
? (2^32582657+1)%13526662966442476828963
*** the PARI stack overflows !
current stack size: 8000000 (7.629 Mbytes)
[hint] you can increase GP stack with allocatemem()

? allocatemem()
*** allocatemem: Warning: doubling stack size; new stack = 16000000
(15.259 Mbytes).
? (2^32582657+1)%13526662966442476828963
%1 = 0

Desmond:gmp-ecpp james$ time ./atkin159.gmp* -q
random seed = 1292417643
iter = 1000
Bmax = 2000
error_shift = 1000
precision = 10000

13526662966442476828963
N[0] = 13526662966442476828963
a = 0
b = 1004476460451487234715
m = 13526662966674952021447
q = 393984299847812659
P = (1025261424, 10368985817635190408101)
P1 = (0, 1)
P2 = (5673503186278490057910, 7335454421726048080324)
N[1] = 393984299847812659
a = 0
b = 373087861890548614
m = 393984298592760729
q = 18761157075845749
P = (130509967, 13537931531828585)
P1 = (0, 1)
P2 = (55954087057946518, 249330802432955570)
N[2] = 18761157075845749
a = 0
b = 13852667664725409
m = 18761156808059167
q = 1423132580449
P = (2389162395, 178014000815193)
P1 = (0, 1)
P2 = (13961564241116338, 14535489368327328)
N[3] = 1423132580449
a = 0
b = 1415057005884
m = 1423134959043
q = 45512647
P = (3228514804, 798490846914)
P1 = (0, 1)
P2 = (264257369526, 562009070315)
proven prime

real 0m2.972s
user 0m0.840s
sys 0m0.009s
Desmond:gmp-ecpp james$

Desmond:~ james$ /Applications/sage/sage ; exit;
----------------------------------------------------------------------
| Sage Version 4.3.2, Release Date: 2010-02-06 |
| Type notebook() for the GUI, and license() for information. |
----------------------------------------------------------------------
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(13526662966442476828963,1483035008190041)
2^4 * 3 * 17 * 19 * 229 * 457 * 3917 * 4423 * 481199