26-digit factor of (M+2)20996011 by ECM by mprime287-OSX
bearnol
[Sat Jan 13 03:59:24 2018]ECM found a factor in curve #500, stage #2
Sigma=2227963549692873, B1=250000, B2=25000000.
2^20996011+1 has a factor: 50965926368055564259063193 (ECM curve 500, B1=250000, B2=25000000)
? (2^20996011+1)%50965926368055564259063193
%1 = 0
James-Wanlesss-iMac:~ james$ time (echo '50965926368055564259063193' | bc | tr -d '\\\n';echo) | math/gmp-ecpp/atkin249.gmp-5.0.1.intelOSX.64.static -q
random seed = 1516469062
error_shift = 1000
precision = 10000
Bmax = 2000
Dmax = 20
N[0] = 50965926368055564259063193
a = 18505953892971969071811529
b = 0
m = 50965926368063733190457978
q = 943397349123041
P = (658161637, 44576531905149005402629758)
P1 = (0, 1)
P2 = (47103663478189996147072225, 15952019236214536492721806)
Bmax = 2000
Dmax = 20
N[1] = 943397349123041
a = 921682005178856
b = 0
m = 943397321779834
q = 1760325797
P = (226809618, 358892277812544)
P1 = (0, 1)
P2 = (373165211300786, 617014802580930)
Bmax = 2000
Dmax = 20
N[2] = 1760325797
a = 968227204
b = 0
m = 1760339906
q = 67705381
P = (201868925, 429381178)
P1 = (0, 1)
P2 = (755818549, 845445377)
proven prime
real 0m6.379s
user 0m2.201s
sys 0m0.027s
James-Wanlesss-iMac:~ james$ time (echo '50965926368055564259063192' | bc | tr -d '\\\n';echo) | math/superfac13.gmp-5.0.1.intelOSX.64.static -e
random seed = 1515906892
base = 955527292941126
number to be tested:
2
2
2
23
8747
B=1000, curve#9, a=1113159421061467
20996011
1508225159789
real 0m0.509s
user 0m0.216s
sys 0m0.007s
James-Wanlesss-iMac:~ james$ time (echo '50965926368055564259063194' | bc | tr -d '\\\n';echo) | math/superfac13.gmp-5.0.1.intelOSX.64.static -e
random seed = 1516512942
base = 172692801098433
number to be tested:
2
3
7
53
97
B=1000, curve#1, a=190697789334064
278911
B=1000, curve#1, a=366386501161368
47639
17764572013
real 0m0.055s
user 0m0.038s
sys 0m0.006s
? FindGroupOrder(p,s)=
{
K = Mod(1,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 = ellinit([0,b*A,0,b^2,0],K);
factor(ellcard(E))
}
%6 = (p,s)->K=Mod(1,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=ellinit([0,b*A,0,b^2,0],K);factor(ellcard(E))
? FindGroupOrder(50965926368055564259063193,2227963549692873)
%7 = [ 2 3]
[ 3 1]
[ 5 1]
[ 7 1]
[ 89 1]
[ 32443 1]
[ 39157 1]
[ 117889 1]
[4552049 1]