#!/usr/local/bin/python # -*- coding: utf-8 -*- import math import random #y^2=x^3+ax+b mod n prime=[2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 251, 257, 263, 269, 271 ] # ax+by=gcd(a,b). This function returns [gcd(a,b),x,y]. Source Wikipedia def extended_gcd(a,b): x,y,lastx,lasty=0,1,1,0 while b!=0: q=a/b a,b=b,a%b x,lastx=(lastx-q*x,x) y,lasty=(lasty-q*y,y) if a<0: return (-a,-lastx,-lasty) else: return (a,lastx,lasty) # pick first a point P=(u,v) with random non-zero coordinates u,v (mod N), then pick a random non-zero A (mod N), # then take B = u^2 - v^3 - Ax (mod N). # http://en.wikipedia.org/wiki/Lenstra_elliptic_curve_factorization def randomCurve(N): A,u,v=random.randrange(N),random.randrange(N),random.randrange(N) B=(v*v-u*u*u-A*u)%N return [(A,B,N),(u,v)] # Given the curve y^2 = x^3 + ax + b over the field K (whose characteristic we assume to be neither 2 nor 3), and points # P = (xP, yP) and Q = (xQ, yQ) on the curve, assume first that xP != xQ. Let the slope of the line s = (yP − yQ)/(xP − xQ); since K # is a field, s is well-defined. Then we can define R = P + Q = (xR, − yR) by # s=(xP-xQ)/(yP-yQ) Mod N # xR=s^2-xP-xQ Mod N # yR=yP+s(xR-xP) Mod N # If xP = xQ, then there are two options: if yP = −yQ, including the case where yP = yQ = 0, then the sum is defined as 0[Identity]. # thus, the inverse of each point on the curve is found by reflecting it across the x-axis. If yP = yQ != 0, then R = P + P = 2P = # (xR, −yR) is given by # s=3xP^2+a/(2yP) Mod N # xR=s^2-2xP Mod N # yR=yP+s(xR-xP) Mod N # http://en.wikipedia.org/wiki/Elliptic_curve#The_group_law''') def addPoint(E,p_1,p_2): if p_1=="Identity": return [p_2,1] if p_2=="Identity": return [p_1,1] a,b,n=E (x_1,y_1)=p_1 (x_2,y_2)=p_2 x_1%=n y_1%=n x_2%=n y_2%=n if x_1 != x_2 : d,u,v=extended_gcd(x_1-x_2,n) s=((y_1-y_2)*u)%n x_3=(s*s-x_1-x_2)%n y_3=(-y_1-s*(x_3-x_1))%n else: if (y_1+y_2)%n==0:return ["Identity",1] else: d,u,v=extended_gcd(2*y_1,n) s=((3*x_1*x_1+a)*u)%n x_3=(s*s-2*x_1)%n y_3=(-y_1-s*(x_3-x_1))%n return [(x_3,y_3),d] # http://en.wikipedia.org/wiki/Elliptic_curve_point_multiplication # Q=0 [Identity element] # while m: # if (m is odd) Q+=P # P+=P # m/=2 # return Q') def mulPoint(E,P,m): Ret="Identity" d=1 while m!=0: if m%2!=0: Ret,d=addPoint(E,Ret,P) if d!=1 : return [Ret,d] # as soon as i got anything otherthan 1 return P,d=addPoint(E,P,P) if d!=1 : return [Ret,d] m>>=1 return [Ret,d] def ellipticFactor(N,m,times=5): for i in xrange(times): E,P=randomCurve(N); Q,d=mulPoint(E,P,m) if d!=1 : return d return N if __name__=="__main__": n=input() for p in prime:#preprocessing while n%p==0: print p n/=p m=int(math.factorial(2000)) while n!=1: k=ellipticFactor(n,m) n/=k print k