def qs(n,verbose=0):
if verbose:
print "Factoring a",int(log(n,10)+1),"digit number"
root2n=isqrt(2*n)
bound=int(5*log(n,10)**2)
factorbase = [2]+[x for x in prime_sieve(bound) if legendre_symbol(n,x)==1]
if verbose:
print "Largest Prime Factor used is",factorbase[-1]
tsqrt=[]
tlog=[]
for p in factorbase:
ms=int(modular_sqrt(n,p))
tsqrt.append(ms)
tlog.append(log(p,10))
xmax = len(factorbase)*60*4
m_val = (xmax * root2n) >> 1
thresh = log(m_val, 10) * 0.735
#ignore small primes
min_prime = int(thresh*3)
fudge = sum(tlog[i] for i,p in enumerate(factorbase) if p < min_prime)/4
thresh -= fudge
roota=int(isqrt(root2n/xmax))
roota=max(3,roota+int(roota&1==0))
smooths=[]
partials={}
polynomials=0
while 1:
while 1:
roota=next_prime(roota)
if legendre_symbol(n,roota)==1:break
polynomials+=1
a = int(roota*roota)
b = modular_sqrt(n,roota)
b = int((b-(b*b-n)*mod_inv(2*b,roota))%a)
c = int((b*b-n)/a)
sievesize = 1<<15
s1={};s2={}
#set up values
for i,p in enumerate(factorbase):
ainv=pow(a,p-2,p)
sol1=ainv*( tsqrt[i]-b) %p
sol2=ainv*(-tsqrt[i]-b) %p
sol1-=((xmax+sol1)/p)*p
sol2-=((xmax+sol2)/p)*p
s1[p]=int(sol1+xmax)
s2[p]=int(sol2+xmax)
#segmented sieve
for low in xrange(-xmax,xmax+1,sievesize+1):
high = min(xmax,low+sievesize)
size=high - low
S=[0.0]*(size+1)
#sieve segment
for i,p in enumerate(factorbase):
if p<min_prime:continue
sol1=s1[p]
sol2=s2[p]
logp=tlog[i]
while sol1<=size or sol2<=size:
if sol1<=size:
S[sol1]+=logp
sol1+=p
if sol2<=size:
S[sol2]+=logp
sol2+=p
s1[p]=int(sol1-size-1)
s2[p]=int(sol2-size-1)
#check segment for smooths
for i in xrange(0,size+1):
if S[i]>thresh:
x=i+low
tofact=nf=a*x*x+2*b*x+c
if nf<0:nf=-nf
for p in factorbase:
while nf%p==0:nf=int(nf/p)
if nf==1:
smooths+=[(a*x+b,(tofact,roota))]
#check for 1 big factor
elif nf in partials:
pairv,pairvals=partials[nf]
smooths+=[( (a*x+b)*pairv, ( tofact*pairvals[0],roota*pairvals[1]*nf) )]
del partials[nf]
else:
partials[nf]=(a*x+b,(tofact,roota))
if len(smooths)>len(factorbase):
break
if verbose:
print 100*len(smooths)/len(factorbase),'%','using',polynomials,'polynomials','\r',
if verbose:
print len(smooths),"relations found using",polynomials,"polynomials"
return algebra(factorbase,smooths,n)
#By listing the first six prime numbers: 2, 3, 5, 7, 11, and 13, we can see that the 6th prime is 13. #What is the 10001st prime number? import tools p=1 for a in range(0,10001): p = tools.next_prime(p) print p
def qs(n, verbose=0):
if verbose:
print "Factoring a", int(log(n, 10) + 1), "digit number"
root2n = isqrt(2 * n)
bound = int(5 * log(n, 10)**2)
factorbase = [2] + [
x for x in prime_sieve(bound) if legendre_symbol(n, x) == 1
]
if verbose:
print "Largest Prime Factor used is", factorbase[-1]
tsqrt = []
tlog = []
for p in factorbase:
ms = int(modular_sqrt(n, p))
tsqrt.append(ms)
tlog.append(log(p, 10))
xmax = len(factorbase) * 60 * 4
m_val = (xmax * root2n) >> 1
thresh = log(m_val, 10) * 0.735
#ignore small primes
min_prime = int(thresh * 3)
fudge = sum(tlog[i] for i, p in enumerate(factorbase) if p < min_prime) / 4
thresh -= fudge
roota = int(isqrt(root2n / xmax))
roota = max(3, roota + int(roota & 1 == 0))
smooths = []
partials = {}
polynomials = 0
while 1:
while 1:
roota = next_prime(roota)
if legendre_symbol(n, roota) == 1: break
polynomials += 1
a = int(roota * roota)
b = modular_sqrt(n, roota)
b = int((b - (b * b - n) * mod_inv(2 * b, roota)) % a)
c = int((b * b - n) / a)
sievesize = 1 << 15
s1 = {}
s2 = {}
#set up values
for i, p in enumerate(factorbase):
ainv = pow(a, p - 2, p)
sol1 = ainv * (tsqrt[i] - b) % p
sol2 = ainv * (-tsqrt[i] - b) % p
sol1 -= ((xmax + sol1) / p) * p
sol2 -= ((xmax + sol2) / p) * p
s1[p] = int(sol1 + xmax)
s2[p] = int(sol2 + xmax)
#segmented sieve
for low in xrange(-xmax, xmax + 1, sievesize + 1):
high = min(xmax, low + sievesize)
size = high - low
S = [0.0] * (size + 1)
#sieve segment
for i, p in enumerate(factorbase):
if p < min_prime: continue
sol1 = s1[p]
sol2 = s2[p]
logp = tlog[i]
while sol1 <= size or sol2 <= size:
if sol1 <= size:
S[sol1] += logp
sol1 += p
if sol2 <= size:
S[sol2] += logp
sol2 += p
s1[p] = int(sol1 - size - 1)
s2[p] = int(sol2 - size - 1)
#check segment for smooths
for i in xrange(0, size + 1):
if S[i] > thresh:
x = i + low
tofact = nf = a * x * x + 2 * b * x + c
if nf < 0: nf = -nf
for p in factorbase:
while nf % p == 0:
nf = int(nf / p)
if nf == 1:
smooths += [(a * x + b, (tofact, roota))]
#check for 1 big factor
elif nf in partials:
pairv, pairvals = partials[nf]
smooths += [((a * x + b) * pairv,
(tofact * pairvals[0],
roota * pairvals[1] * nf))]
del partials[nf]
else:
partials[nf] = (a * x + b, (tofact, roota))
if len(smooths) > len(factorbase):
break
if verbose:
print 100 * len(smooths) / len(
factorbase), '%', 'using', polynomials, 'polynomials', '\r',
if verbose:
print len(smooths), "relations found using", polynomials, "polynomials"
return algebra(factorbase, smooths, n)