def pollards_pminus1(N, B=None, steps=None):
if steps == None:
steps = Steps()
if B == None:
steps.add_cuberoot(N)
B = math.pow(N, 1.0 / 3.0)
primes_list = get_primes()
a, i = 2, 0
a_set = set([])
while True:
pi = primes_list[i]
if pi > B:
break
steps.add_log(B)
steps.add_log(pi)
e = int(math.floor(math.log(B) / math.log(pi)))
steps.add_exp(pi, e)
f = pi**e
steps.add_exp(a, f)
steps.add_mod()
a_hold = powering.power_mod(a, f, N) #(a**f) % N
if a_hold == 0:
break
a = a_hold
a_set.add(a)
i += 1
for a in a_set:
g, g_steps = gcd(a - 1, N)
steps.append(g_steps)
if 1 < g and g < N:
return g, N / g, steps
return None, None, steps
def lehmans_var_of_fermat(N):
s3n = int(math.ceil(N**(1.0 / 3.0)))
steps = Steps()
for k in xrange(1, s3n + 1):
steps.add_sqrt(k * N)
sk = 2.0 * math.sqrt(k * N)
steps.add_exp(sk + N, 1.0 / 6.0)
steps.add_sqrt(k)
for a in xrange(
int(math.ceil(sk)),
int(
math.floor(sk + N**(1.0 / 6.0) / (4.0 * math.sqrt(k))) +
1)):
b = a * a - 4 * k * N
isq, isq_steps = is_square(b)
steps.append(isq_steps)
if isq:
my_gcd, gcd_steps = gcd(a + math.sqrt(b), N)
steps.append(gcd_steps)
return my_gcd, N / my_gcd, steps
raise Exception("should not get here ")