def fermats_diff_of_squares(N):
if N % 2 == 0:
raise ValueError("Must be given odd N")
steps = Steps()
steps.add_sqrt(N)
x = int(math.floor(math.sqrt(N)))
t, r = 2 * x + 1, x * x - N
isq, isq_steps = is_square(r)
steps.append(isq_steps)
while (not isq):
r += t
t += 2
isq, isq_steps = is_square(r)
steps.append(isq_steps)
x = (t - 1) / 2
steps.add_sqrt(r)
y = int(math.sqrt(r))
return x - y, x + y, steps
def harts_one_line(N, L=-1):
steps = Steps()
if L == -1:
L = N
for i in xrange(1, L):
steps.add_sqrt(N * i)
s = int(math.ceil(math.sqrt(N * i)))
steps.add_mod()
m = (s * s) % N
isq, isq_steps = is_square(m)
steps.append(isq_steps)
if (isq):
break
steps.add_sqrt(m)
t = math.sqrt(m)
f1, gcd_steps = gcd(s - t, N)
f2 = N / f1
steps.append(gcd_steps)
return f1, f2, steps
def trial_division(N, when_to_stop=None):
primes_list = get_primes()
m, pi, p = N, 0, 2
steps = Steps()
factors = []
if when_to_stop == None:
when_to_stop = math.sqrt(N)
steps.add_sqrt(N)
while (p <= when_to_stop):
steps.add_mod()
if m % p == 0:
m = m / p
factors.append(p)
else:
pi += 1
p = primes_list[pi]
num_unfactored = 0
if m != 1:
factors.append(m)
if m not in primes_list:
num_unfactored = 1
return factors, steps, num_unfactored
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 ")
def is_square(x):
steps = Steps()
steps.add_sqrt(x)
return x >= 0 and int(math.sqrt(x)) == math.sqrt(x), steps