def test_is_power(self):
self.assertRaisesRegexp(ValueError, "is_power: Must have k >= 1.", is_power, 10, 0)
self.assertRaisesRegexp(ValueError, "is_power: Must have n >= 1.", is_power, 0, 3)
for k in [2, 3, 5, 7, 11]:
self.assertEqual(is_power(k), False)
for k in xrange(10):
a = random.randint(10, 100)
b = random.randint(10, 100)
n = a**b
x, y = is_power(n)
self.assertTrue(x**y == n)
for k in xrange(10):
a = random.randint(10, 100)
b = random.randint(10, 100)
n = a**b
x, y = is_power(n, b)
self.assertTrue(x**y == n)
values = [
4, 8, 9, 16, 25, 27, 32, 36, 49, 64, 81, 100, 121, 125, 128, 144, 169, 196, 216, 225, 243, 256, 289, 324,
343, 361, 400, 441, 484, 512, 529, 576, 625, 676, 729, 784, 841, 900, 961, 1000
]
powers = [e for e in xrange(1, 1001) if is_power(e)]
self.assertEqual(values, powers)
self.assertEqual(is_power([(2, 4), (3, 2)]), (12, 2))
self.assertEqual(is_power([(2, 4), (3, 3)]), False)
self.assertEqual(is_power([(3, 4), (5, 4)]), (15, 4))
self.assertEqual(is_power([(3, 4), (5, 4)], 2), (225, 2))
def factor(n, skip_trial_division=False, use_cfrac=False):
r"""Returns the factorization of the integer `n`.
Parameters
----------
n : int
Integer to be factored.
skip_trial_division : bool, optional (default=False)
If False, performs a preliminary trial division using the
precomputed list of primes. If True, then this step is skipped.
use_cfrac : bool, optional (default=False)
If True, applies the cfrac method to search for factors instead of
the Pollard rho method. If False, then the Pollard rho method is
used to search for factors.
Returns
-------
factorization : list
A list of pairs (p, e) consisting of the primes p and exponents e
such that ``p**e`` divides `n`.
Notes
-----
This method uses a variety of techniques to obtain a factorization of
`n`. First, we perform trial division using a precomputed list of
primes to find all small prime factors. After this, we apply Pollard's
rho method to find any remaining small prime divisors.
References
----------
.. [1] H. Riesel, "Prime Numbers and Computer Methods for
Factorization", 2nd edition, Birkhauser Verlag, Basel, Switzerland,
1994.
Examples
--------
>>> factor(100)
[(2, 2), (5, 2)]
>>> factor(537869)
[(37, 1), (14537, 1)]
>>> factor(2**91 - 1)
[(127, 1), (911, 1), (8191, 1), (112901153L, 1), (23140471537L, 1)]
>>> factor(10**22 + 1)
[(89, 1), (101, 1), (1052788969L, 1), (1056689261L, 1)]
"""
fac = collections.defaultdict(int)
_is_prime = primality.is_probable_prime
if use_cfrac:
_factor = algorithms.cfrac
else:
_factor = algorithms.pollard_rho_brent
if n == 0:
raise ValueError("Prime factorization of 0 not defined")
# Detect perfect powers.
ab = is_power(n)
if ab:
(n, multiplier) = ab
else:
(n, multiplier) = (n, 1)
# Take care of the sign
if n < 0:
fac[-1] = 1
n *= -1
# Strip off all small factors of n
for p in _PRIME_LIST:
if n == 1:
break
elif p*p > n:
fac[n] += 1
n = 1
break
while n % p == 0:
fac[p] += 1
n //= p
# Next, strip off remaining factors.
while n > 1:
if _is_prime(n):
fac[n] += 1
n = 1
else:
d = _factor(n)
while not _is_prime(d):
d = _factor(d)
while n % d == 0:
fac[d] += 1
n = n // d
factorization = sorted([(p, fac[p]*multiplier) for p in fac])
return factorization