def carmichael_lambda(n):
"""Returns the smallest positive integer m such that a**m = 1 (mod n) for
all integers a coprime to n.
Parameters
* n: int or list (n > 1)
The modulus. This value can be an int or a factorization.
Returns:
* m: int
Raises:
* ValueError: If n <= 0 or n is not an integer or factorization.
Examples:
>>> carmichael_lambda(100)
20
>>> carmichael_lambda([(2, 2), (5, 2)])
20
>>> carmichael_lambda(113)
112
>>> carmichael_lambda(0)
Traceback (most recent call last):
...
ValueError: carmichael_lambda: Must have n > 0.
Details:
The Carmichael lambda function gives the exponent of the multiplicative
group of integers modulo n. For each n, lambda(n) divides euler_phi(n).
This method uses the standard definition of the Carmichael lambda
function. In particular, lambda(n) is the least common multiple of the
values of lambda(p**e) for each prime power in the factorization of n.
See "Fundamental Number Theory with Applications" by Mollin for details.
"""
if isinstance(n, (int, long)):
if n == 1:
return 1
elif n <= 0:
raise ValueError("carmichael_lambda: Must have n > 0.")
n_factorization = factor.factor(n)
elif isinstance(n, list) and n[0][0] > 0:
n_factorization = n
else:
raise ValueError("carmichael_lambda: Input must be a positive integer or a factorization.")
terms = []
for (p, e) in n_factorization:
if p != 2:
term = p**(e - 1)*(p - 1)
else:
if e <= 2:
term = 1 << (e - 1)
else:
term = 1 << (e - 2)
terms.append(term)
value = lcm_list(terms)
return value
def carmichael_lambda(n):
r"""Returns the value of the Carmichael lambda function at `n`.
This function returns the smallest positive integer `m` such that
``a**m = 1 (mod n)`` for all integers `a` coprime to `n`. This integer
`m` is also called the least universal exponent for `n`.
Parameters
----------
n : integer or factorization (n > 1)
The value of `n` can be an integer or the factorization of an
integer given as a list of (prime, exponent) pairs.
Returns
-------
m : int
The least universal exponent for `n`.
Raises
------
ValueError
If ``n <= 0`` or if `n` is a valid factorization.
Notes
-----
The Carmichael lambda function gives the exponent of the multiplicative
group of integers modulo n. For each n, lambda(n) divides euler_phi(n).
This method uses the standard definition of the Carmichael lambda
function. In particular, lambda(n) is the least common multiple of the
values of lambda(p**e) for each prime power in the factorization of n.
See "Fundamental Number Theory with Applications" by Mollin for
details.
Examples
--------
>>> carmichael_lambda(100)
20
>>> carmichael_lambda([(2, 2), (5, 2)])
20
>>> carmichael_lambda(113)
112
>>> carmichael_lambda(0)
Traceback (most recent call last):
...
ValueError: carmichael_lambda: Must have n > 0.
"""
if isinstance(n, (int, long)):
if n == 1:
return 1
elif n <= 0:
raise ValueError("carmichael_lambda: Must have n > 0.")
n_factorization = rosemary.number_theory.factorization.factorization.factor(
n)
elif isinstance(n, list) and n[0][0] > 0:
n_factorization = n
else:
raise ValueError(
"carmichael_lambda: Input must be a positive integer or a factorization."
)
terms = []
for (p, e) in n_factorization:
if p != 2:
term = p**(e - 1) * (p - 1)
else:
if e <= 2:
term = 1 << (e - 1)
else:
term = 1 << (e - 2)
terms.append(term)
value = lcm_list(terms)
return value
def test_lcm_list(self):
self.assertEqual(lcm_list(range(1, 31)), 2329089562800)
def test_lcm_list(self):
self.assertEqual(lcm_list(range(1, 31)), 2329089562800)
