# Source code for sympy.ntheory.factor_

"""
Integer factorization
"""
from __future__ import print_function, division

import random
import math

from .primetest import isprime
from .generate import sieve, primerange, nextprime
from sympy.core import sympify
from sympy.core.evalf import bitcount
from sympy.core.logic import fuzzy_and
from sympy.core.numbers import igcd, ilcm, Rational
from sympy.core.power import integer_nthroot, Pow
from sympy.core.mul import Mul
from sympy.core.compatibility import as_int, SYMPY_INTS, range
from sympy.core.singleton import S
from sympy.core.function import Function
from sympy.core.expr import Expr

small_trailing = [i and max(int(not i % 2**j) and j for j in range(1, 8))
for i in range(256)]

[docs]def smoothness(n):
"""
Return the B-smooth and B-power smooth values of n.

The smoothness of n is the largest prime factor of n; the power-
smoothness is the largest divisor raised to its multiplicity.

>>> from sympy.ntheory.factor_ import smoothness
>>> smoothness(2**7*3**2)
(3, 128)
>>> smoothness(2**4*13)
(13, 16)
>>> smoothness(2)
(2, 2)

========

factorint, smoothness_p
"""

if n == 1:
return (1, 1)  # not prime, but otherwise this causes headaches
facs = factorint(n)
return max(facs), max(m**facs[m] for m in facs)

[docs]def smoothness_p(n, m=-1, power=0, visual=None):
"""
Return a list of [m, (p, (M, sm(p + m), psm(p + m)))...]
where:

1. p**M is the base-p divisor of n
2. sm(p + m) is the smoothness of p + m (m = -1 by default)
3. psm(p + m) is the power smoothness of p + m

The list is sorted according to smoothness (default) or by power smoothness
if power=1.

The smoothness of the numbers to the left (m = -1) or right (m = 1) of a
factor govern the results that are obtained from the p +/- 1 type factoring
methods.

>>> from sympy.ntheory.factor_ import smoothness_p, factorint
>>> smoothness_p(10431, m=1)
(1, [(3, (2, 2, 4)), (19, (1, 5, 5)), (61, (1, 31, 31))])
>>> smoothness_p(10431)
(-1, [(3, (2, 2, 2)), (19, (1, 3, 9)), (61, (1, 5, 5))])
>>> smoothness_p(10431, power=1)
(-1, [(3, (2, 2, 2)), (61, (1, 5, 5)), (19, (1, 3, 9))])

If visual=True then an annotated string will be returned:

>>> print(smoothness_p(21477639576571, visual=1))
p**i=4410317**1 has p-1 B=1787, B-pow=1787
p**i=4869863**1 has p-1 B=2434931, B-pow=2434931

This string can also be generated directly from a factorization dictionary
and vice versa:

>>> factorint(17*9)
{3: 2, 17: 1}
>>> smoothness_p(_)
'p**i=3**2 has p-1 B=2, B-pow=2\\np**i=17**1 has p-1 B=2, B-pow=16'
>>> smoothness_p(_)
{3: 2, 17: 1}

The table of the output logic is:

====== ====== ======= =======
|              Visual
------ ----------------------
Input  True   False   other
====== ====== ======= =======
dict    str    tuple   str
str     str    tuple   dict
tuple   str    tuple   str
n       str    tuple   tuple
mul     str    tuple   tuple
====== ====== ======= =======

========

factorint, smoothness
"""
from sympy.utilities import flatten

# visual must be True, False or other (stored as None)
if visual in (1, 0):
visual = bool(visual)
elif visual not in (True, False):
visual = None

if type(n) is str:
if visual:
return n
d = {}
for li in n.splitlines():
k, v = [int(i) for i in
li.split('has')[0].split('=')[1].split('**')]
d[k] = v
if visual is not True and visual is not False:
return d
return smoothness_p(d, visual=False)
elif type(n) is not tuple:
facs = factorint(n, visual=False)

if power:
k = -1
else:
k = 1
if type(n) is not tuple:
rv = (m, sorted([(f,
tuple([M] + list(smoothness(f + m))))
for f, M in [i for i in facs.items()]],
key=lambda x: (x[1][k], x[0])))
else:
rv = n

if visual is False or (visual is not True) and (type(n) in [int, Mul]):
return rv
lines = []
for dat in rv[1]:
dat = flatten(dat)
dat.insert(2, m)
lines.append('p**i=%i**%i has p%+i B=%i, B-pow=%i' % tuple(dat))
return '\n'.join(lines)

[docs]def trailing(n):
"""Count the number of trailing zero digits in the binary
representation of n, i.e. determine the largest power of 2
that divides n.

Examples
========

>>> from sympy import trailing
>>> trailing(128)
7
>>> trailing(63)
0
"""
n = abs(int(n))
if not n:
return 0
low_byte = n & 0xff
if low_byte:
return small_trailing[low_byte]

# 2**m is quick for z up through 2**30
z = bitcount(n) - 1
if isinstance(z, SYMPY_INTS):
if n == 1 << z:
return z

t = 0
p = 8
while not n & 1:
while not n & ((1 << p) - 1):
n >>= p
t += p
p *= 2
p //= 2
return t

[docs]def multiplicity(p, n):
"""
Find the greatest integer m such that p**m divides n.

Examples
========

>>> from sympy.ntheory import multiplicity
>>> from sympy.core.numbers import Rational as R
>>> [multiplicity(5, n) for n in [8, 5, 25, 125, 250]]
[0, 1, 2, 3, 3]
>>> multiplicity(3, R(1, 9))
-2

"""
try:
p, n = as_int(p), as_int(n)
except ValueError:
if all(isinstance(i, (SYMPY_INTS, Rational)) for i in (p, n)):
try:
p = Rational(p)
n = Rational(n)
if p.q == 1:
if n.p == 1:
return -multiplicity(p.p, n.q)
return S.Zero
elif p.p == 1:
return multiplicity(p.q, n.q)
else:
like = min(
multiplicity(p.p, n.p),
multiplicity(p.q, n.q))
cross = min(
multiplicity(p.q, n.p),
multiplicity(p.p, n.q))
return like - cross
except AttributeError:
pass
raise ValueError('expecting ints or fractions, got %s and %s' % (p, n))

if n == 0:
raise ValueError('no such integer exists: multiplicity of %s is not-defined' %(n))
if p == 2:
return trailing(n)
if p < 2:
raise ValueError('p must be an integer, 2 or larger, but got %s' % p)
if p == n:
return 1

m = 0
n, rem = divmod(n, p)
while not rem:
m += 1
if m > 5:
# The multiplicity could be very large. Better
# to increment in powers of two
e = 2
while 1:
ppow = p**e
if ppow < n:
nnew, rem = divmod(n, ppow)
if not rem:
m += e
e *= 2
n = nnew
continue
return m + multiplicity(p, n)
n, rem = divmod(n, p)
return m

[docs]def perfect_power(n, candidates=None, big=True, factor=True):
"""
Return (b, e) such that n == b**e if n is a
perfect power; otherwise return False.

By default, the base is recursively decomposed and the exponents
collected so the largest possible e is sought. If big=False
then the smallest possible e (thus prime) will be chosen.

If candidates for exponents are given, they are assumed to be sorted
and the first one that is larger than the computed maximum will signal
failure for the routine.

If factor=True then simultaneous factorization of n is attempted
since finding a factor indicates the only possible root for n. This
is True by default since only a few small factors will be tested in
the course of searching for the perfect power.

Examples
========

>>> from sympy import perfect_power
>>> perfect_power(16)
(2, 4)
>>> perfect_power(16, big = False)
(4, 2)
"""
n = int(n)
if n < 3:
return False
logn = math.log(n, 2)
max_possible = int(logn) + 2  # only check values less than this
not_square = n % 10 in [2, 3, 7, 8]  # squares cannot end in 2, 3, 7, 8
if not candidates:
candidates = primerange(2 + not_square, max_possible)

afactor = 2 + n % 2
for e in candidates:
if e < 3:
if e == 1 or e == 2 and not_square:
continue
if e > max_possible:
return False

# see if there is a factor present
if factor:
if n % afactor == 0:
# find what the potential power is
if afactor == 2:
e = trailing(n)
else:
e = multiplicity(afactor, n)
# if it's a trivial power we are done
if e == 1:
return False

# maybe the bth root of n is exact
r, exact = integer_nthroot(n, e)
if not exact:
# then remove this factor and check to see if
# any of e's factors are a common exponent; if
# not then it's not a perfect power
n //= afactor**e
m = perfect_power(n, candidates=primefactors(e), big=big)
if m is False:
return False
else:
r, m = m
# adjust the two exponents so the bases can
# be combined
g = igcd(m, e)
if g == 1:
return False
m //= g
e //= g
r, e = r**m*afactor**e, g
if not big:
e0 = primefactors(e)
if len(e0) > 1 or e0[0] != e:
e0 = e0[0]
r, e = r**(e//e0), e0
return r, e
else:
# get the next factor ready for the next pass through the loop
afactor = nextprime(afactor)

# Weed out downright impossible candidates
if logn/e < 40:
b = 2.0**(logn/e)
if abs(int(b + 0.5) - b) > 0.01:
continue

# now see if the plausible e makes a perfect power
r, exact = integer_nthroot(n, e)
if exact:
if big:
m = perfect_power(r, big=big, factor=factor)
if m is not False:
r, e = m[0], e*m[1]
return int(r), e
else:
return False

[docs]def pollard_rho(n, s=2, a=1, retries=5, seed=1234, max_steps=None, F=None):
r"""
Use Pollard's rho method to try to extract a nontrivial factor
of n. The returned factor may be a composite number. If no
factor is found, None is returned.

The algorithm generates pseudo-random values of x with a generator
function, replacing x with F(x). If F is not supplied then the
function x**2 + a is used. The first value supplied to F(x) is s.
Upon failure (if retries is > 0) a new a and s will be
supplied; the a will be ignored if F was supplied.

The sequence of numbers generated by such functions generally have a
a lead-up to some number and then loop around back to that number and
begin to repeat the sequence, e.g. 1, 2, 3, 4, 5, 3, 4, 5 -- this leader
and loop look a bit like the Greek letter rho, and thus the name, 'rho'.

For a given function, very different leader-loop values can be obtained
so it is a good idea to allow for retries:

>>> from sympy.ntheory.generate import cycle_length
>>> n = 16843009
>>> F = lambda x:(2048*pow(x, 2, n) + 32767) % n
>>> for s in range(5):
...     print('loop length = %4i; leader length = %3i' % next(cycle_length(F, s)))
...
loop length = 2489; leader length =  42
loop length =   78; leader length = 120
loop length = 1482; leader length =  99
loop length = 1482; leader length = 285
loop length = 1482; leader length = 100

Here is an explicit example where there is a two element leadup to
a sequence of 3 numbers (11, 14, 4) that then repeat:

>>> x=2
>>> for i in range(9):
...     x=(x**2+12)%17
...     print(x)
...
16
13
11
14
4
11
14
4
11
>>> next(cycle_length(lambda x: (x**2+12)%17, 2))
(3, 2)
>>> list(cycle_length(lambda x: (x**2+12)%17, 2, values=True))
[16, 13, 11, 14, 4]

Instead of checking the differences of all generated values for a gcd
with n, only the kth and 2*kth numbers are checked, e.g. 1st and 2nd,
2nd and 4th, 3rd and 6th until it has been detected that the loop has been
traversed. Loops may be many thousands of steps long before rho finds a
factor or reports failure. If max_steps is specified, the iteration
is cancelled with a failure after the specified number of steps.

Examples
========

>>> from sympy import pollard_rho
>>> n=16843009
>>> F=lambda x:(2048*pow(x,2,n) + 32767) % n
>>> pollard_rho(n, F=F)
257

Use the default setting with a bad value of a and no retries:

>>> pollard_rho(n, a=n-2, retries=0)

If retries is > 0 then perhaps the problem will correct itself when
new values are generated for a:

>>> pollard_rho(n, a=n-2, retries=1)
257

References
==========

- Richard Crandall & Carl Pomerance (2005), "Prime Numbers:
A Computational Perspective", Springer, 2nd edition, 229-231

"""
n = int(n)
if n < 5:
raise ValueError('pollard_rho should receive n > 4')
prng = random.Random(seed + retries)
V = s
for i in range(retries + 1):
U = V
if not F:
F = lambda x: (pow(x, 2, n) + a) % n
j = 0
while 1:
if max_steps and (j > max_steps):
break
j += 1
U = F(U)
V = F(F(V))  # V is 2x further along than U
g = igcd(U - V, n)
if g == 1:
continue
if g == n:
break
return int(g)
V = prng.randint(0, n - 1)
a = prng.randint(1, n - 3)  # for x**2 + a, a%n should not be 0 or -2
F = None
return None

[docs]def pollard_pm1(n, B=10, a=2, retries=0, seed=1234):
"""
Use Pollard's p-1 method to try to extract a nontrivial factor
of n. Either a divisor (perhaps composite) or None is returned.

The value of a is the base that is used in the test gcd(a**M - 1, n).
The default is 2.  If retries > 0 then if no factor is found after the
first attempt, a new a will be generated randomly (using the seed)
and the process repeated.

Note: the value of M is lcm(1..B) = reduce(ilcm, range(2, B + 1)).

A search is made for factors next to even numbers having a power smoothness
less than B. Choosing a larger B increases the likelihood of finding a
larger factor but takes longer. Whether a factor of n is found or not
depends on a and the power smoothness of the even mumber just less than
the factor p (hence the name p - 1).

Although some discussion of what constitutes a good a some
descriptions are hard to interpret. At the modular.math site referenced
below it is stated that if gcd(a**M - 1, n) = N then a**M % q**r is 1
for every prime power divisor of N. But consider the following:

>>> from sympy.ntheory.factor_ import smoothness_p, pollard_pm1
>>> n=257*1009
>>> smoothness_p(n)
(-1, [(257, (1, 2, 256)), (1009, (1, 7, 16))])

So we should (and can) find a root with B=16:

>>> pollard_pm1(n, B=16, a=3)
1009

If we attempt to increase B to 256 we find that it doesn't work:

>>> pollard_pm1(n, B=256)
>>>

But if the value of a is changed we find that only multiples of
257 work, e.g.:

>>> pollard_pm1(n, B=256, a=257)
1009

Checking different a values shows that all the ones that didn't
work had a gcd value not equal to n but equal to one of the
factors:

>>> from sympy.core.numbers import ilcm, igcd
>>> from sympy import factorint, Pow
>>> M = 1
>>> for i in range(2, 256):
...     M = ilcm(M, i)
...
>>> set([igcd(pow(a, M, n) - 1, n) for a in range(2, 256) if
...      igcd(pow(a, M, n) - 1, n) != n])
{1009}

But does aM % d for every divisor of n give 1?

>>> aM = pow(255, M, n)
>>> [(d, aM%Pow(*d.args)) for d in factorint(n, visual=True).args]
[(257**1, 1), (1009**1, 1)]

No, only one of them. So perhaps the principle is that a root will
be found for a given value of B provided that:

1) the power smoothness of the p - 1 value next to the root
does not exceed B
2) a**M % p != 1 for any of the divisors of n.

By trying more than one a it is possible that one of them
will yield a factor.

Examples
========

With the default smoothness bound, this number can't be cracked:

>>> from sympy.ntheory import pollard_pm1, primefactors
>>> pollard_pm1(21477639576571)

Increasing the smoothness bound helps:

>>> pollard_pm1(21477639576571, B=2000)
4410317

Looking at the smoothness of the factors of this number we find:

>>> from sympy.utilities import flatten
>>> from sympy.ntheory.factor_ import smoothness_p, factorint
>>> print(smoothness_p(21477639576571, visual=1))
p**i=4410317**1 has p-1 B=1787, B-pow=1787
p**i=4869863**1 has p-1 B=2434931, B-pow=2434931

The B and B-pow are the same for the p - 1 factorizations of the divisors
because those factorizations had a very large prime factor:

>>> factorint(4410317 - 1)
{2: 2, 617: 1, 1787: 1}
>>> factorint(4869863-1)
{2: 1, 2434931: 1}

Note that until B reaches the B-pow value of 1787, the number is not cracked;

>>> pollard_pm1(21477639576571, B=1786)
>>> pollard_pm1(21477639576571, B=1787)
4410317

The B value has to do with the factors of the number next to the divisor,
not the divisors themselves. A worst case scenario is that the number next
to the factor p has a large prime divisisor or is a perfect power. If these
conditions apply then the power-smoothness will be about p/2 or p. The more
realistic is that there will be a large prime factor next to p requiring
a B value on the order of p/2. Although primes may have been searched for
up to this level, the p/2 is a factor of p - 1, something that we don't
know. The modular.math reference below states that 15% of numbers in the
range of 10**15 to 15**15 + 10**4 are 10**6 power smooth so a B of 10**6
will fail 85% of the time in that range. From 10**8 to 10**8 + 10**3 the
percentages are nearly reversed...but in that range the simple trial
division is quite fast.

References
==========

- Richard Crandall & Carl Pomerance (2005), "Prime Numbers:
A Computational Perspective", Springer, 2nd edition, 236-238
- http://modular.math.washington.edu/edu/2007/spring/ent/ent-html/node81.html
- http://www.cs.toronto.edu/~yuvalf/Factorization.pdf
"""

n = int(n)
if n < 4 or B < 3:
raise ValueError('pollard_pm1 should receive n > 3 and B > 2')
prng = random.Random(seed + B)

# computing a**lcm(1,2,3,..B) % n for B > 2
# it looks weird, but it's right: primes run [2, B]
# and the answer's not right until the loop is done.
for i in range(retries + 1):
aM = a
for p in sieve.primerange(2, B + 1):
e = int(math.log(B, p))
aM = pow(aM, pow(p, e), n)
g = igcd(aM - 1, n)
if 1 < g < n:
return int(g)

# get a new a:
# since the exponent, lcm(1..B), is even, if we allow 'a' to be 'n-1'
# then (n - 1)**even % n will be 1 which will give a g of 0 and 1 will
# give a zero, too, so we set the range as [2, n-2]. Some references
# say 'a' should be coprime to n, but either will detect factors.
a = prng.randint(2, n - 2)

def _trial(factors, n, candidates, verbose=False):
"""
Helper function for integer factorization. Trial factors n
against all integers given in the sequence candidates
and updates the dict factors in-place. Returns the reduced
value of n and a flag indicating whether any factors were found.
"""
if verbose:
factors0 = list(factors.keys())
nfactors = len(factors)
for d in candidates:
if n % d == 0:
m = multiplicity(d, n)
n //= d**m
factors[d] = m
if verbose:
for k in sorted(set(factors).difference(set(factors0))):
print(factor_msg % (k, factors[k]))
return int(n), len(factors) != nfactors

def _check_termination(factors, n, limitp1, use_trial, use_rho, use_pm1,
verbose):
"""
Helper function for integer factorization. Checks if n
is a prime or a perfect power, and in those cases updates
the factorization and raises StopIteration.
"""

if verbose:
print('Check for termination')

# since we've already been factoring there is no need to do
# simultaneous factoring with the power check
p = perfect_power(n, factor=False)
if p is not False:
base, exp = p
if limitp1:
limit = limitp1 - 1
else:
limit = limitp1
facs = factorint(base, limit, use_trial, use_rho, use_pm1,
verbose=False)
for b, e in facs.items():
if verbose:
print(factor_msg % (b, e))
factors[b] = exp*e
raise StopIteration

if isprime(n):
factors[int(n)] = 1
raise StopIteration

if n == 1:
raise StopIteration

trial_int_msg = "Trial division with ints [%i ... %i] and fail_max=%i"
trial_msg = "Trial division with primes [%i ... %i]"
rho_msg = "Pollard's rho with retries %i, max_steps %i and seed %i"
pm1_msg = "Pollard's p-1 with smoothness bound %i and seed %i"
factor_msg = '\t%i ** %i'
fermat_msg = 'Close factors satisying Fermat condition found.'
complete_msg = 'Factorization is complete.'

def _factorint_small(factors, n, limit, fail_max):
"""
Return the value of n and either a 0 (indicating that factorization up
to the limit was complete) or else the next near-prime that would have
been tested.

Factoring stops if there are fail_max unsuccessful tests in a row.

If factors of n were found they will be in the factors dictionary as
{factor: multiplicity} and the returned value of n will have had those
factors removed. The factors dictionary is modified in-place.

"""

def done(n, d):
"""return n, d if the sqrt(n) wasn't reached yet, else
n, 0 indicating that factoring is done.
"""
if d*d <= n:
return n, d
return n, 0

d = 2
m = trailing(n)
if m:
factors[d] = m
n >>= m
d = 3
if limit < d:
if n > 1:
factors[n] = 1
return done(n, d)
# reduce
m = 0
while n % d == 0:
n //= d
m += 1
if m == 20:
mm = multiplicity(d, n)
m += mm
n //= d**mm
break
if m:
factors[d] = m

# when d*d exceeds maxx or n we are done; if limit**2 is greater
# than n then maxx is set to zero so the value of n will flag the finish
if limit*limit > n:
maxx = 0
else:
maxx = limit*limit

dd = maxx or n
d = 5
fails = 0
while fails < fail_max:
if d*d > dd:
break
# d = 6*i - 1
# reduce
m = 0
while n % d == 0:
n //= d
m += 1
if m == 20:
mm = multiplicity(d, n)
m += mm
n //= d**mm
break
if m:
factors[d] = m
dd = maxx or n
fails = 0
else:
fails += 1
d += 2
if d*d > dd:
break
# d = 6*i - 1
# reduce
m = 0
while n % d == 0:
n //= d
m += 1
if m == 20:
mm = multiplicity(d, n)
m += mm
n //= d**mm
break
if m:
factors[d] = m
dd = maxx or n
fails = 0
else:
fails += 1
# d = 6*(i+1) - 1
d += 4

return done(n, d)

[docs]def factorint(n, limit=None, use_trial=True, use_rho=True, use_pm1=True,
verbose=False, visual=None, multiple=False):
r"""
Given a positive integer n, factorint(n) returns a dict containing
the prime factors of n as keys and their respective multiplicities
as values. For example:

>>> from sympy.ntheory import factorint
>>> factorint(2000)    # 2000 = (2**4) * (5**3)
{2: 4, 5: 3}
>>> factorint(65537)   # This number is prime
{65537: 1}

For input less than 2, factorint behaves as follows:

- factorint(1) returns the empty factorization, {}
- factorint(0) returns {0:1}
- factorint(-n) adds -1:1 to the factors and then factors n

Partial Factorization:

If limit (> 3) is specified, the search is stopped after performing
trial division up to (and including) the limit (or taking a
corresponding number of rho/p-1 steps). This is useful if one has
a large number and only is interested in finding small factors (if
any). Note that setting a limit does not prevent larger factors
from being found early; it simply means that the largest factor may
be composite. Since checking for perfect power is relatively cheap, it is
done regardless of the limit setting.

This number, for example, has two small factors and a huge
semi-prime factor that cannot be reduced easily:

>>> from sympy.ntheory import isprime
>>> from sympy.core.compatibility import long
>>> a = 1407633717262338957430697921446883
>>> f = factorint(a, limit=10000)
>>> f == {991: 1, long(202916782076162456022877024859): 1, 7: 1}
True
>>> isprime(max(f))
False

This number has a small factor and a residual perfect power whose
base is greater than the limit:

>>> factorint(3*101**7, limit=5)
{3: 1, 101: 7}

List of Factors:

If multiple is set to True then a list containing the
prime factors including multiplicities is returned.

>>> factorint(24, multiple=True)
[2, 2, 2, 3]

Visual Factorization:

If visual is set to True, then it will return a visual
factorization of the integer.  For example:

>>> from sympy import pprint
>>> pprint(factorint(4200, visual=True))
3  1  2  1
2 *3 *5 *7

Note that this is achieved by using the evaluate=False flag in Mul
and Pow. If you do other manipulations with an expression where
evaluate=False, it may evaluate.  Therefore, you should use the
visual option only for visualization, and use the normal dictionary
returned by visual=False if you want to perform operations on the
factors.

You can easily switch between the two forms by sending them back to
factorint:

>>> from sympy import Mul, Pow
>>> regular = factorint(1764); regular
{2: 2, 3: 2, 7: 2}
>>> pprint(factorint(regular))
2  2  2
2 *3 *7

>>> visual = factorint(1764, visual=True); pprint(visual)
2  2  2
2 *3 *7
>>> print(factorint(visual))
{2: 2, 3: 2, 7: 2}

If you want to send a number to be factored in a partially factored form
you can do so with a dictionary or unevaluated expression:

>>> factorint(factorint({4: 2, 12: 3})) # twice to toggle to dict form
{2: 10, 3: 3}
>>> factorint(Mul(4, 12, evaluate=False))
{2: 4, 3: 1}

The table of the output logic is:

====== ====== ======= =======
Visual
------ ----------------------
Input  True   False   other
====== ====== ======= =======
dict    mul    dict    mul
n       mul    dict    dict
mul     mul    dict    dict
====== ====== ======= =======

Notes
=====

Algorithm:

The function switches between multiple algorithms. Trial division
quickly finds small factors (of the order 1-5 digits), and finds
all large factors if given enough time. The Pollard rho and p-1
algorithms are used to find large factors ahead of time; they
will often find factors of the order of 10 digits within a few
seconds:

>>> factors = factorint(12345678910111213141516)
>>> for base, exp in sorted(factors.items()):
...     print('%s %s' % (base, exp))
...
2 2
2507191691 1
1231026625769 1

Any of these methods can optionally be disabled with the following
boolean parameters:

- use_trial: Toggle use of trial division
- use_rho: Toggle use of Pollard's rho method
- use_pm1: Toggle use of Pollard's p-1 method

factorint also periodically checks if the remaining part is
a prime number or a perfect power, and in those cases stops.

For unevaluated factorial, it uses Legendre's formula(theorem).

If verbose is set to True, detailed progress is printed.

========

smoothness, smoothness_p, divisors

"""
if multiple:
fac = factorint(n, limit=limit, use_trial=use_trial,
use_rho=use_rho, use_pm1=use_pm1,
verbose=verbose, visual=False, multiple=False)
factorlist = sum(([p] * fac[p] if fac[p] > 0 else [S(1)/p]*(-1*fac[p])
for p in sorted(fac)), [])
return factorlist

factordict = {}
if visual and not isinstance(n, Mul) and not isinstance(n, dict):
factordict = factorint(n, limit=limit, use_trial=use_trial,
use_rho=use_rho, use_pm1=use_pm1,
verbose=verbose, visual=False)
elif isinstance(n, Mul):
factordict = dict([(int(k), int(v)) for k, v in
list(n.as_powers_dict().items())])
elif isinstance(n, dict):
factordict = n
if factordict and (isinstance(n, Mul) or isinstance(n, dict)):
# check it
for k in list(factordict.keys()):
if isprime(k):
continue
e = factordict.pop(k)
d = factorint(k, limit=limit, use_trial=use_trial, use_rho=use_rho,
use_pm1=use_pm1, verbose=verbose, visual=False)
for k, v in d.items():
if k in factordict:
factordict[k] += v*e
else:
factordict[k] = v*e
if visual or (type(n) is dict and
visual is not True and
visual is not False):
if factordict == {}:
return S.One
if -1 in factordict:
factordict.pop(-1)
args = [S.NegativeOne]
else:
args = []
args.extend([Pow(*i, evaluate=False)
for i in sorted(factordict.items())])
return Mul(*args, evaluate=False)
elif isinstance(n, dict) or isinstance(n, Mul):
return factordict

assert use_trial or use_rho or use_pm1

# for unevaluated factorial, if n < 20!, direct computation is faster
# since it uses lookup table
from sympy.functions.combinatorial.factorials import factorial
if isinstance(n, factorial) and n.args[0].is_Integer and n.args[0] >= 20:
x = n.args[0]
factors = {}
for p in sieve.primerange(2, x):
m = 0
d = p
q = x // p
while q != 0:
m += q
d *= p
q = x // d
factors[p] = m
if factors and verbose:
for k in sorted(factors):
print(factor_msg % (k, factors[k]))
if verbose:
print(complete_msg)
return factors

n = as_int(n)
if limit:
limit = int(limit)

# special cases
if n < 0:
factors = factorint(
-n, limit=limit, use_trial=use_trial, use_rho=use_rho,
use_pm1=use_pm1, verbose=verbose, visual=False)
factors[-1] = 1
return factors

if limit and limit < 2:
if n == 1:
return {}
return {n: 1}
elif n < 10:
# doing this we are assured of getting a limit > 2
# when we have to compute it later
return [{0: 1}, {}, {2: 1}, {3: 1}, {2: 2}, {5: 1},
{2: 1, 3: 1}, {7: 1}, {2: 3}, {3: 2}][n]

factors = {}

# do simplistic factorization
if verbose:
sn = str(n)
if len(sn) > 50:
print('Factoring %s' % sn[:5] + \
'..(%i other digits)..' % (len(sn) - 10) + sn[-5:])
else:
print('Factoring', n)

if use_trial:
# this is the preliminary factorization for small factors
small = 2**15
fail_max = 600
small = min(small, limit or small)
if verbose:
print(trial_int_msg % (2, small, fail_max))
n, next_p = _factorint_small(factors, n, small, fail_max)
else:
next_p = 2
if factors and verbose:
for k in sorted(factors):
print(factor_msg % (k, factors[k]))
if next_p == 0:
if n > 1:
factors[int(n)] = 1
if verbose:
print(complete_msg)
return factors

# continue with more advanced factorization methods

# first check if the simplistic run didn't finish
# because of the limit and check for a perfect
# power before exiting
try:
if limit and next_p > limit:
if verbose:
print('Exceeded limit:', limit)

_check_termination(factors, n, limit, use_trial, use_rho, use_pm1,
verbose)

if n > 1:
factors[int(n)] = 1
return factors
else:
# Before quitting (or continuing on)...

# ...do a Fermat test since it's so easy and we need the
# square root anyway. Finding 2 factors is easy if they are
# "close enough." This is the big root equivalent of dividing by
# 2, 3, 5.
sqrt_n = integer_nthroot(n, 2)[0]
a = sqrt_n + 1
a2 = a**2
b2 = a2 - n
for i in range(3):
b, fermat = integer_nthroot(b2, 2)
if fermat:
break
b2 += 2*a + 1  # equiv to (a+1)**2 - n
a += 1
if fermat:
if verbose:
print(fermat_msg)
if limit:
limit -= 1
for r in [a - b, a + b]:
facs = factorint(r, limit=limit, use_trial=use_trial,
use_rho=use_rho, use_pm1=use_pm1,
verbose=verbose)
factors.update(facs)
raise StopIteration

# ...see if factorization can be terminated
_check_termination(factors, n, limit, use_trial, use_rho, use_pm1,
verbose)

except StopIteration:
if verbose:
print(complete_msg)
return factors

# these are the limits for trial division which will
# be attempted in parallel with pollard methods
low, high = next_p, 2*next_p

limit = limit or sqrt_n
# add 1 to make sure limit is reached in primerange calls
limit += 1

while 1:

try:
high_ = high
if limit < high_:
high_ = limit

# Trial division
if use_trial:
if verbose:
print(trial_msg % (low, high_))
ps = sieve.primerange(low, high_)
n, found_trial = _trial(factors, n, ps, verbose)
if found_trial:
_check_termination(factors, n, limit, use_trial, use_rho,
use_pm1, verbose)
else:
found_trial = False

if high > limit:
if verbose:
print('Exceeded limit:', limit)
if n > 1:
factors[int(n)] = 1
raise StopIteration

# Only used advanced methods when no small factors were found
if (use_pm1 or use_rho):
high_root = max(int(math.log(high_**0.7)), low, 3)

# Pollard p-1
if use_pm1:
if verbose:
print(pm1_msg % (high_root, high_))
c = pollard_pm1(n, B=high_root, seed=high_)
if c:
# factor it and let _trial do the update
ps = factorint(c, limit=limit - 1,
use_trial=use_trial,
use_rho=use_rho,
use_pm1=use_pm1,
verbose=verbose)
n, _ = _trial(factors, n, ps, verbose=False)
_check_termination(factors, n, limit, use_trial,
use_rho, use_pm1, verbose)

# Pollard rho
if use_rho:
max_steps = high_root
if verbose:
print(rho_msg % (1, max_steps, high_))
c = pollard_rho(n, retries=1, max_steps=max_steps,
seed=high_)
if c:
# factor it and let _trial do the update
ps = factorint(c, limit=limit - 1,
use_trial=use_trial,
use_rho=use_rho,
use_pm1=use_pm1,
verbose=verbose)
n, _ = _trial(factors, n, ps, verbose=False)
_check_termination(factors, n, limit, use_trial,
use_rho, use_pm1, verbose)

except StopIteration:
if verbose:
print(complete_msg)
return factors

low, high = high, high*2

def factorrat(rat, limit=None, use_trial=True, use_rho=True, use_pm1=True,
verbose=False, visual=None, multiple=False):
r"""
Given a Rational r, factorrat(r) returns a dict containing
the prime factors of r as keys and their respective multiplicities
as values. For example:

>>> from sympy.ntheory import factorrat
>>> from sympy.core.symbol import S
>>> factorrat(S(8)/9)    # 8/9 = (2**3) * (3**-2)
{2: 3, 3: -2}
>>> factorrat(S(-1)/987)    # -1/789 = -1 * (3**-1) * (7**-1) * (47**-1)
{-1: 1, 3: -1, 7: -1, 47: -1}

Please see the docstring for factorint for detailed explanations
and examples of the following keywords:

- limit: Integer limit up to which trial division is done
- use_trial: Toggle use of trial division
- use_rho: Toggle use of Pollard's rho method
- use_pm1: Toggle use of Pollard's p-1 method
- verbose: Toggle detailed printing of progress
- multiple: Toggle returning a list of factors or dict
- visual: Toggle product form of output
"""
from collections import defaultdict
if multiple:
fac = factorrat(rat, limit=limit, use_trial=use_trial,
use_rho=use_rho, use_pm1=use_pm1,
verbose=verbose, visual=False,multiple=False)
factorlist = sum(([p] * fac[p] if fac[p] > 0 else [S(1)/p]*(-1*fac[p])
for p, _ in sorted(fac.items(),
key=lambda elem: elem[0]
if elem[1] > 0
else 1/elem[0])), [])
return factorlist

f = factorint(rat.p, limit=limit, use_trial=use_trial,
use_rho=use_rho, use_pm1=use_pm1,
verbose=verbose).copy()
f = defaultdict(int, f)
for p, e in factorint(rat.q, limit=limit,
use_trial=use_trial,
use_rho=use_rho,
use_pm1=use_pm1,
verbose=verbose).items():
f[p] += -e

if len(f) > 1 and 1 in f:
del f[1]
if not visual:
return dict(f)
else:
if -1 in f:
f.pop(-1)
args = [S.NegativeOne]
else:
args = []
args.extend([Pow(*i, evaluate=False)
for i in sorted(f.items())])
return Mul(*args, evaluate=False)

[docs]def primefactors(n, limit=None, verbose=False):
"""Return a sorted list of n's prime factors, ignoring multiplicity
and any composite factor that remains if the limit was set too low
for complete factorization. Unlike factorint(), primefactors() does
not return -1 or 0.

Examples
========

>>> from sympy.ntheory import primefactors, factorint, isprime
>>> primefactors(6)
[2, 3]
>>> primefactors(-5)
[5]

>>> sorted(factorint(123456).items())
[(2, 6), (3, 1), (643, 1)]
>>> primefactors(123456)
[2, 3, 643]

>>> sorted(factorint(10000000001, limit=200).items())
[(101, 1), (99009901, 1)]
>>> isprime(99009901)
False
>>> primefactors(10000000001, limit=300)
[101]

========

divisors
"""
n = int(n)
factors = sorted(factorint(n, limit=limit, verbose=verbose).keys())
s = [f for f in factors[:-1:] if f not in [-1, 0, 1]]
if factors and isprime(factors[-1]):
s += [factors[-1]]
return s

def _divisors(n):
"""Helper function for divisors which generates the divisors."""

factordict = factorint(n)
ps = sorted(factordict.keys())

def rec_gen(n=0):
if n == len(ps):
yield 1
else:
pows = [1]
for j in range(factordict[ps[n]]):
pows.append(pows[-1] * ps[n])
for q in rec_gen(n + 1):
for p in pows:
yield p * q

for p in rec_gen():
yield p

[docs]def divisors(n, generator=False):
r"""
Return all divisors of n sorted from 1..n by default.
If generator is True an unordered generator is returned.

The number of divisors of n can be quite large if there are many
prime factors (counting repeated factors). If only the number of
factors is desired use divisor_count(n).

Examples
========

>>> from sympy import divisors, divisor_count
>>> divisors(24)
[1, 2, 3, 4, 6, 8, 12, 24]
>>> divisor_count(24)
8

>>> list(divisors(120, generator=True))
[1, 2, 4, 8, 3, 6, 12, 24, 5, 10, 20, 40, 15, 30, 60, 120]

This is a slightly modified version of Tim Peters referenced at:
http://stackoverflow.com/questions/1010381/python-factorization

========

primefactors, factorint, divisor_count
"""

n = as_int(abs(n))
if isprime(n):
return [1, n]
if n == 1:
return [1]
if n == 0:
return []
rv = _divisors(n)
if not generator:
return sorted(rv)
return rv

[docs]def divisor_count(n, modulus=1):
"""
Return the number of divisors of n. If modulus is not 1 then only
those that are divisible by modulus are counted.

References
==========

- http://www.mayer.dial.pipex.com/maths/formulae.htm

>>> from sympy import divisor_count
>>> divisor_count(6)
4

========

factorint, divisors, totient
"""

if not modulus:
return 0
elif modulus != 1:
n, r = divmod(n, modulus)
if r:
return 0
if n == 0:
return 0
return Mul(*[v + 1 for k, v in factorint(n).items() if k > 1])

def _udivisors(n):
"""Helper function for udivisors which generates the unitary divisors."""

factorpows = [p**e for p, e in factorint(n).items()]
for i in range(2**len(factorpows)):
d, j, k = 1, i, 0
while j:
if (j & 1):
d *= factorpows[k]
j >>= 1
k += 1
yield d

[docs]def udivisors(n, generator=False):
r"""
Return all unitary divisors of n sorted from 1..n by default.
If generator is True an unordered generator is returned.

The number of unitary divisors of n can be quite large if there are many
prime factors. If only the number of unitary divisors is desired use
udivisor_count(n).

References
==========

- http://en.wikipedia.org/wiki/Unitary_divisor
- http://mathworld.wolfram.com/UnitaryDivisor.html

Examples
========

>>> from sympy.ntheory.factor_ import udivisors, udivisor_count
>>> udivisors(15)
[1, 3, 5, 15]
>>> udivisor_count(15)
4

>>> sorted(udivisors(120, generator=True))
[1, 3, 5, 8, 15, 24, 40, 120]

========

primefactors, factorint, divisors, divisor_count, udivisor_count
"""

n = as_int(abs(n))
if isprime(n):
return [1, n]
if n == 1:
return [1]
if n == 0:
return []
rv = _udivisors(n)
if not generator:
return sorted(rv)
return rv

[docs]def udivisor_count(n):
"""
Return the number of unitary divisors of n.

References
==========

- http://mathworld.wolfram.com/UnitaryDivisorFunction.html

>>> from sympy.ntheory.factor_ import udivisor_count
>>> udivisor_count(120)
8

========

factorint, divisors, udivisors, divisor_count, totient
"""

if n == 0:
return 0
return 2**len([p for p in factorint(n) if p > 1])

def _antidivisors(n):
"""Helper function for antidivisors which generates the antidivisors."""

for d in _divisors(n):
y = 2*d
if n > y and n % y:
yield y
for d in _divisors(2*n-1):
if n > d >= 2 and n % d:
yield d
for d in _divisors(2*n+1):
if n > d >= 2 and n % d:
yield d

[docs]def antidivisors(n, generator=False):
r"""
Return all antidivisors of n sorted from 1..n by default.

Antidivisors [1]_ of n are numbers that do not divide n by the largest
possible margin.  If generator is True an unordered generator is returned.

References
==========

.. [1] definition is described in http://oeis.org/A066272/a066272a.html

Examples
========

>>> from sympy.ntheory.factor_ import antidivisors
>>> antidivisors(24)
[7, 16]

>>> sorted(antidivisors(128, generator=True))
[3, 5, 15, 17, 51, 85]

========

primefactors, factorint, divisors, divisor_count, antidivisor_count
"""

n = as_int(abs(n))
if n <= 2:
return []
rv = _antidivisors(n)
if not generator:
return sorted(rv)
return rv

[docs]def antidivisor_count(n):
"""
Return the number of antidivisors [1]_ of n.

References
==========

.. [1] formula from https://oeis.org/A066272

Examples
========

>>> from sympy.ntheory.factor_ import antidivisor_count
>>> antidivisor_count(13)
4
>>> antidivisor_count(27)
5

========

factorint, divisors, antidivisors, divisor_count, totient
"""

n = as_int(abs(n))
if n <= 2:
return 0
return divisor_count(2*n-1) + divisor_count(2*n+1) + \
divisor_count(n) - divisor_count(n, 2) - 5

[docs]class totient(Function):
r"""
Calculate the Euler totient function phi(n)

totient(n) or \phi(n) is the number of positive integers \leq n
that are relatively prime to n.

References
==========

.. [1] https://en.wikipedia.org/wiki/Euler%27s_totient_function
.. [2] http://mathworld.wolfram.com/TotientFunction.html

Examples
========

>>> from sympy.ntheory import totient
>>> totient(1)
1
>>> totient(25)
20

========

divisor_count
"""
@classmethod
def eval(cls, n):
n = sympify(n)
if n.is_Integer:
if n < 1:
raise ValueError("n must be a positive integer")
factors = factorint(n)
t = 1
for p, k in factors.items():
t *= (p - 1) * p**(k - 1)
return t
elif not isinstance(n, Expr) or (n.is_integer is False) or (n.is_positive is False):
raise ValueError("n must be a positive integer")

def _eval_is_integer(self):
return fuzzy_and([self.args[0].is_integer, self.args[0].is_positive])

[docs]class reduced_totient(Function):
r"""
Calculate the Carmichael reduced totient function lambda(n)

reduced_totient(n) or \lambda(n) is the smallest m > 0 such that
k^m \equiv 1 \mod n for all k relatively prime to n.

References
==========

.. [1] https://en.wikipedia.org/wiki/Carmichael_function
.. [2] http://mathworld.wolfram.com/CarmichaelFunction.html

Examples
========

>>> from sympy.ntheory import reduced_totient
>>> reduced_totient(1)
1
>>> reduced_totient(8)
2
>>> reduced_totient(30)
4

========

totient
"""
@classmethod
def eval(cls, n):
n = sympify(n)
if n.is_Integer:
if n < 1:
raise ValueError("n must be a positive integer")
factors = factorint(n)
t = 1
for p, k in factors.items():
if p == 2 and k > 2:
t = ilcm(t, 2**(k - 2))
else:
t = ilcm(t, (p - 1) * p**(k - 1))
return t

def _eval_is_integer(self):
return fuzzy_and([self.args[0].is_integer, self.args[0].is_positive])

[docs]class divisor_sigma(Function):
r"""
Calculate the divisor function \sigma_k(n) for positive integer n

divisor_sigma(n, k) is equal to sum([x**k for x in divisors(n)])

If n's prime factorization is:

.. math ::
n = \prod_{i=1}^\omega p_i^{m_i},

then

.. math ::
\sigma_k(n) = \prod_{i=1}^\omega (1+p_i^k+p_i^{2k}+\cdots
+ p_i^{m_ik}).

Parameters
==========

k : power of divisors in the sum

for k = 0, 1:
divisor_sigma(n, 0) is equal to divisor_count(n)
divisor_sigma(n, 1) is equal to sum(divisors(n))

Default for k is 1.

References
==========

.. [1] http://en.wikipedia.org/wiki/Divisor_function

Examples
========

>>> from sympy.ntheory import divisor_sigma
>>> divisor_sigma(18, 0)
6
>>> divisor_sigma(39, 1)
56
>>> divisor_sigma(12, 2)
210
>>> divisor_sigma(37)
38

========

divisor_count, totient, divisors, factorint
"""

@classmethod
def eval(cls, n, k=1):
n = sympify(n)
k = sympify(k)
if n.is_prime:
return 1 + n**k
if n.is_Integer:
if n <= 0:
raise ValueError("n must be a positive integer")
else:
return Mul(*[(p**(k*(e + 1)) - 1)/(p**k - 1) if k != 0
else e + 1 for p, e in factorint(n).items()])

[docs]def core(n, t=2):
r"""
Calculate core(n,t) = core_t(n) of a positive integer n

core_2(n) is equal to the squarefree part of n

If n's prime factorization is:

.. math ::
n = \prod_{i=1}^\omega p_i^{m_i},

then

.. math ::
core_t(n) = \prod_{i=1}^\omega p_i^{m_i \mod t}.

Parameters
==========

t : core(n,t) calculates the t-th power free part of n

core(n, 2) is the squarefree part of n
core(n, 3) is the cubefree part of n

Default for t is 2.

References
==========

.. [1] http://en.wikipedia.org/wiki/Square-free_integer#Squarefree_core

Examples
========

>>> from sympy.ntheory.factor_ import core
>>> core(24, 2)
6
>>> core(9424, 3)
1178
>>> core(379238)
379238
>>> core(15**11, 10)
15

========

factorint, sympy.solvers.diophantine.square_factor
"""

n = as_int(n)
t = as_int(t)
if n <= 0:
raise ValueError("n must be a positive integer")
elif t <= 1:
raise ValueError("t must be >= 2")
else:
y = 1
for p, e in factorint(n).items():
y *= p**(e % t)
return y

[docs]def digits(n, b=10):
"""
Return a list of the digits of n in base b. The first element in the list
is b (or -b if n is negative).

Examples
========

>>> from sympy.ntheory.factor_ import digits
>>> digits(35)
[10, 3, 5]
>>> digits(27, 2)
[2, 1, 1, 0, 1, 1]
>>> digits(65536, 256)
[256, 1, 0, 0]
>>> digits(-3958, 27)
[-27, 5, 11, 16]
"""

b = as_int(b)
n = as_int(n)
if b <= 1:
raise ValueError("b must be >= 2")
else:
x, y = abs(n), []
while x >= b:
x, r = divmod(x, b)
y.append(r)
y.append(x)
y.append(-b if n < 0 else b)
y.reverse()
return y

[docs]class udivisor_sigma(Function):
r"""
Calculate the unitary divisor function \sigma_k^*(n) for positive integer n

udivisor_sigma(n, k) is equal to sum([x**k for x in udivisors(n)])

If n's prime factorization is:

.. math ::
n = \prod_{i=1}^\omega p_i^{m_i},

then

.. math ::
\sigma_k^*(n) = \prod_{i=1}^\omega (1+ p_i^{m_ik}).

Parameters
==========

k : power of divisors in the sum

for k = 0, 1:
udivisor_sigma(n, 0) is equal to udivisor_count(n)
udivisor_sigma(n, 1) is equal to sum(udivisors(n))

Default for k is 1.

References
==========

.. [1] http://mathworld.wolfram.com/UnitaryDivisorFunction.html

Examples
========

>>> from sympy.ntheory.factor_ import udivisor_sigma
>>> udivisor_sigma(18, 0)
4
>>> udivisor_sigma(74, 1)
114
>>> udivisor_sigma(36, 3)
47450
>>> udivisor_sigma(111)
152

========

divisor_count, totient, divisors, udivisors, udivisor_count, divisor_sigma,
factorint
"""

@classmethod
def eval(cls, n, k=1):
n = sympify(n)
k = sympify(k)
if n.is_prime:
return 1 + n**k
if n.is_Integer:
if n <= 0:
raise ValueError("n must be a positive integer")
else:
return Mul(*[1+p**(k*e) for p, e in factorint(n).items()])

r"""
Calculate the number of distinct prime factors for a positive integer n.

If n's prime factorization is:

.. math ::
n = \prod_{i=1}^k p_i^{m_i},

then primenu(n) or \nu(n) is:

.. math ::
\nu(n) = k.

References
==========

.. [1] http://mathworld.wolfram.com/PrimeFactor.html

Examples
========

0
3

========

factorint
"""

@classmethod
def eval(cls, n):
n = sympify(n)
if n.is_Integer:
if n <= 0:
raise ValueError("n must be a positive integer")
else:
return len(factorint(n).keys())

[docs]class primeomega(Function):
r"""
Calculate the number of prime factors counting multiplicities for a
positive integer n.

If n's prime factorization is:

.. math ::
n = \prod_{i=1}^k p_i^{m_i},

then primeomega(n)  or \Omega(n) is:

.. math ::
\Omega(n) = \sum_{i=1}^k m_i.

References
==========

.. [1] http://mathworld.wolfram.com/PrimeFactor.html

Examples
========

>>> from sympy.ntheory.factor_ import primeomega
>>> primeomega(1)
0
>>> primeomega(20)
3

========

factorint
"""

@classmethod
def eval(cls, n):
n = sympify(n)
if n.is_Integer:
if n <= 0:
raise ValueError("n must be a positive integer")
else:
return sum(factorint(n).values())
`