def test_factorint_verbose(capsys):
out = """Factoring 10201
Trial division with ints [2 ... 100] and fail_max=600
Exceeded limit: 100
Check for termination
\t101 ** 1
Factorization is complete.
"""
factorint(10201, limit=100, verbose=True)
assert capsys.readouterr()[0] == out
out = """Factoring 34376910917
Trial division with ints [2 ... 32768] and fail_max=600
Check for termination
Trial division with primes [1805 ... 3610]
Trial division with primes [3610 ... 7220]
Trial division with primes [7220 ... 14440]
Trial division with primes [14440 ... 28880]
Trial division with primes [28880 ... 57760]
Trial division with primes [57760 ... 115520]
Trial division with primes [115520 ... 185411]
\t131101 ** 1
Check for termination
Factorization is complete.
"""
factorint(131101*262217, use_pm1=0, use_rho=0, verbose=True)
assert capsys.readouterr()[0] == out
def test_factorint_verbose(capsys):
out = """Factoring 10201
Trial division with ints [2 ... 100] and fail_max=600
Exceeded limit: 100
Check for termination
\t101 ** 1
Factorization is complete.
"""
factorint(10201, limit=100, verbose=True)
assert capsys.readouterr()[0] == out
out = """Factoring 34376910917
Trial division with ints [2 ... 32768] and fail_max=600
Check for termination
Trial division with primes [1805 ... 3610]
Trial division with primes [3610 ... 7220]
Trial division with primes [7220 ... 14440]
Trial division with primes [14440 ... 28880]
Trial division with primes [28880 ... 57760]
Trial division with primes [57760 ... 115520]
Trial division with primes [115520 ... 185411]
\t131101 ** 1
Check for termination
Factorization is complete.
"""
factorint(131101*262217, use_pm1=0, use_rho=0, verbose=True)
assert capsys.readouterr()[0] == out
def test_visual_factorint():
assert factorint(1, visual=1) == 1
forty2 = factorint(42, visual=True)
assert type(forty2) == Mul
assert str(forty2) == '2**1*3**1*7**1'
assert factorint(1, visual=True) is S.One
no = dict(evaluate=False)
assert factorint(42**2, visual=True) == Mul(Pow(2, 2,
**no), Pow(3, 2, **no),
Pow(7, 2, **no), **no)
assert -1 in factorint(-42, visual=True).args
def test_visual_factorint():
assert factorint(1, visual=1) == 1
forty2 = factorint(42, visual=True)
assert type(forty2) == Mul
assert str(forty2) == '2**1*3**1*7**1'
assert factorint(1, visual=True) is Integer(1)
no = {'evaluate': False}
assert factorint(42**2, visual=True) == Mul(Pow(2, 2, **no),
Pow(3, 2, **no),
Pow(7, 2, **no), **no)
assert -1 in factorint(-42, visual=True).args
def test_visual_factorint():
assert factorint(1, visual=1) == 1
forty2 = factorint(42, visual=True)
assert type(forty2) == Mul
assert str(forty2) == '2**1*3**1*7**1'
assert factorint(1, visual=True) is Integer(1)
no = {'evaluate': False}
assert factorint(42**2, visual=True) == Mul(Pow(2, 2, **no),
Pow(3, 2, **no),
Pow(7, 2, **no), **no)
assert -1 in factorint(-42, visual=True).args
def gf_csolve(f, n):
"""
To solve f(x) congruent 0 mod(n).
n is divided into canonical factors and f(x) cong 0 mod(p**e) will be
solved for each factor. Applying the Chinese Remainder Theorem to the
results returns the final answers.
Examples
========
Solve [1, 1, 7] congruent 0 mod(189):
>>> gf_csolve([1, 1, 7], 189)
[13, 49, 76, 112, 139, 175]
References
==========
.. [1] 'An introduction to the Theory of Numbers' 5th Edition by
Ivan Niven, Zuckerman and Montgomery.
"""
from diofant.polys.domains import ZZ
P = factorint(n)
X = [csolve_prime(f, p, e) for p, e in P.items()]
pools = list(map(tuple, X))
perms = [[]]
for pool in pools:
perms = [x + [y] for x in perms for y in pool]
dist_factors = [pow(p, e) for p, e in P.items()]
return sorted([gf_crt(per, dist_factors, ZZ) for per in perms])
def dup_zz_cyclotomic_poly(n, K):
"""Efficiently generate n-th cyclotomic polnomial. """
h = [K.one, -K.one]
for p, k in factorint(n).items():
h = dup_quo(dup_inflate(h, p, K), h, K)
h = dup_inflate(h, p**(k - 1), K)
return h
def _dup_cyclotomic_decompose(n, K):
H = [[K.one, -K.one]]
for p, k in factorint(n).items():
Q = [dup_quo(dup_inflate(h, p, K), h, K) for h in H]
H.extend(Q)
for i in range(1, k):
Q = [dup_inflate(q, p, K) for q in Q]
H.extend(Q)
return H
def test_square_factor():
assert square_factor(1) == square_factor(-1) == 1
assert square_factor(0) == 1
assert square_factor(5) == square_factor(-5) == 1
assert square_factor(4) == square_factor(-4) == 2
assert square_factor(12) == square_factor(-12) == 2
assert square_factor(6) == 1
assert square_factor(18) == 3
assert square_factor(52) == 2
assert square_factor(49) == 7
assert square_factor(392) == 14
assert square_factor(factorint(-12)) == 2
def dup_zz_irreducible_p(f, K):
"""Test irreducibility using Eisenstein's criterion. """
lc = dup_LC(f, K)
tc = dup_TC(f, K)
e_fc = dup_content(f[1:], K)
if e_fc:
e_ff = factorint(int(e_fc))
for p in e_ff.keys():
if (lc % p) and (tc % p**2):
return True
def gf_irred_p_rabin(f, p, K):
"""
Rabin's polynomial irreducibility test over finite fields.
Examples
========
>>> from diofant.polys.domains import ZZ
>>> gf_irred_p_rabin(ZZ.map([1, 4, 2, 2, 3, 2, 4, 1, 4, 0, 4]), 5, ZZ)
True
>>> gf_irred_p_rabin(ZZ.map([3, 2, 4]), 5, ZZ)
False
"""
n = gf_degree(f)
if n <= 1:
return True
_, f = gf_monic(f, p, K)
x = [K.one, K.zero]
indices = { n//d for d in factorint(n) }
b = gf_frobenius_monomial_base(f, p, K)
h = b[1]
for i in range(1, n):
if i in indices:
g = gf_sub(h, x, p, K)
if gf_gcd(f, g, p, K) != [K.one]:
return False
h = gf_frobenius_map(h, f, b, p, K)
return h == x
def test_sympyissue_4356():
assert factorint(1030903) == {53: 2, 367: 1}
def test_factorint():
assert primefactors(123456) == [2, 3, 643]
assert primefactors(10000000001, limit=300) == [101]
assert factorint(0) == {0: 1}
assert factorint(1) == {}
assert factorint(-1) == {-1: 1}
assert factorint(-2) == {-1: 1, 2: 1}
assert factorint(-16) == {-1: 1, 2: 4}
assert factorint(2) == {2: 1}
assert factorint(126) == {2: 1, 3: 2, 7: 1}
assert factorint(123456) == {2: 6, 3: 1, 643: 1}
assert factorint(5951757) == {3: 1, 7: 1, 29: 2, 337: 1}
assert factorint(64015937) == {7993: 1, 8009: 1}
assert factorint(2**(2**6) + 1) == {274177: 1, 67280421310721: 1}
assert multiproduct(factorint(fac(200))) == fac(200)
for b, e in factorint(fac(150)).items():
assert e == fac_multiplicity(150, b)
assert factorint(103005006059**7) == {103005006059: 7}
assert factorint(31337**191) == {31337: 191}
assert factorint(2**1000 * 3**500 * 257**127 * 383**60) == \
{2: 1000, 3: 500, 257: 127, 383: 60}
assert len(factorint(fac(10000))) == 1229
assert factorint(12932983746293756928584532764589230) == \
{2: 1, 5: 1, 73: 1, 727719592270351: 1, 63564265087747: 1, 383: 1}
assert factorint(727719592270351) == {727719592270351: 1}
assert factorint(2**64 + 1, use_trial=False) == factorint(2**64 + 1)
for n in range(60000):
assert multiproduct(factorint(n)) == n
assert pollard_rho(2**64 + 1, seed=1) == 274177
assert pollard_rho(19, seed=1) is None
n = 16843009
assert pollard_rho(n, F=lambda x: (2048*pow(x, 2, n) + 32767) % n) == 257
assert factorint(3, limit=2) == {3: 1}
assert factorint(12345) == {3: 1, 5: 1, 823: 1}
assert factorint(
12345, limit=3) == {4115: 1, 3: 1} # the 5 is greater than the limit
assert factorint(1, limit=1) == {}
assert factorint(0, 3) == {0: 1}
assert factorint(12, limit=1) == {12: 1}
assert factorint(30, limit=2) == {2: 1, 15: 1}
assert factorint(16, limit=2) == {2: 4}
assert factorint(124, limit=3) == {2: 2, 31: 1}
assert factorint(4*31**2, limit=3) == {2: 2, 31: 2}
assert factorint(10201, limit=100) == {101: 2}
p1 = nextprime(2**32)
p2 = nextprime(2**16)
p3 = nextprime(p2)
assert factorint(p1*p2*p3) == {p1: 1, p2: 1, p3: 1}
assert factorint(13*17*19, limit=15) == {13: 1, 17*19: 1}
assert factorint(1951*15013*15053, limit=2000) == {225990689: 1, 1951: 1}
assert factorint(primorial(17) + 1, use_pm1=0) == {19026377261: 1,
3467: 1, 277: 1,
105229: 1}
assert factorint(34376910917, use_pm1=0, use_rho=0) == {131101: 1,
262217: 1}
assert factorint(34376910917, use_pm1=0) == {131101: 1, 262217: 1}
assert factorint(34376910917, use_rho=0) == {131101: 1, 262217: 1}
# when prime b is closer than approx sqrt(8*p) to prime p then they are
# "close" and have a trivial factorization
a = nextprime(2**2**8) # 78 digits
b = nextprime(a + 2**2**4)
assert 'Fermat' in capture(lambda: factorint(a*b, verbose=1))
pytest.raises(ValueError, lambda: pollard_rho(4))
pytest.raises(ValueError, lambda: pollard_pm1(3))
pytest.raises(ValueError, lambda: pollard_pm1(10, B=2))
# verbose coverage
n = nextprime(2**16)*nextprime(2**17)*nextprime(1901)
assert 'with primes' in capture(lambda: factorint(n, verbose=1))
capture(lambda: factorint(nextprime(2**16)*1012, verbose=1))
n = nextprime(2**17)
capture(lambda: factorint(n**3, verbose=1)) # perfect power termination
capture(lambda: factorint(2*n, verbose=1)) # factoring complete msg
# exceed 1st
n = nextprime(2**17)
n *= nextprime(n)
assert '1000' in capture(lambda: factorint(n, limit=1000, verbose=1))
n *= nextprime(n)
assert len(factorint(n)) == 3
assert len(factorint(n, limit=p1)) == 3
n *= nextprime(2*n)
# exceed 2nd
assert '2001' in capture(lambda: factorint(n, limit=2000, verbose=1))
assert capture(
lambda: factorint(n, limit=4000, verbose=1)).count('Pollard') == 2
# non-prime pm1 result
n = nextprime(8069)
n *= nextprime(2*n)*nextprime(2*n, 2)
capture(lambda: factorint(n, verbose=1)) # non-prime pm1 result
# factor fermat composite
p1 = nextprime(2**17)
p2 = nextprime(2*p1)
assert factorint((p1*p2**2)**3) == {p1: 3, p2: 6}
# Test for non integer input
pytest.raises(ValueError, lambda: factorint(4.5))
def test_factorint_invariant(x):
assert Mul(*[v**p for v, p in factorint(x).items()]) == x
def test_sympyissue_4356():
assert factorint(1030903) == {53: 2, 367: 1}
def test_factorint_invariant(x):
assert Mul(*[v**p for v, p in factorint(x).items()]) == x
def test_factorint():
assert primefactors(123456) == [2, 3, 643]
assert primefactors(10000000001, limit=300) == [101]
assert factorint(0) == {0: 1}
assert factorint(1) == {}
assert factorint(-1) == {-1: 1}
assert factorint(-2) == {-1: 1, 2: 1}
assert factorint(-16) == {-1: 1, 2: 4}
assert factorint(2) == {2: 1}
assert factorint(126) == {2: 1, 3: 2, 7: 1}
assert factorint(123456) == {2: 6, 3: 1, 643: 1}
assert factorint(5951757) == {3: 1, 7: 1, 29: 2, 337: 1}
assert factorint(64015937) == {7993: 1, 8009: 1}
assert factorint(2**(2**6) + 1) == {274177: 1, 67280421310721: 1}
assert multiproduct(factorint(fac(200))) == fac(200)
for b, e in factorint(fac(150)).items():
assert e == fac_multiplicity(150, b)
assert factorint(103005006059**7) == {103005006059: 7}
assert factorint(31337**191) == {31337: 191}
assert factorint(2**1000 * 3**500 * 257**127 * 383**60) == \
{2: 1000, 3: 500, 257: 127, 383: 60}
assert len(factorint(fac(10000))) == 1229
assert factorint(12932983746293756928584532764589230) == \
{2: 1, 5: 1, 73: 1, 727719592270351: 1, 63564265087747: 1, 383: 1}
assert factorint(727719592270351) == {727719592270351: 1}
assert factorint(2**64 + 1, use_trial=False) == factorint(2**64 + 1)
for n in range(60000):
assert multiproduct(factorint(n)) == n
assert pollard_rho(2**64 + 1, seed=1) == 274177
assert pollard_rho(19, seed=1) is None
n = 16843009
assert pollard_rho(n, F=lambda x: (2048*pow(x, 2, n) + 32767) % n) == 257
assert factorint(3, limit=2) == {3: 1}
assert factorint(12345) == {3: 1, 5: 1, 823: 1}
assert factorint(
12345, limit=3) == {4115: 1, 3: 1} # the 5 is greater than the limit
assert factorint(1, limit=1) == {}
assert factorint(0, 3) == {0: 1}
assert factorint(12, limit=1) == {12: 1}
assert factorint(30, limit=2) == {2: 1, 15: 1}
assert factorint(16, limit=2) == {2: 4}
assert factorint(124, limit=3) == {2: 2, 31: 1}
assert factorint(4*31**2, limit=3) == {2: 2, 31: 2}
assert factorint(10201, limit=100) == {101: 2}
p1 = nextprime(2**32)
p2 = nextprime(2**16)
p3 = nextprime(p2)
assert factorint(p1*p2*p3) == {p1: 1, p2: 1, p3: 1}
assert factorint(13*17*19, limit=15) == {13: 1, 17*19: 1}
assert factorint(1951*15013*15053, limit=2000) == {225990689: 1, 1951: 1}
assert factorint(primorial(17) + 1, use_pm1=0) == {19026377261: 1,
3467: 1, 277: 1,
105229: 1}
assert factorint(34376910917, use_pm1=0, use_rho=0) == {131101: 1,
262217: 1}
assert factorint(34376910917, use_pm1=0) == {131101: 1, 262217: 1}
assert factorint(34376910917, use_rho=0) == {131101: 1, 262217: 1}
# when prime b is closer than approx sqrt(8*p) to prime p then they are
# "close" and have a trivial factorization
a = nextprime(2**2**8) # 78 digits
b = nextprime(a + 2**2**4)
assert 'Fermat' in capture(lambda: factorint(a*b, verbose=1))
pytest.raises(ValueError, lambda: pollard_rho(4))
pytest.raises(ValueError, lambda: pollard_pm1(3))
pytest.raises(ValueError, lambda: pollard_pm1(10, B=2))
# verbose coverage
n = nextprime(2**16)*nextprime(2**17)*nextprime(1901)
assert 'with primes' in capture(lambda: factorint(n, verbose=1))
capture(lambda: factorint(nextprime(2**16)*1012, verbose=1))
n = nextprime(2**17)
capture(lambda: factorint(n**3, verbose=1)) # perfect power termination
capture(lambda: factorint(2*n, verbose=1)) # factoring complete msg
# exceed 1st
n = nextprime(2**17)
n *= nextprime(n)
assert '1000' in capture(lambda: factorint(n, limit=1000, verbose=1))
n *= nextprime(n)
assert len(factorint(n)) == 3
assert len(factorint(n, limit=p1)) == 3
n *= nextprime(2*n)
# exceed 2nd
assert '2001' in capture(lambda: factorint(n, limit=2000, verbose=1))
assert capture(
lambda: factorint(n, limit=4000, verbose=1)).count('Pollard') == 2
# non-prime pm1 result
n = nextprime(8069)
n *= nextprime(2*n)*nextprime(2*n, 2)
capture(lambda: factorint(n, verbose=1)) # non-prime pm1 result
# factor fermat composite
p1 = nextprime(2**17)
p2 = nextprime(2*p1)
assert factorint((p1*p2**2)**3) == {p1: 3, p2: 6}
# Test for non integer input
pytest.raises(ValueError, lambda: factorint(4.5))
