2655 lines
74 KiB
Python
2655 lines
74 KiB
Python
![]() |
"""
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Integer factorization
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"""
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from collections import defaultdict
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from functools import reduce
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import random
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import math
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from sympy.core import sympify
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from sympy.core.containers import Dict
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from sympy.core.evalf import bitcount
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from sympy.core.expr import Expr
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from sympy.core.function import Function
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from sympy.core.logic import fuzzy_and
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from sympy.core.mul import Mul
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from sympy.core.numbers import igcd, ilcm, Rational, Integer
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from sympy.core.power import integer_nthroot, Pow, integer_log
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from sympy.core.singleton import S
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from sympy.external.gmpy import SYMPY_INTS
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from .primetest import isprime
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from .generate import sieve, primerange, nextprime
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from .digits import digits
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from sympy.utilities.iterables import flatten
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from sympy.utilities.misc import as_int, filldedent
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from .ecm import _ecm_one_factor
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# Note: This list should be updated whenever new Mersenne primes are found.
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# Refer: https://www.mersenne.org/
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MERSENNE_PRIME_EXPONENTS = (2, 3, 5, 7, 13, 17, 19, 31, 61, 89, 107, 127, 521, 607, 1279, 2203,
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2281, 3217, 4253, 4423, 9689, 9941, 11213, 19937, 21701, 23209, 44497, 86243, 110503, 132049,
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216091, 756839, 859433, 1257787, 1398269, 2976221, 3021377, 6972593, 13466917, 20996011, 24036583,
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25964951, 30402457, 32582657, 37156667, 42643801, 43112609, 57885161, 74207281, 77232917, 82589933)
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# compute more when needed for i in Mersenne prime exponents
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PERFECT = [6] # 2**(i-1)*(2**i-1)
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MERSENNES = [3] # 2**i - 1
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def _ismersenneprime(n):
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global MERSENNES
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j = len(MERSENNES)
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while n > MERSENNES[-1] and j < len(MERSENNE_PRIME_EXPONENTS):
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# conservatively grow the list
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MERSENNES.append(2**MERSENNE_PRIME_EXPONENTS[j] - 1)
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j += 1
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return n in MERSENNES
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def _isperfect(n):
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global PERFECT
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if n % 2 == 0:
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j = len(PERFECT)
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while n > PERFECT[-1] and j < len(MERSENNE_PRIME_EXPONENTS):
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# conservatively grow the list
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t = 2**(MERSENNE_PRIME_EXPONENTS[j] - 1)
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PERFECT.append(t*(2*t - 1))
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j += 1
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return n in PERFECT
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small_trailing = [0] * 256
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for j in range(1,8):
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small_trailing[1<<j::1<<(j+1)] = [j] * (1<<(7-j))
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def smoothness(n):
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"""
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Return the B-smooth and B-power smooth values of n.
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The smoothness of n is the largest prime factor of n; the power-
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smoothness is the largest divisor raised to its multiplicity.
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Examples
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========
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>>> from sympy.ntheory.factor_ import smoothness
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>>> smoothness(2**7*3**2)
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(3, 128)
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>>> smoothness(2**4*13)
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(13, 16)
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>>> smoothness(2)
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(2, 2)
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See Also
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========
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factorint, smoothness_p
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"""
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if n == 1:
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return (1, 1) # not prime, but otherwise this causes headaches
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facs = factorint(n)
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return max(facs), max(m**facs[m] for m in facs)
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def smoothness_p(n, m=-1, power=0, visual=None):
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"""
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Return a list of [m, (p, (M, sm(p + m), psm(p + m)))...]
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where:
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1. p**M is the base-p divisor of n
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2. sm(p + m) is the smoothness of p + m (m = -1 by default)
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3. psm(p + m) is the power smoothness of p + m
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The list is sorted according to smoothness (default) or by power smoothness
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if power=1.
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The smoothness of the numbers to the left (m = -1) or right (m = 1) of a
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factor govern the results that are obtained from the p +/- 1 type factoring
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methods.
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>>> from sympy.ntheory.factor_ import smoothness_p, factorint
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>>> smoothness_p(10431, m=1)
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(1, [(3, (2, 2, 4)), (19, (1, 5, 5)), (61, (1, 31, 31))])
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>>> smoothness_p(10431)
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(-1, [(3, (2, 2, 2)), (19, (1, 3, 9)), (61, (1, 5, 5))])
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>>> smoothness_p(10431, power=1)
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(-1, [(3, (2, 2, 2)), (61, (1, 5, 5)), (19, (1, 3, 9))])
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If visual=True then an annotated string will be returned:
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>>> print(smoothness_p(21477639576571, visual=1))
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p**i=4410317**1 has p-1 B=1787, B-pow=1787
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p**i=4869863**1 has p-1 B=2434931, B-pow=2434931
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This string can also be generated directly from a factorization dictionary
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and vice versa:
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>>> factorint(17*9)
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{3: 2, 17: 1}
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>>> smoothness_p(_)
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'p**i=3**2 has p-1 B=2, B-pow=2\\np**i=17**1 has p-1 B=2, B-pow=16'
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>>> smoothness_p(_)
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{3: 2, 17: 1}
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The table of the output logic is:
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====== ====== ======= =======
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| Visual
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------ ----------------------
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Input True False other
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====== ====== ======= =======
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dict str tuple str
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str str tuple dict
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tuple str tuple str
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n str tuple tuple
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mul str tuple tuple
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====== ====== ======= =======
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See Also
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========
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factorint, smoothness
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"""
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# visual must be True, False or other (stored as None)
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if visual in (1, 0):
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visual = bool(visual)
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elif visual not in (True, False):
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visual = None
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if isinstance(n, str):
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if visual:
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return n
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d = {}
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for li in n.splitlines():
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k, v = [int(i) for i in
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li.split('has')[0].split('=')[1].split('**')]
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d[k] = v
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if visual is not True and visual is not False:
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return d
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return smoothness_p(d, visual=False)
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elif not isinstance(n, tuple):
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facs = factorint(n, visual=False)
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if power:
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k = -1
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else:
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k = 1
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if isinstance(n, tuple):
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rv = n
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else:
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rv = (m, sorted([(f,
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tuple([M] + list(smoothness(f + m))))
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for f, M in list(facs.items())],
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key=lambda x: (x[1][k], x[0])))
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if visual is False or (visual is not True) and (type(n) in [int, Mul]):
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return rv
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lines = []
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for dat in rv[1]:
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dat = flatten(dat)
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dat.insert(2, m)
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lines.append('p**i=%i**%i has p%+i B=%i, B-pow=%i' % tuple(dat))
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return '\n'.join(lines)
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def trailing(n):
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"""Count the number of trailing zero digits in the binary
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representation of n, i.e. determine the largest power of 2
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that divides n.
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Examples
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========
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>>> from sympy import trailing
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>>> trailing(128)
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7
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>>> trailing(63)
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0
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"""
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n = abs(int(n))
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if not n:
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return 0
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low_byte = n & 0xff
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if low_byte:
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return small_trailing[low_byte]
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# 2**m is quick for z up through 2**30
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z = bitcount(n) - 1
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if isinstance(z, SYMPY_INTS):
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if n == 1 << z:
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return z
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if z < 300:
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# fixed 8-byte reduction
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t = 8
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n >>= 8
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while not n & 0xff:
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n >>= 8
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t += 8
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return t + small_trailing[n & 0xff]
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# binary reduction important when there might be a large
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# number of trailing 0s
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t = 0
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p = 8
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while not n & 1:
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while not n & ((1 << p) - 1):
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n >>= p
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t += p
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p *= 2
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p //= 2
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return t
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def multiplicity(p, n):
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"""
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Find the greatest integer m such that p**m divides n.
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Examples
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========
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>>> from sympy import multiplicity, Rational
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>>> [multiplicity(5, n) for n in [8, 5, 25, 125, 250]]
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[0, 1, 2, 3, 3]
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>>> multiplicity(3, Rational(1, 9))
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-2
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Note: when checking for the multiplicity of a number in a
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large factorial it is most efficient to send it as an unevaluated
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factorial or to call ``multiplicity_in_factorial`` directly:
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>>> from sympy.ntheory import multiplicity_in_factorial
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>>> from sympy import factorial
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>>> p = factorial(25)
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>>> n = 2**100
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>>> nfac = factorial(n, evaluate=False)
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>>> multiplicity(p, nfac)
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52818775009509558395695966887
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>>> _ == multiplicity_in_factorial(p, n)
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True
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"""
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try:
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p, n = as_int(p), as_int(n)
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except ValueError:
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from sympy.functions.combinatorial.factorials import factorial
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if all(isinstance(i, (SYMPY_INTS, Rational)) for i in (p, n)):
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p = Rational(p)
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n = Rational(n)
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if p.q == 1:
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if n.p == 1:
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return -multiplicity(p.p, n.q)
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return multiplicity(p.p, n.p) - multiplicity(p.p, n.q)
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elif p.p == 1:
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return multiplicity(p.q, n.q)
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else:
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like = min(
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multiplicity(p.p, n.p),
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multiplicity(p.q, n.q))
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cross = min(
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multiplicity(p.q, n.p),
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multiplicity(p.p, n.q))
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return like - cross
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elif (isinstance(p, (SYMPY_INTS, Integer)) and
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isinstance(n, factorial) and
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isinstance(n.args[0], Integer) and
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n.args[0] >= 0):
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return multiplicity_in_factorial(p, n.args[0])
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raise ValueError('expecting ints or fractions, got %s and %s' % (p, n))
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if n == 0:
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raise ValueError('no such integer exists: multiplicity of %s is not-defined' %(n))
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if p == 2:
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return trailing(n)
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if p < 2:
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raise ValueError('p must be an integer, 2 or larger, but got %s' % p)
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if p == n:
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return 1
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m = 0
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n, rem = divmod(n, p)
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while not rem:
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m += 1
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if m > 5:
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# The multiplicity could be very large. Better
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# to increment in powers of two
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e = 2
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while 1:
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ppow = p**e
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if ppow < n:
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nnew, rem = divmod(n, ppow)
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if not rem:
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m += e
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e *= 2
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n = nnew
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continue
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return m + multiplicity(p, n)
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n, rem = divmod(n, p)
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return m
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def multiplicity_in_factorial(p, n):
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"""return the largest integer ``m`` such that ``p**m`` divides ``n!``
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without calculating the factorial of ``n``.
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Examples
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|
========
|
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>>> from sympy.ntheory import multiplicity_in_factorial
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>>> from sympy import factorial
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>>> multiplicity_in_factorial(2, 3)
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1
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An instructive use of this is to tell how many trailing zeros
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a given factorial has. For example, there are 6 in 25!:
|
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>>> factorial(25)
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15511210043330985984000000
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>>> multiplicity_in_factorial(10, 25)
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6
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|
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For large factorials, it is much faster/feasible to use
|
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this function rather than computing the actual factorial:
|
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>>> multiplicity_in_factorial(factorial(25), 2**100)
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52818775009509558395695966887
|
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"""
|
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p, n = as_int(p), as_int(n)
|
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if p <= 0:
|
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raise ValueError('expecting positive integer got %s' % p )
|
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if n < 0:
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raise ValueError('expecting non-negative integer got %s' % n )
|
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factors = factorint(p)
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# keep only the largest of a given multiplicity since those
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# of a given multiplicity will be goverened by the behavior
|
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# of the largest factor
|
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test = defaultdict(int)
|
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for k, v in factors.items():
|
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test[v] = max(k, test[v])
|
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keep = set(test.values())
|
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# remove others from factors
|
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for k in list(factors.keys()):
|
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if k not in keep:
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factors.pop(k)
|
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|
|
||
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mp = S.Infinity
|
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for i in factors:
|
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# multiplicity of i in n! is
|
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mi = (n - (sum(digits(n, i)) - i))//(i - 1)
|
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# multiplicity of p in n! depends on multiplicity
|
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|
# of prime `i` in p, so we floor divide by factors[i]
|
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# and keep it if smaller than the multiplicity of p
|
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# seen so far
|
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mp = min(mp, mi//factors[i])
|
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return mp
|
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|
||
|
|
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|
def perfect_power(n, candidates=None, big=True, factor=True):
|
||
|
"""
|
||
|
Return ``(b, e)`` such that ``n`` == ``b**e`` if ``n`` is a unique
|
||
|
perfect power with ``e > 1``, else ``False`` (e.g. 1 is not a
|
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|
perfect power). A ValueError is raised if ``n`` is not Rational.
|
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|
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|
By default, the base is recursively decomposed and the exponents
|
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collected so the largest possible ``e`` is sought. If ``big=False``
|
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|
then the smallest possible ``e`` (thus prime) will be chosen.
|
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|
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If ``factor=True`` then simultaneous factorization of ``n`` is
|
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attempted since finding a factor indicates the only possible root
|
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|
for ``n``. This is True by default since only a few small factors will
|
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be tested in the course of searching for the perfect power.
|
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|
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|
The use of ``candidates`` is primarily for internal use; if provided,
|
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|
False will be returned if ``n`` cannot be written as a power with one
|
||
|
of the candidates as an exponent and factoring (beyond testing for
|
||
|
a factor of 2) will not be attempted.
|
||
|
|
||
|
Examples
|
||
|
========
|
||
|
|
||
|
>>> from sympy import perfect_power, Rational
|
||
|
>>> perfect_power(16)
|
||
|
(2, 4)
|
||
|
>>> perfect_power(16, big=False)
|
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(4, 2)
|
||
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|
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|
Negative numbers can only have odd perfect powers:
|
||
|
|
||
|
>>> perfect_power(-4)
|
||
|
False
|
||
|
>>> perfect_power(-8)
|
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|
(-2, 3)
|
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|
|
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|
Rationals are also recognized:
|
||
|
|
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|
>>> perfect_power(Rational(1, 2)**3)
|
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|
(1/2, 3)
|
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>>> perfect_power(Rational(-3, 2)**3)
|
||
|
(-3/2, 3)
|
||
|
|
||
|
Notes
|
||
|
=====
|
||
|
|
||
|
To know whether an integer is a perfect power of 2 use
|
||
|
|
||
|
>>> is2pow = lambda n: bool(n and not n & (n - 1))
|
||
|
>>> [(i, is2pow(i)) for i in range(5)]
|
||
|
[(0, False), (1, True), (2, True), (3, False), (4, True)]
|
||
|
|
||
|
It is not necessary to provide ``candidates``. When provided
|
||
|
it will be assumed that they are ints. The first one that is
|
||
|
larger than the computed maximum possible exponent will signal
|
||
|
failure for the routine.
|
||
|
|
||
|
>>> perfect_power(3**8, [9])
|
||
|
False
|
||
|
>>> perfect_power(3**8, [2, 4, 8])
|
||
|
(3, 8)
|
||
|
>>> perfect_power(3**8, [4, 8], big=False)
|
||
|
(9, 4)
|
||
|
|
||
|
See Also
|
||
|
========
|
||
|
sympy.core.power.integer_nthroot
|
||
|
sympy.ntheory.primetest.is_square
|
||
|
"""
|
||
|
if isinstance(n, Rational) and not n.is_Integer:
|
||
|
p, q = n.as_numer_denom()
|
||
|
if p is S.One:
|
||
|
pp = perfect_power(q)
|
||
|
if pp:
|
||
|
pp = (n.func(1, pp[0]), pp[1])
|
||
|
else:
|
||
|
pp = perfect_power(p)
|
||
|
if pp:
|
||
|
num, e = pp
|
||
|
pq = perfect_power(q, [e])
|
||
|
if pq:
|
||
|
den, _ = pq
|
||
|
pp = n.func(num, den), e
|
||
|
return pp
|
||
|
|
||
|
n = as_int(n)
|
||
|
if n < 0:
|
||
|
pp = perfect_power(-n)
|
||
|
if pp:
|
||
|
b, e = pp
|
||
|
if e % 2:
|
||
|
return -b, e
|
||
|
return False
|
||
|
|
||
|
if n <= 3:
|
||
|
# no unique exponent for 0, 1
|
||
|
# 2 and 3 have exponents of 1
|
||
|
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
|
||
|
min_possible = 2 + not_square
|
||
|
if not candidates:
|
||
|
candidates = primerange(min_possible, max_possible)
|
||
|
else:
|
||
|
candidates = sorted([i for i in candidates
|
||
|
if min_possible <= i < max_possible])
|
||
|
if n%2 == 0:
|
||
|
e = trailing(n)
|
||
|
candidates = [i for i in candidates if e%i == 0]
|
||
|
if big:
|
||
|
candidates = reversed(candidates)
|
||
|
for e in candidates:
|
||
|
r, ok = integer_nthroot(n, e)
|
||
|
if ok:
|
||
|
return (r, e)
|
||
|
return False
|
||
|
|
||
|
def _factors():
|
||
|
rv = 2 + n % 2
|
||
|
while True:
|
||
|
yield rv
|
||
|
rv = nextprime(rv)
|
||
|
|
||
|
for fac, e in zip(_factors(), candidates):
|
||
|
# see if there is a factor present
|
||
|
if factor and n % fac == 0:
|
||
|
# find what the potential power is
|
||
|
if fac == 2:
|
||
|
e = trailing(n)
|
||
|
else:
|
||
|
e = multiplicity(fac, n)
|
||
|
# if it's a trivial power we are done
|
||
|
if e == 1:
|
||
|
return False
|
||
|
|
||
|
# maybe the e-th root of n is exact
|
||
|
r, exact = integer_nthroot(n, e)
|
||
|
if not exact:
|
||
|
# Having a factor, we know that e is the maximal
|
||
|
# possible value for a root of n.
|
||
|
# If n = fac**e*m can be written as a perfect
|
||
|
# power then see if m can be written as r**E where
|
||
|
# gcd(e, E) != 1 so n = (fac**(e//E)*r)**E
|
||
|
m = n//fac**e
|
||
|
rE = perfect_power(m, candidates=divisors(e, generator=True))
|
||
|
if not rE:
|
||
|
return False
|
||
|
else:
|
||
|
r, E = rE
|
||
|
r, e = fac**(e//E)*r, E
|
||
|
if not big:
|
||
|
e0 = primefactors(e)
|
||
|
if e0[0] != e:
|
||
|
r, e = r**(e//e0[0]), e0[0]
|
||
|
return r, e
|
||
|
|
||
|
# 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:
|
||
|
r, e = m[0], e*m[1]
|
||
|
return int(r), e
|
||
|
|
||
|
return False
|
||
|
|
||
|
|
||
|
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
|
||
|
==========
|
||
|
|
||
|
.. [1] 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
|
||
|
|
||
|
|
||
|
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 number 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 does not 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 did not
|
||
|
work had a gcd value not equal to ``n`` but equal to one of the
|
||
|
factors:
|
||
|
|
||
|
>>> from sympy import ilcm, igcd, 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 cannot be cracked:
|
||
|
|
||
|
>>> from sympy.ntheory import pollard_pm1
|
||
|
>>> 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.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 do not
|
||
|
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
|
||
|
==========
|
||
|
|
||
|
.. [1] Richard Crandall & Carl Pomerance (2005), "Prime Numbers:
|
||
|
A Computational Perspective", Springer, 2nd edition, 236-238
|
||
|
.. [2] https://web.archive.org/web/20150716201437/http://modular.math.washington.edu/edu/2007/spring/ent/ent-html/node81.html
|
||
|
.. [3] https://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"
|
||
|
ecm_msg = "Elliptic Curve with B1 bound %i, B2 bound %i, num_curves %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) was not 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)
|
||
|
|
||
|
|
||
|
def factorint(n, limit=None, use_trial=True, use_rho=True, use_pm1=True,
|
||
|
use_ecm=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
|
||
|
>>> a = 1407633717262338957430697921446883
|
||
|
>>> f = factorint(a, limit=10000)
|
||
|
>>> f == {991: 1, int(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
|
||
|
>>> 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.
|
||
|
|
||
|
See Also
|
||
|
========
|
||
|
|
||
|
smoothness, smoothness_p, divisors
|
||
|
|
||
|
"""
|
||
|
if isinstance(n, Dict):
|
||
|
n = dict(n)
|
||
|
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.One/p]*(-fac[p])
|
||
|
for p in sorted(fac)), [])
|
||
|
return factorlist
|
||
|
|
||
|
factordict = {}
|
||
|
if visual and not isinstance(n, (Mul, 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 = {int(k): int(v) for k, v in
|
||
|
n.as_powers_dict().items()}
|
||
|
elif isinstance(n, dict):
|
||
|
factordict = n
|
||
|
if factordict and isinstance(n, (Mul, dict)):
|
||
|
# check it
|
||
|
for key in list(factordict.keys()):
|
||
|
if isprime(key):
|
||
|
continue
|
||
|
e = factordict.pop(key)
|
||
|
d = factorint(key, 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, Mul)):
|
||
|
return factordict
|
||
|
|
||
|
assert use_trial or use_rho or use_pm1 or use_ecm
|
||
|
|
||
|
from sympy.functions.combinatorial.factorials import factorial
|
||
|
if isinstance(n, factorial):
|
||
|
x = as_int(n.args[0])
|
||
|
if x >= 20:
|
||
|
factors = {}
|
||
|
m = 2 # to initialize the if condition below
|
||
|
for p in sieve.primerange(2, x + 1):
|
||
|
if m > 1:
|
||
|
m, q = 0, x // p
|
||
|
while q != 0:
|
||
|
m += q
|
||
|
q //= p
|
||
|
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
|
||
|
else:
|
||
|
# if n < 20!, direct computation is faster
|
||
|
# since it uses a lookup table
|
||
|
n = n.func(x)
|
||
|
|
||
|
n = as_int(n)
|
||
|
if limit:
|
||
|
limit = int(limit)
|
||
|
use_ecm = False
|
||
|
|
||
|
# 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)
|
||
|
for k, v in facs.items():
|
||
|
factors[k] = factors.get(k, 0) + v
|
||
|
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
|
||
|
iteration = 0
|
||
|
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 not found_trial:
|
||
|
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,
|
||
|
use_ecm=use_ecm,
|
||
|
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,
|
||
|
use_ecm=use_ecm,
|
||
|
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
|
||
|
#Use subexponential algorithms if use_ecm
|
||
|
#Use pollard algorithms for finding small factors for 3 iterations
|
||
|
#if after small factors the number of digits of n is >= 20 then use ecm
|
||
|
iteration += 1
|
||
|
if use_ecm and iteration >= 3 and len(str(n)) >= 25:
|
||
|
break
|
||
|
low, high = high, high*2
|
||
|
B1 = 10000
|
||
|
B2 = 100*B1
|
||
|
num_curves = 50
|
||
|
while(1):
|
||
|
if verbose:
|
||
|
print(ecm_msg % (B1, B2, num_curves))
|
||
|
while(1):
|
||
|
try:
|
||
|
factor = _ecm_one_factor(n, B1, B2, num_curves)
|
||
|
ps = factorint(factor, limit=limit - 1,
|
||
|
use_trial=use_trial,
|
||
|
use_rho=use_rho,
|
||
|
use_pm1=use_pm1,
|
||
|
use_ecm=use_ecm,
|
||
|
verbose=verbose)
|
||
|
n, _ = _trial(factors, n, ps, verbose=False)
|
||
|
_check_termination(factors, n, limit, use_trial,
|
||
|
use_rho, use_pm1, verbose)
|
||
|
except ValueError:
|
||
|
break
|
||
|
except StopIteration:
|
||
|
if verbose:
|
||
|
print(complete_msg)
|
||
|
return factors
|
||
|
B1 *= 5
|
||
|
B2 = 100*B1
|
||
|
num_curves *= 4
|
||
|
|
||
|
|
||
|
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 import factorrat, 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
|
||
|
"""
|
||
|
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.One/p]*(-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)
|
||
|
|
||
|
|
||
|
|
||
|
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]
|
||
|
|
||
|
See Also
|
||
|
========
|
||
|
|
||
|
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, proper=False):
|
||
|
"""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
|
||
|
|
||
|
if proper:
|
||
|
for p in rec_gen():
|
||
|
if p != n:
|
||
|
yield p
|
||
|
else:
|
||
|
yield from rec_gen()
|
||
|
|
||
|
|
||
|
def divisors(n, generator=False, proper=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]
|
||
|
|
||
|
Notes
|
||
|
=====
|
||
|
|
||
|
This is a slightly modified version of Tim Peters referenced at:
|
||
|
https://stackoverflow.com/questions/1010381/python-factorization
|
||
|
|
||
|
See Also
|
||
|
========
|
||
|
|
||
|
primefactors, factorint, divisor_count
|
||
|
"""
|
||
|
|
||
|
n = as_int(abs(n))
|
||
|
if isprime(n):
|
||
|
if proper:
|
||
|
return [1]
|
||
|
return [1, n]
|
||
|
if n == 1:
|
||
|
if proper:
|
||
|
return []
|
||
|
return [1]
|
||
|
if n == 0:
|
||
|
return []
|
||
|
rv = _divisors(n, proper)
|
||
|
if not generator:
|
||
|
return sorted(rv)
|
||
|
return rv
|
||
|
|
||
|
|
||
|
def divisor_count(n, modulus=1, proper=False):
|
||
|
"""
|
||
|
Return the number of divisors of ``n``. If ``modulus`` is not 1 then only
|
||
|
those that are divisible by ``modulus`` are counted. If ``proper`` is True
|
||
|
then the divisor of ``n`` will not be counted.
|
||
|
|
||
|
Examples
|
||
|
========
|
||
|
|
||
|
>>> from sympy import divisor_count
|
||
|
>>> divisor_count(6)
|
||
|
4
|
||
|
>>> divisor_count(6, 2)
|
||
|
2
|
||
|
>>> divisor_count(6, proper=True)
|
||
|
3
|
||
|
|
||
|
See Also
|
||
|
========
|
||
|
|
||
|
factorint, divisors, totient, proper_divisor_count
|
||
|
|
||
|
"""
|
||
|
|
||
|
if not modulus:
|
||
|
return 0
|
||
|
elif modulus != 1:
|
||
|
n, r = divmod(n, modulus)
|
||
|
if r:
|
||
|
return 0
|
||
|
if n == 0:
|
||
|
return 0
|
||
|
n = Mul(*[v + 1 for k, v in factorint(n).items() if k > 1])
|
||
|
if n and proper:
|
||
|
n -= 1
|
||
|
return n
|
||
|
|
||
|
|
||
|
def proper_divisors(n, generator=False):
|
||
|
"""
|
||
|
Return all divisors of n except n, sorted by default.
|
||
|
If generator is ``True`` an unordered generator is returned.
|
||
|
|
||
|
Examples
|
||
|
========
|
||
|
|
||
|
>>> from sympy import proper_divisors, proper_divisor_count
|
||
|
>>> proper_divisors(24)
|
||
|
[1, 2, 3, 4, 6, 8, 12]
|
||
|
>>> proper_divisor_count(24)
|
||
|
7
|
||
|
>>> list(proper_divisors(120, generator=True))
|
||
|
[1, 2, 4, 8, 3, 6, 12, 24, 5, 10, 20, 40, 15, 30, 60]
|
||
|
|
||
|
See Also
|
||
|
========
|
||
|
|
||
|
factorint, divisors, proper_divisor_count
|
||
|
|
||
|
"""
|
||
|
return divisors(n, generator=generator, proper=True)
|
||
|
|
||
|
|
||
|
def proper_divisor_count(n, modulus=1):
|
||
|
"""
|
||
|
Return the number of proper divisors of ``n``.
|
||
|
|
||
|
Examples
|
||
|
========
|
||
|
|
||
|
>>> from sympy import proper_divisor_count
|
||
|
>>> proper_divisor_count(6)
|
||
|
3
|
||
|
>>> proper_divisor_count(6, modulus=2)
|
||
|
1
|
||
|
|
||
|
See Also
|
||
|
========
|
||
|
|
||
|
divisors, proper_divisors, divisor_count
|
||
|
|
||
|
"""
|
||
|
return divisor_count(n, modulus=modulus, proper=True)
|
||
|
|
||
|
|
||
|
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
|
||
|
|
||
|
|
||
|
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).
|
||
|
|
||
|
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]
|
||
|
|
||
|
See Also
|
||
|
========
|
||
|
|
||
|
primefactors, factorint, divisors, divisor_count, udivisor_count
|
||
|
|
||
|
References
|
||
|
==========
|
||
|
|
||
|
.. [1] https://en.wikipedia.org/wiki/Unitary_divisor
|
||
|
.. [2] https://mathworld.wolfram.com/UnitaryDivisor.html
|
||
|
|
||
|
"""
|
||
|
|
||
|
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
|
||
|
|
||
|
|
||
|
def udivisor_count(n):
|
||
|
"""
|
||
|
Return the number of unitary divisors of ``n``.
|
||
|
|
||
|
Parameters
|
||
|
==========
|
||
|
|
||
|
n : integer
|
||
|
|
||
|
Examples
|
||
|
========
|
||
|
|
||
|
>>> from sympy.ntheory.factor_ import udivisor_count
|
||
|
>>> udivisor_count(120)
|
||
|
8
|
||
|
|
||
|
See Also
|
||
|
========
|
||
|
|
||
|
factorint, divisors, udivisors, divisor_count, totient
|
||
|
|
||
|
References
|
||
|
==========
|
||
|
|
||
|
.. [1] https://mathworld.wolfram.com/UnitaryDivisorFunction.html
|
||
|
|
||
|
"""
|
||
|
|
||
|
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
|
||
|
|
||
|
|
||
|
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.
|
||
|
|
||
|
Examples
|
||
|
========
|
||
|
|
||
|
>>> from sympy.ntheory.factor_ import antidivisors
|
||
|
>>> antidivisors(24)
|
||
|
[7, 16]
|
||
|
|
||
|
>>> sorted(antidivisors(128, generator=True))
|
||
|
[3, 5, 15, 17, 51, 85]
|
||
|
|
||
|
See Also
|
||
|
========
|
||
|
|
||
|
primefactors, factorint, divisors, divisor_count, antidivisor_count
|
||
|
|
||
|
References
|
||
|
==========
|
||
|
|
||
|
.. [1] definition is described in https://oeis.org/A066272/a066272a.html
|
||
|
|
||
|
"""
|
||
|
|
||
|
n = as_int(abs(n))
|
||
|
if n <= 2:
|
||
|
return []
|
||
|
rv = _antidivisors(n)
|
||
|
if not generator:
|
||
|
return sorted(rv)
|
||
|
return rv
|
||
|
|
||
|
|
||
|
def antidivisor_count(n):
|
||
|
"""
|
||
|
Return the number of antidivisors [1]_ of ``n``.
|
||
|
|
||
|
Parameters
|
||
|
==========
|
||
|
|
||
|
n : integer
|
||
|
|
||
|
Examples
|
||
|
========
|
||
|
|
||
|
>>> from sympy.ntheory.factor_ import antidivisor_count
|
||
|
>>> antidivisor_count(13)
|
||
|
4
|
||
|
>>> antidivisor_count(27)
|
||
|
5
|
||
|
|
||
|
See Also
|
||
|
========
|
||
|
|
||
|
factorint, divisors, antidivisors, divisor_count, totient
|
||
|
|
||
|
References
|
||
|
==========
|
||
|
|
||
|
.. [1] formula from https://oeis.org/A066272
|
||
|
|
||
|
"""
|
||
|
|
||
|
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
|
||
|
|
||
|
|
||
|
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.
|
||
|
|
||
|
Parameters
|
||
|
==========
|
||
|
|
||
|
n : integer
|
||
|
|
||
|
Examples
|
||
|
========
|
||
|
|
||
|
>>> from sympy.ntheory import totient
|
||
|
>>> totient(1)
|
||
|
1
|
||
|
>>> totient(25)
|
||
|
20
|
||
|
>>> totient(45) == totient(5)*totient(9)
|
||
|
True
|
||
|
|
||
|
See Also
|
||
|
========
|
||
|
|
||
|
divisor_count
|
||
|
|
||
|
References
|
||
|
==========
|
||
|
|
||
|
.. [1] https://en.wikipedia.org/wiki/Euler%27s_totient_function
|
||
|
.. [2] https://mathworld.wolfram.com/TotientFunction.html
|
||
|
|
||
|
"""
|
||
|
@classmethod
|
||
|
def eval(cls, n):
|
||
|
if n.is_Integer:
|
||
|
if n < 1:
|
||
|
raise ValueError("n must be a positive integer")
|
||
|
factors = factorint(n)
|
||
|
return cls._from_factors(factors)
|
||
|
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])
|
||
|
|
||
|
@classmethod
|
||
|
def _from_distinct_primes(self, *args):
|
||
|
"""Subroutine to compute totient from the list of assumed
|
||
|
distinct primes
|
||
|
|
||
|
Examples
|
||
|
========
|
||
|
|
||
|
>>> from sympy.ntheory.factor_ import totient
|
||
|
>>> totient._from_distinct_primes(5, 7)
|
||
|
24
|
||
|
"""
|
||
|
return reduce(lambda i, j: i * (j-1), args, 1)
|
||
|
|
||
|
@classmethod
|
||
|
def _from_factors(self, factors):
|
||
|
"""Subroutine to compute totient from already-computed factors
|
||
|
|
||
|
Examples
|
||
|
========
|
||
|
|
||
|
>>> from sympy.ntheory.factor_ import totient
|
||
|
>>> totient._from_factors({5: 2})
|
||
|
20
|
||
|
"""
|
||
|
t = 1
|
||
|
for p, k in factors.items():
|
||
|
t *= (p - 1) * p**(k - 1)
|
||
|
return t
|
||
|
|
||
|
|
||
|
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.
|
||
|
|
||
|
Examples
|
||
|
========
|
||
|
|
||
|
>>> from sympy.ntheory import reduced_totient
|
||
|
>>> reduced_totient(1)
|
||
|
1
|
||
|
>>> reduced_totient(8)
|
||
|
2
|
||
|
>>> reduced_totient(30)
|
||
|
4
|
||
|
|
||
|
See Also
|
||
|
========
|
||
|
|
||
|
totient
|
||
|
|
||
|
References
|
||
|
==========
|
||
|
|
||
|
.. [1] https://en.wikipedia.org/wiki/Carmichael_function
|
||
|
.. [2] https://mathworld.wolfram.com/CarmichaelFunction.html
|
||
|
|
||
|
"""
|
||
|
@classmethod
|
||
|
def eval(cls, n):
|
||
|
if n.is_Integer:
|
||
|
if n < 1:
|
||
|
raise ValueError("n must be a positive integer")
|
||
|
factors = factorint(n)
|
||
|
return cls._from_factors(factors)
|
||
|
|
||
|
@classmethod
|
||
|
def _from_factors(self, factors):
|
||
|
"""Subroutine to compute totient from already-computed factors
|
||
|
"""
|
||
|
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
|
||
|
|
||
|
@classmethod
|
||
|
def _from_distinct_primes(self, *args):
|
||
|
"""Subroutine to compute totient from the list of assumed
|
||
|
distinct primes
|
||
|
"""
|
||
|
args = [p - 1 for p in args]
|
||
|
return ilcm(*args)
|
||
|
|
||
|
def _eval_is_integer(self):
|
||
|
return fuzzy_and([self.args[0].is_integer, self.args[0].is_positive])
|
||
|
|
||
|
|
||
|
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
|
||
|
==========
|
||
|
|
||
|
n : integer
|
||
|
|
||
|
k : integer, optional
|
||
|
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.
|
||
|
|
||
|
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
|
||
|
|
||
|
See Also
|
||
|
========
|
||
|
|
||
|
divisor_count, totient, divisors, factorint
|
||
|
|
||
|
References
|
||
|
==========
|
||
|
|
||
|
.. [1] https://en.wikipedia.org/wiki/Divisor_function
|
||
|
|
||
|
"""
|
||
|
|
||
|
@classmethod
|
||
|
def eval(cls, n, k=S.One):
|
||
|
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")
|
||
|
elif k.is_Integer:
|
||
|
k = int(k)
|
||
|
return Integer(math.prod(
|
||
|
(p**(k*(e + 1)) - 1)//(p**k - 1) if k != 0
|
||
|
else e + 1 for p, e in factorint(n).items()))
|
||
|
else:
|
||
|
return Mul(*[(p**(k*(e + 1)) - 1)/(p**k - 1) if k != 0
|
||
|
else e + 1 for p, e in factorint(n).items()])
|
||
|
|
||
|
if n.is_integer: # symbolic case
|
||
|
args = []
|
||
|
for p, e in (_.as_base_exp() for _ in Mul.make_args(n)):
|
||
|
if p.is_prime and e.is_positive:
|
||
|
args.append((p**(k*(e + 1)) - 1)/(p**k - 1) if
|
||
|
k != 0 else e + 1)
|
||
|
else:
|
||
|
return
|
||
|
return Mul(*args)
|
||
|
|
||
|
|
||
|
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
|
||
|
==========
|
||
|
|
||
|
n : integer
|
||
|
|
||
|
t : integer
|
||
|
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.
|
||
|
|
||
|
Examples
|
||
|
========
|
||
|
|
||
|
>>> from sympy.ntheory.factor_ import core
|
||
|
>>> core(24, 2)
|
||
|
6
|
||
|
>>> core(9424, 3)
|
||
|
1178
|
||
|
>>> core(379238)
|
||
|
379238
|
||
|
>>> core(15**11, 10)
|
||
|
15
|
||
|
|
||
|
See Also
|
||
|
========
|
||
|
|
||
|
factorint, sympy.solvers.diophantine.diophantine.square_factor
|
||
|
|
||
|
References
|
||
|
==========
|
||
|
|
||
|
.. [1] https://en.wikipedia.org/wiki/Square-free_integer#Squarefree_core
|
||
|
|
||
|
"""
|
||
|
|
||
|
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
|
||
|
|
||
|
|
||
|
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.
|
||
|
|
||
|
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
|
||
|
|
||
|
See Also
|
||
|
========
|
||
|
|
||
|
divisor_count, totient, divisors, udivisors, udivisor_count, divisor_sigma,
|
||
|
factorint
|
||
|
|
||
|
References
|
||
|
==========
|
||
|
|
||
|
.. [1] https://mathworld.wolfram.com/UnitaryDivisorFunction.html
|
||
|
|
||
|
"""
|
||
|
|
||
|
@classmethod
|
||
|
def eval(cls, n, k=S.One):
|
||
|
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()])
|
||
|
|
||
|
|
||
|
class primenu(Function):
|
||
|
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.
|
||
|
|
||
|
Examples
|
||
|
========
|
||
|
|
||
|
>>> from sympy.ntheory.factor_ import primenu
|
||
|
>>> primenu(1)
|
||
|
0
|
||
|
>>> primenu(30)
|
||
|
3
|
||
|
|
||
|
See Also
|
||
|
========
|
||
|
|
||
|
factorint
|
||
|
|
||
|
References
|
||
|
==========
|
||
|
|
||
|
.. [1] https://mathworld.wolfram.com/PrimeFactor.html
|
||
|
|
||
|
"""
|
||
|
|
||
|
@classmethod
|
||
|
def eval(cls, n):
|
||
|
if n.is_Integer:
|
||
|
if n <= 0:
|
||
|
raise ValueError("n must be a positive integer")
|
||
|
else:
|
||
|
return len(factorint(n).keys())
|
||
|
|
||
|
|
||
|
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.
|
||
|
|
||
|
Examples
|
||
|
========
|
||
|
|
||
|
>>> from sympy.ntheory.factor_ import primeomega
|
||
|
>>> primeomega(1)
|
||
|
0
|
||
|
>>> primeomega(20)
|
||
|
3
|
||
|
|
||
|
See Also
|
||
|
========
|
||
|
|
||
|
factorint
|
||
|
|
||
|
References
|
||
|
==========
|
||
|
|
||
|
.. [1] https://mathworld.wolfram.com/PrimeFactor.html
|
||
|
|
||
|
"""
|
||
|
|
||
|
@classmethod
|
||
|
def eval(cls, n):
|
||
|
if n.is_Integer:
|
||
|
if n <= 0:
|
||
|
raise ValueError("n must be a positive integer")
|
||
|
else:
|
||
|
return sum(factorint(n).values())
|
||
|
|
||
|
|
||
|
def mersenne_prime_exponent(nth):
|
||
|
"""Returns the exponent ``i`` for the nth Mersenne prime (which
|
||
|
has the form `2^i - 1`).
|
||
|
|
||
|
Examples
|
||
|
========
|
||
|
|
||
|
>>> from sympy.ntheory.factor_ import mersenne_prime_exponent
|
||
|
>>> mersenne_prime_exponent(1)
|
||
|
2
|
||
|
>>> mersenne_prime_exponent(20)
|
||
|
4423
|
||
|
"""
|
||
|
n = as_int(nth)
|
||
|
if n < 1:
|
||
|
raise ValueError("nth must be a positive integer; mersenne_prime_exponent(1) == 2")
|
||
|
if n > 51:
|
||
|
raise ValueError("There are only 51 perfect numbers; nth must be less than or equal to 51")
|
||
|
return MERSENNE_PRIME_EXPONENTS[n - 1]
|
||
|
|
||
|
|
||
|
def is_perfect(n):
|
||
|
"""Returns True if ``n`` is a perfect number, else False.
|
||
|
|
||
|
A perfect number is equal to the sum of its positive, proper divisors.
|
||
|
|
||
|
Examples
|
||
|
========
|
||
|
|
||
|
>>> from sympy.ntheory.factor_ import is_perfect, divisors, divisor_sigma
|
||
|
>>> is_perfect(20)
|
||
|
False
|
||
|
>>> is_perfect(6)
|
||
|
True
|
||
|
>>> 6 == divisor_sigma(6) - 6 == sum(divisors(6)[:-1])
|
||
|
True
|
||
|
|
||
|
References
|
||
|
==========
|
||
|
|
||
|
.. [1] https://mathworld.wolfram.com/PerfectNumber.html
|
||
|
.. [2] https://en.wikipedia.org/wiki/Perfect_number
|
||
|
|
||
|
"""
|
||
|
|
||
|
n = as_int(n)
|
||
|
if _isperfect(n):
|
||
|
return True
|
||
|
|
||
|
# all perfect numbers for Mersenne primes with exponents
|
||
|
# less than or equal to 43112609 are known
|
||
|
iknow = MERSENNE_PRIME_EXPONENTS.index(43112609)
|
||
|
if iknow <= len(PERFECT) - 1 and n <= PERFECT[iknow]:
|
||
|
# there may be gaps between this and larger known values
|
||
|
# so only conclude in the range for which all values
|
||
|
# are known
|
||
|
return False
|
||
|
if n%2 == 0:
|
||
|
last2 = n % 100
|
||
|
if last2 != 28 and last2 % 10 != 6:
|
||
|
return False
|
||
|
r, b = integer_nthroot(1 + 8*n, 2)
|
||
|
if not b:
|
||
|
return False
|
||
|
m, x = divmod(1 + r, 4)
|
||
|
if x:
|
||
|
return False
|
||
|
e, b = integer_log(m, 2)
|
||
|
if not b:
|
||
|
return False
|
||
|
else:
|
||
|
if n < 10**2000: # https://www.lirmm.fr/~ochem/opn/
|
||
|
return False
|
||
|
if n % 105 == 0: # not divis by 105
|
||
|
return False
|
||
|
if not any(n%m == r for m, r in [(12, 1), (468, 117), (324, 81)]):
|
||
|
return False
|
||
|
# there are many criteria that the factor structure of n
|
||
|
# must meet; since we will have to factor it to test the
|
||
|
# structure we will have the factors and can then check
|
||
|
# to see whether it is a perfect number or not. So we
|
||
|
# skip the structure checks and go straight to the final
|
||
|
# test below.
|
||
|
rv = divisor_sigma(n) - n
|
||
|
if rv == n:
|
||
|
if n%2 == 0:
|
||
|
raise ValueError(filldedent('''
|
||
|
This even number is perfect and is associated with a
|
||
|
Mersenne Prime, 2^%s - 1. It should be
|
||
|
added to SymPy.''' % (e + 1)))
|
||
|
else:
|
||
|
raise ValueError(filldedent('''In 1888, Sylvester stated: "
|
||
|
...a prolonged meditation on the subject has satisfied
|
||
|
me that the existence of any one such [odd perfect number]
|
||
|
-- its escape, so to say, from the complex web of conditions
|
||
|
which hem it in on all sides -- would be little short of a
|
||
|
miracle." I guess SymPy just found that miracle and it
|
||
|
factors like this: %s''' % factorint(n)))
|
||
|
|
||
|
|
||
|
def is_mersenne_prime(n):
|
||
|
"""Returns True if ``n`` is a Mersenne prime, else False.
|
||
|
|
||
|
A Mersenne prime is a prime number having the form `2^i - 1`.
|
||
|
|
||
|
Examples
|
||
|
========
|
||
|
|
||
|
>>> from sympy.ntheory.factor_ import is_mersenne_prime
|
||
|
>>> is_mersenne_prime(6)
|
||
|
False
|
||
|
>>> is_mersenne_prime(127)
|
||
|
True
|
||
|
|
||
|
References
|
||
|
==========
|
||
|
|
||
|
.. [1] https://mathworld.wolfram.com/MersennePrime.html
|
||
|
|
||
|
"""
|
||
|
|
||
|
n = as_int(n)
|
||
|
if _ismersenneprime(n):
|
||
|
return True
|
||
|
if not isprime(n):
|
||
|
return False
|
||
|
r, b = integer_log(n + 1, 2)
|
||
|
if not b:
|
||
|
return False
|
||
|
raise ValueError(filldedent('''
|
||
|
This Mersenne Prime, 2^%s - 1, should
|
||
|
be added to SymPy's known values.''' % r))
|
||
|
|
||
|
|
||
|
def abundance(n):
|
||
|
"""Returns the difference between the sum of the positive
|
||
|
proper divisors of a number and the number.
|
||
|
|
||
|
Examples
|
||
|
========
|
||
|
|
||
|
>>> from sympy.ntheory import abundance, is_perfect, is_abundant
|
||
|
>>> abundance(6)
|
||
|
0
|
||
|
>>> is_perfect(6)
|
||
|
True
|
||
|
>>> abundance(10)
|
||
|
-2
|
||
|
>>> is_abundant(10)
|
||
|
False
|
||
|
"""
|
||
|
return divisor_sigma(n, 1) - 2 * n
|
||
|
|
||
|
|
||
|
def is_abundant(n):
|
||
|
"""Returns True if ``n`` is an abundant number, else False.
|
||
|
|
||
|
A abundant number is smaller than the sum of its positive proper divisors.
|
||
|
|
||
|
Examples
|
||
|
========
|
||
|
|
||
|
>>> from sympy.ntheory.factor_ import is_abundant
|
||
|
>>> is_abundant(20)
|
||
|
True
|
||
|
>>> is_abundant(15)
|
||
|
False
|
||
|
|
||
|
References
|
||
|
==========
|
||
|
|
||
|
.. [1] https://mathworld.wolfram.com/AbundantNumber.html
|
||
|
|
||
|
"""
|
||
|
n = as_int(n)
|
||
|
if is_perfect(n):
|
||
|
return False
|
||
|
return n % 6 == 0 or bool(abundance(n) > 0)
|
||
|
|
||
|
|
||
|
def is_deficient(n):
|
||
|
"""Returns True if ``n`` is a deficient number, else False.
|
||
|
|
||
|
A deficient number is greater than the sum of its positive proper divisors.
|
||
|
|
||
|
Examples
|
||
|
========
|
||
|
|
||
|
>>> from sympy.ntheory.factor_ import is_deficient
|
||
|
>>> is_deficient(20)
|
||
|
False
|
||
|
>>> is_deficient(15)
|
||
|
True
|
||
|
|
||
|
References
|
||
|
==========
|
||
|
|
||
|
.. [1] https://mathworld.wolfram.com/DeficientNumber.html
|
||
|
|
||
|
"""
|
||
|
n = as_int(n)
|
||
|
if is_perfect(n):
|
||
|
return False
|
||
|
return bool(abundance(n) < 0)
|
||
|
|
||
|
|
||
|
def is_amicable(m, n):
|
||
|
"""Returns True if the numbers `m` and `n` are "amicable", else False.
|
||
|
|
||
|
Amicable numbers are two different numbers so related that the sum
|
||
|
of the proper divisors of each is equal to that of the other.
|
||
|
|
||
|
Examples
|
||
|
========
|
||
|
|
||
|
>>> from sympy.ntheory.factor_ import is_amicable, divisor_sigma
|
||
|
>>> is_amicable(220, 284)
|
||
|
True
|
||
|
>>> divisor_sigma(220) == divisor_sigma(284)
|
||
|
True
|
||
|
|
||
|
References
|
||
|
==========
|
||
|
|
||
|
.. [1] https://en.wikipedia.org/wiki/Amicable_numbers
|
||
|
|
||
|
"""
|
||
|
if m == n:
|
||
|
return False
|
||
|
a, b = (divisor_sigma(i) for i in (m, n))
|
||
|
return a == b == (m + n)
|
||
|
|
||
|
|
||
|
def dra(n, b):
|
||
|
"""
|
||
|
Returns the additive digital root of a natural number ``n`` in base ``b``
|
||
|
which is a single digit value obtained by an iterative process of summing
|
||
|
digits, on each iteration using the result from the previous iteration to
|
||
|
compute a digit sum.
|
||
|
|
||
|
Examples
|
||
|
========
|
||
|
|
||
|
>>> from sympy.ntheory.factor_ import dra
|
||
|
>>> dra(3110, 12)
|
||
|
8
|
||
|
|
||
|
References
|
||
|
==========
|
||
|
|
||
|
.. [1] https://en.wikipedia.org/wiki/Digital_root
|
||
|
|
||
|
"""
|
||
|
|
||
|
num = abs(as_int(n))
|
||
|
b = as_int(b)
|
||
|
if b <= 1:
|
||
|
raise ValueError("Base should be an integer greater than 1")
|
||
|
|
||
|
if num == 0:
|
||
|
return 0
|
||
|
|
||
|
return (1 + (num - 1) % (b - 1))
|
||
|
|
||
|
|
||
|
def drm(n, b):
|
||
|
"""
|
||
|
Returns the multiplicative digital root of a natural number ``n`` in a given
|
||
|
base ``b`` which is a single digit value obtained by an iterative process of
|
||
|
multiplying digits, on each iteration using the result from the previous
|
||
|
iteration to compute the digit multiplication.
|
||
|
|
||
|
Examples
|
||
|
========
|
||
|
|
||
|
>>> from sympy.ntheory.factor_ import drm
|
||
|
>>> drm(9876, 10)
|
||
|
0
|
||
|
|
||
|
>>> drm(49, 10)
|
||
|
8
|
||
|
|
||
|
References
|
||
|
==========
|
||
|
|
||
|
.. [1] https://mathworld.wolfram.com/MultiplicativeDigitalRoot.html
|
||
|
|
||
|
"""
|
||
|
|
||
|
n = abs(as_int(n))
|
||
|
b = as_int(b)
|
||
|
if b <= 1:
|
||
|
raise ValueError("Base should be an integer greater than 1")
|
||
|
while n > b:
|
||
|
mul = 1
|
||
|
while n > 1:
|
||
|
n, r = divmod(n, b)
|
||
|
if r == 0:
|
||
|
return 0
|
||
|
mul *= r
|
||
|
n = mul
|
||
|
return n
|