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"""
Primality testing
"""
from sympy.core.numbers import igcd
from sympy.core.power import integer_nthroot
from sympy.core.sympify import sympify
from sympy.external.gmpy import HAS_GMPY
from sympy.utilities.misc import as_int
from mpmath.libmp import bitcount as _bitlength
def _int_tuple(*i):
return tuple(int(_) for _ in i)
def is_euler_pseudoprime(n, b):
"""Returns True if n is prime or an Euler pseudoprime to base b, else False.
Euler Pseudoprime : In arithmetic, an odd composite integer n is called an
euler pseudoprime to base a, if a and n are coprime and satisfy the modular
arithmetic congruence relation :
a ^ (n-1)/2 = + 1(mod n) or
a ^ (n-1)/2 = - 1(mod n)
(where mod refers to the modulo operation).
Examples
========
>>> from sympy.ntheory.primetest import is_euler_pseudoprime
>>> is_euler_pseudoprime(2, 5)
True
References
==========
.. [1] https://en.wikipedia.org/wiki/Euler_pseudoprime
"""
from sympy.ntheory.factor_ import trailing
if not mr(n, [b]):
return False
n = as_int(n)
r = n - 1
c = pow(b, r >> trailing(r), n)
if c == 1:
return True
while True:
if c == n - 1:
return True
c = pow(c, 2, n)
if c == 1:
return False
def is_square(n, prep=True):
"""Return True if n == a * a for some integer a, else False.
If n is suspected of *not* being a square then this is a
quick method of confirming that it is not.
Examples
========
>>> from sympy.ntheory.primetest import is_square
>>> is_square(25)
True
>>> is_square(2)
False
References
==========
.. [1] https://mersenneforum.org/showpost.php?p=110896
See Also
========
sympy.core.power.integer_nthroot
"""
if prep:
n = as_int(n)
if n < 0:
return False
if n in (0, 1):
return True
# def magic(n):
# s = {x**2 % n for x in range(n)}
# return sum(1 << bit for bit in s)
# >>> print(hex(magic(128)))
# 0x2020212020202130202021202030213
# >>> print(hex(magic(99)))
# 0x209060049048220348a410213
# >>> print(hex(magic(91)))
# 0x102e403012a0c9862c14213
# >>> print(hex(magic(85)))
# 0x121065188e001c46298213
if not 0x2020212020202130202021202030213 & (1 << (n & 127)):
return False # e.g. 2, 3
m = n % (99 * 91 * 85)
if not 0x209060049048220348a410213 & (1 << (m % 99)):
return False # e.g. 17, 68
if not 0x102e403012a0c9862c14213 & (1 << (m % 91)):
return False # e.g. 97, 388
if not 0x121065188e001c46298213 & (1 << (m % 85)):
return False # e.g. 793, 1408
# n is either:
# a) odd = 4*even + 1 (and square if even = k*(k + 1))
# b) even with
# odd multiplicity of 2 --> not square, e.g. 39040
# even multiplicity of 2, e.g. 4, 16, 36, ..., 16324
# removal of factors of 2 to give an odd, and rejection if
# any(i%2 for i in divmod(odd - 1, 4))
# will give an odd number in form 4*even + 1.
# Use of `trailing` to check the power of 2 is not done since it
# does not apply to a large percentage of arbitrary numbers
# and the integer_nthroot is able to quickly resolve these cases.
return integer_nthroot(n, 2)[1]
def _test(n, base, s, t):
"""Miller-Rabin strong pseudoprime test for one base.
Return False if n is definitely composite, True if n is
probably prime, with a probability greater than 3/4.
"""
# do the Fermat test
b = pow(base, t, n)
if b == 1 or b == n - 1:
return True
else:
for j in range(1, s):
b = pow(b, 2, n)
if b == n - 1:
return True
# see I. Niven et al. "An Introduction to Theory of Numbers", page 78
if b == 1:
return False
return False
def mr(n, bases):
"""Perform a Miller-Rabin strong pseudoprime test on n using a
given list of bases/witnesses.
References
==========
.. [1] Richard Crandall & Carl Pomerance (2005), "Prime Numbers:
A Computational Perspective", Springer, 2nd edition, 135-138
A list of thresholds and the bases they require are here:
https://en.wikipedia.org/wiki/Miller%E2%80%93Rabin_primality_test#Deterministic_variants
Examples
========
>>> from sympy.ntheory.primetest import mr
>>> mr(1373651, [2, 3])
False
>>> mr(479001599, [31, 73])
True
"""
from sympy.ntheory.factor_ import trailing
from sympy.polys.domains import ZZ
n = as_int(n)
if n < 2:
return False
# remove powers of 2 from n-1 (= t * 2**s)
s = trailing(n - 1)
t = n >> s
for base in bases:
# Bases >= n are wrapped, bases < 2 are invalid
if base >= n:
base %= n
if base >= 2:
base = ZZ(base)
if not _test(n, base, s, t):
return False
return True
def _lucas_sequence(n, P, Q, k):
"""Return the modular Lucas sequence (U_k, V_k, Q_k).
Given a Lucas sequence defined by P, Q, returns the kth values for
U and V, along with Q^k, all modulo n. This is intended for use with
possibly very large values of n and k, where the combinatorial functions
would be completely unusable.
The modular Lucas sequences are used in numerous places in number theory,
especially in the Lucas compositeness tests and the various n + 1 proofs.
Examples
========
>>> from sympy.ntheory.primetest import _lucas_sequence
>>> N = 10**2000 + 4561
>>> sol = U, V, Qk = _lucas_sequence(N, 3, 1, N//2); sol
(0, 2, 1)
"""
D = P*P - 4*Q
if n < 2:
raise ValueError("n must be >= 2")
if k < 0:
raise ValueError("k must be >= 0")
if D == 0:
raise ValueError("D must not be zero")
if k == 0:
return _int_tuple(0, 2, Q)
U = 1
V = P
Qk = Q
b = _bitlength(k)
if Q == 1:
# Optimization for extra strong tests.
while b > 1:
U = (U*V) % n
V = (V*V - 2) % n
b -= 1
if (k >> (b - 1)) & 1:
U, V = U*P + V, V*P + U*D
if U & 1:
U += n
if V & 1:
V += n
U, V = U >> 1, V >> 1
elif P == 1 and Q == -1:
# Small optimization for 50% of Selfridge parameters.
while b > 1:
U = (U*V) % n
if Qk == 1:
V = (V*V - 2) % n
else:
V = (V*V + 2) % n
Qk = 1
b -= 1
if (k >> (b-1)) & 1:
U, V = U + V, V + U*D
if U & 1:
U += n
if V & 1:
V += n
U, V = U >> 1, V >> 1
Qk = -1
else:
# The general case with any P and Q.
while b > 1:
U = (U*V) % n
V = (V*V - 2*Qk) % n
Qk *= Qk
b -= 1
if (k >> (b - 1)) & 1:
U, V = U*P + V, V*P + U*D
if U & 1:
U += n
if V & 1:
V += n
U, V = U >> 1, V >> 1
Qk *= Q
Qk %= n
return _int_tuple(U % n, V % n, Qk)
def _lucas_selfridge_params(n):
"""Calculates the Selfridge parameters (D, P, Q) for n. This is
method A from page 1401 of Baillie and Wagstaff.
References
==========
.. [1] "Lucas Pseudoprimes", Baillie and Wagstaff, 1980.
http://mpqs.free.fr/LucasPseudoprimes.pdf
"""
from sympy.ntheory.residue_ntheory import jacobi_symbol
D = 5
while True:
g = igcd(abs(D), n)
if g > 1 and g != n:
return (0, 0, 0)
if jacobi_symbol(D, n) == -1:
break
if D > 0:
D = -D - 2
else:
D = -D + 2
return _int_tuple(D, 1, (1 - D)/4)
def _lucas_extrastrong_params(n):
"""Calculates the "extra strong" parameters (D, P, Q) for n.
References
==========
.. [1] OEIS A217719: Extra Strong Lucas Pseudoprimes
https://oeis.org/A217719
.. [1] https://en.wikipedia.org/wiki/Lucas_pseudoprime
"""
from sympy.ntheory.residue_ntheory import jacobi_symbol
P, Q, D = 3, 1, 5
while True:
g = igcd(D, n)
if g > 1 and g != n:
return (0, 0, 0)
if jacobi_symbol(D, n) == -1:
break
P += 1
D = P*P - 4
return _int_tuple(D, P, Q)
def is_lucas_prp(n):
"""Standard Lucas compositeness test with Selfridge parameters. Returns
False if n is definitely composite, and True if n is a Lucas probable
prime.
This is typically used in combination with the Miller-Rabin test.
References
==========
- "Lucas Pseudoprimes", Baillie and Wagstaff, 1980.
http://mpqs.free.fr/LucasPseudoprimes.pdf
- OEIS A217120: Lucas Pseudoprimes
https://oeis.org/A217120
- https://en.wikipedia.org/wiki/Lucas_pseudoprime
Examples
========
>>> from sympy.ntheory.primetest import isprime, is_lucas_prp
>>> for i in range(10000):
... if is_lucas_prp(i) and not isprime(i):
... print(i)
323
377
1159
1829
3827
5459
5777
9071
9179
"""
n = as_int(n)
if n == 2:
return True
if n < 2 or (n % 2) == 0:
return False
if is_square(n, False):
return False
D, P, Q = _lucas_selfridge_params(n)
if D == 0:
return False
U, V, Qk = _lucas_sequence(n, P, Q, n+1)
return U == 0
def is_strong_lucas_prp(n):
"""Strong Lucas compositeness test with Selfridge parameters. Returns
False if n is definitely composite, and True if n is a strong Lucas
probable prime.
This is often used in combination with the Miller-Rabin test, and
in particular, when combined with M-R base 2 creates the strong BPSW test.
References
==========
- "Lucas Pseudoprimes", Baillie and Wagstaff, 1980.
http://mpqs.free.fr/LucasPseudoprimes.pdf
- OEIS A217255: Strong Lucas Pseudoprimes
https://oeis.org/A217255
- https://en.wikipedia.org/wiki/Lucas_pseudoprime
- https://en.wikipedia.org/wiki/Baillie-PSW_primality_test
Examples
========
>>> from sympy.ntheory.primetest import isprime, is_strong_lucas_prp
>>> for i in range(20000):
... if is_strong_lucas_prp(i) and not isprime(i):
... print(i)
5459
5777
10877
16109
18971
"""
from sympy.ntheory.factor_ import trailing
n = as_int(n)
if n == 2:
return True
if n < 2 or (n % 2) == 0:
return False
if is_square(n, False):
return False
D, P, Q = _lucas_selfridge_params(n)
if D == 0:
return False
# remove powers of 2 from n+1 (= k * 2**s)
s = trailing(n + 1)
k = (n+1) >> s
U, V, Qk = _lucas_sequence(n, P, Q, k)
if U == 0 or V == 0:
return True
for r in range(1, s):
V = (V*V - 2*Qk) % n
if V == 0:
return True
Qk = pow(Qk, 2, n)
return False
def is_extra_strong_lucas_prp(n):
"""Extra Strong Lucas compositeness test. Returns False if n is
definitely composite, and True if n is a "extra strong" Lucas probable
prime.
The parameters are selected using P = 3, Q = 1, then incrementing P until
(D|n) == -1. The test itself is as defined in Grantham 2000, from the
Mo and Jones preprint. The parameter selection and test are the same as
used in OEIS A217719, Perl's Math::Prime::Util, and the Lucas pseudoprime
page on Wikipedia.
With these parameters, there are no counterexamples below 2^64 nor any
known above that range. It is 20-50% faster than the strong test.
Because of the different parameters selected, there is no relationship
between the strong Lucas pseudoprimes and extra strong Lucas pseudoprimes.
In particular, one is not a subset of the other.
References
==========
- "Frobenius Pseudoprimes", Jon Grantham, 2000.
https://www.ams.org/journals/mcom/2001-70-234/S0025-5718-00-01197-2/
- OEIS A217719: Extra Strong Lucas Pseudoprimes
https://oeis.org/A217719
- https://en.wikipedia.org/wiki/Lucas_pseudoprime
Examples
========
>>> from sympy.ntheory.primetest import isprime, is_extra_strong_lucas_prp
>>> for i in range(20000):
... if is_extra_strong_lucas_prp(i) and not isprime(i):
... print(i)
989
3239
5777
10877
"""
# Implementation notes:
# 1) the parameters differ from Thomas R. Nicely's. His parameter
# selection leads to pseudoprimes that overlap M-R tests, and
# contradict Baillie and Wagstaff's suggestion of (D|n) = -1.
# 2) The MathWorld page as of June 2013 specifies Q=-1. The Lucas
# sequence must have Q=1. See Grantham theorem 2.3, any of the
# references on the MathWorld page, or run it and see Q=-1 is wrong.
from sympy.ntheory.factor_ import trailing
n = as_int(n)
if n == 2:
return True
if n < 2 or (n % 2) == 0:
return False
if is_square(n, False):
return False
D, P, Q = _lucas_extrastrong_params(n)
if D == 0:
return False
# remove powers of 2 from n+1 (= k * 2**s)
s = trailing(n + 1)
k = (n+1) >> s
U, V, Qk = _lucas_sequence(n, P, Q, k)
if U == 0 and (V == 2 or V == n - 2):
return True
for r in range(1, s):
if V == 0:
return True
V = (V*V - 2) % n
return False
def isprime(n):
"""
Test if n is a prime number (True) or not (False). For n < 2^64 the
answer is definitive; larger n values have a small probability of actually
being pseudoprimes.
Negative numbers (e.g. -2) are not considered prime.
The first step is looking for trivial factors, which if found enables
a quick return. Next, if the sieve is large enough, use bisection search
on the sieve. For small numbers, a set of deterministic Miller-Rabin
tests are performed with bases that are known to have no counterexamples
in their range. Finally if the number is larger than 2^64, a strong
BPSW test is performed. While this is a probable prime test and we
believe counterexamples exist, there are no known counterexamples.
Examples
========
>>> from sympy.ntheory import isprime
>>> isprime(13)
True
>>> isprime(13.0) # limited precision
False
>>> isprime(15)
False
Notes
=====
This routine is intended only for integer input, not numerical
expressions which may represent numbers. Floats are also
rejected as input because they represent numbers of limited
precision. While it is tempting to permit 7.0 to represent an
integer there are errors that may "pass silently" if this is
allowed:
>>> from sympy import Float, S
>>> int(1e3) == 1e3 == 10**3
True
>>> int(1e23) == 1e23
True
>>> int(1e23) == 10**23
False
>>> near_int = 1 + S(1)/10**19
>>> near_int == int(near_int)
False
>>> n = Float(near_int, 10) # truncated by precision
>>> n == int(n)
True
>>> n = Float(near_int, 20)
>>> n == int(n)
False
See Also
========
sympy.ntheory.generate.primerange : Generates all primes in a given range
sympy.ntheory.generate.primepi : Return the number of primes less than or equal to n
sympy.ntheory.generate.prime : Return the nth prime
References
==========
- https://en.wikipedia.org/wiki/Strong_pseudoprime
- "Lucas Pseudoprimes", Baillie and Wagstaff, 1980.
http://mpqs.free.fr/LucasPseudoprimes.pdf
- https://en.wikipedia.org/wiki/Baillie-PSW_primality_test
"""
try:
n = as_int(n)
except ValueError:
return False
# Step 1, do quick composite testing via trial division. The individual
# modulo tests benchmark faster than one or two primorial igcds for me.
# The point here is just to speedily handle small numbers and many
# composites. Step 2 only requires that n <= 2 get handled here.
if n in [2, 3, 5]:
return True
if n < 2 or (n % 2) == 0 or (n % 3) == 0 or (n % 5) == 0:
return False
if n < 49:
return True
if (n % 7) == 0 or (n % 11) == 0 or (n % 13) == 0 or (n % 17) == 0 or \
(n % 19) == 0 or (n % 23) == 0 or (n % 29) == 0 or (n % 31) == 0 or \
(n % 37) == 0 or (n % 41) == 0 or (n % 43) == 0 or (n % 47) == 0:
return False
if n < 2809:
return True
if n < 31417:
return pow(2, n, n) == 2 and n not in [7957, 8321, 13747, 18721, 19951, 23377]
# bisection search on the sieve if the sieve is large enough
from sympy.ntheory.generate import sieve as s
if n <= s._list[-1]:
l, u = s.search(n)
return l == u
# If we have GMPY2, skip straight to step 3 and do a strong BPSW test.
# This should be a bit faster than our step 2, and for large values will
# be a lot faster than our step 3 (C+GMP vs. Python).
if HAS_GMPY == 2:
from gmpy2 import is_strong_prp, is_strong_selfridge_prp
return is_strong_prp(n, 2) and is_strong_selfridge_prp(n)
# Step 2: deterministic Miller-Rabin testing for numbers < 2^64. See:
# https://miller-rabin.appspot.com/
# for lists. We have made sure the M-R routine will successfully handle
# bases larger than n, so we can use the minimal set.
# In September 2015 deterministic numbers were extended to over 2^81.
# https://arxiv.org/pdf/1509.00864.pdf
# https://oeis.org/A014233
if n < 341531:
return mr(n, [9345883071009581737])
if n < 885594169:
return mr(n, [725270293939359937, 3569819667048198375])
if n < 350269456337:
return mr(n, [4230279247111683200, 14694767155120705706, 16641139526367750375])
if n < 55245642489451:
return mr(n, [2, 141889084524735, 1199124725622454117, 11096072698276303650])
if n < 7999252175582851:
return mr(n, [2, 4130806001517, 149795463772692060, 186635894390467037, 3967304179347715805])
if n < 585226005592931977:
return mr(n, [2, 123635709730000, 9233062284813009, 43835965440333360, 761179012939631437, 1263739024124850375])
if n < 18446744073709551616:
return mr(n, [2, 325, 9375, 28178, 450775, 9780504, 1795265022])
if n < 318665857834031151167461:
return mr(n, [2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37])
if n < 3317044064679887385961981:
return mr(n, [2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41])
# We could do this instead at any point:
#if n < 18446744073709551616:
# return mr(n, [2]) and is_extra_strong_lucas_prp(n)
# Here are tests that are safe for MR routines that don't understand
# large bases.
#if n < 9080191:
# return mr(n, [31, 73])
#if n < 19471033:
# return mr(n, [2, 299417])
#if n < 38010307:
# return mr(n, [2, 9332593])
#if n < 316349281:
# return mr(n, [11000544, 31481107])
#if n < 4759123141:
# return mr(n, [2, 7, 61])
#if n < 105936894253:
# return mr(n, [2, 1005905886, 1340600841])
#if n < 31858317218647:
# return mr(n, [2, 642735, 553174392, 3046413974])
#if n < 3071837692357849:
# return mr(n, [2, 75088, 642735, 203659041, 3613982119])
#if n < 18446744073709551616:
# return mr(n, [2, 325, 9375, 28178, 450775, 9780504, 1795265022])
# Step 3: BPSW.
#
# Time for isprime(10**2000 + 4561), no gmpy or gmpy2 installed
# 44.0s old isprime using 46 bases
# 5.3s strong BPSW + one random base
# 4.3s extra strong BPSW + one random base
# 4.1s strong BPSW
# 3.2s extra strong BPSW
# Classic BPSW from page 1401 of the paper. See alternate ideas below.
return mr(n, [2]) and is_strong_lucas_prp(n)
# Using extra strong test, which is somewhat faster
#return mr(n, [2]) and is_extra_strong_lucas_prp(n)
# Add a random M-R base
#import random
#return mr(n, [2, random.randint(3, n-1)]) and is_strong_lucas_prp(n)
def is_gaussian_prime(num):
r"""Test if num is a Gaussian prime number.
References
==========
.. [1] https://oeis.org/wiki/Gaussian_primes
"""
num = sympify(num)
a, b = num.as_real_imag()
a = as_int(a, strict=False)
b = as_int(b, strict=False)
if a == 0:
b = abs(b)
return isprime(b) and b % 4 == 3
elif b == 0:
a = abs(a)
return isprime(a) and a % 4 == 3
return isprime(a**2 + b**2)
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