def test_isprime():
s = Sieve()
s.extend(100000)
assert repr(s) == '<Sieve with 9592 primes sieved: 2, 3, 5, ... 99989, 99991>'
ps = set(s.primerange(2, 100001))
for n in range(100001):
# if (n in ps) != isprime(n): print n
assert (n in ps) == isprime(n)
assert isprime(179424673)
# Some Mersenne primes
assert isprime(2**61 - 1)
assert isprime(2**89 - 1)
assert isprime(2**607 - 1)
assert not isprime(2**601 - 1)
# Arnault's number
assert isprime(int('''
803837457453639491257079614341942108138837688287558145837488917522297\
427376533365218650233616396004545791504202360320876656996676098728404\
396540823292873879185086916685732826776177102938969773947016708230428\
687109997439976544144845341155872450633409279022275296229414984230688\
1685404326457534018329786111298960644845216191652872597534901'''))
# pseudoprime that passes the base set [2, 3, 7, 61, 24251]
assert not isprime(9188353522314541)
assert _mr_safe_helper(
"if n < 170584961: return mr(n, [350, 3958281543])") == \
' # [350, 3958281543] stot = 1 clear [2, 3, 5, 7, 29, 67, 679067]'
assert _mr_safe_helper(
"if n < 3474749660383: return mr(n, [2, 3, 5, 7, 11, 13])") == \
' # [2, 3, 5, 7, 11, 13] stot = 7 clear == bases'
def test_isprime():
s = Sieve()
s.extend(100000)
assert repr(s) == '<Sieve with 9592 primes sieved: 2, 3, 5, ... 99989, 99991>'
ps = set(s.primerange(2, 100001))
for n in range(100001):
# if (n in ps) != isprime(n): print n
assert (n in ps) == isprime(n)
assert isprime(179424673)
# Some Mersenne primes
assert isprime(2**61 - 1)
assert isprime(2**89 - 1)
assert isprime(2**607 - 1)
assert not isprime(2**601 - 1)
# Arnault's number
assert isprime(int('''
803837457453639491257079614341942108138837688287558145837488917522297\
427376533365218650233616396004545791504202360320876656996676098728404\
396540823292873879185086916685732826776177102938969773947016708230428\
687109997439976544144845341155872450633409279022275296229414984230688\
1685404326457534018329786111298960644845216191652872597534901'''))
# pseudoprime that passes the base set [2, 3, 7, 61, 24251]
assert not isprime(9188353522314541)
assert _mr_safe_helper(
"if n < 170584961: return mr(n, [350, 3958281543])") == \
' # [350, 3958281543] stot = 1 clear [2, 3, 5, 7, 29, 67, 679067]'
assert _mr_safe_helper(
"if n < 3474749660383: return mr(n, [2, 3, 5, 7, 11, 13])") == \
' # [2, 3, 5, 7, 11, 13] stot = 7 clear == bases'
def test_randprime():
random.seed(1234)
assert randprime(2, 3) == 2
assert randprime(1, 3) == 2
assert randprime(3, 5) == 3
pytest.raises(ValueError, lambda: randprime(20, 22))
for a in [100, 300, 500, 250000]:
for b in [100, 300, 500, 250000]:
p = randprime(a, a + b)
assert a <= p < (a + b) and isprime(p)
assert randprime(70, 7) is None
def test_randprime():
random.seed(1234)
assert randprime(2, 3) == 2
assert randprime(1, 3) == 2
assert randprime(3, 5) == 3
pytest.raises(ValueError, lambda: randprime(20, 22))
for a in [100, 300, 500, 250000]:
for b in [100, 300, 500, 250000]:
p = randprime(a, a + b)
assert a <= p < (a + b) and isprime(p)
assert randprime(70, 7) is None
def _check_cycles_alt_sym(perm):
"""
Checks for cycles of prime length p with n/2 < p < n-2.
Here `n` is the degree of the permutation. This is a helper function for
the function is_alt_sym from diofant.combinatorics.perm_groups.
Examples
========
>>> from diofant.combinatorics.util import _check_cycles_alt_sym
>>> from diofant.combinatorics.permutations import Permutation
>>> a = Permutation([[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10], [11, 12]])
>>> _check_cycles_alt_sym(a)
False
>>> b = Permutation([[0, 1, 2, 3, 4, 5, 6], [7, 8, 9, 10]])
>>> _check_cycles_alt_sym(b)
True
See Also
========
diofant.combinatorics.perm_groups.PermutationGroup.is_alt_sym
"""
n = perm.size
af = perm.array_form
current_len = 0
total_len = 0
used = set()
for i in range(n // 2):
if i not in used and i < n // 2 - total_len:
current_len = 1
used.add(i)
j = i
while (af[j] != i):
current_len += 1
j = af[j]
used.add(j)
total_len += current_len
if current_len > n // 2 and current_len < n - 2 and isprime(
current_len):
return True
return False
def _inv_totient_estimate(m):
"""
Find ``(L, U)`` such that ``L <= phi^-1(m) <= U``.
Examples
========
>>> _inv_totient_estimate(192)
(192, 840)
>>> _inv_totient_estimate(400)
(400, 1750)
"""
primes = [ d + 1 for d in divisors(m) if isprime(d + 1) ]
a, b = 1, 1
for p in primes:
a *= p
b *= p - 1
L = m
U = int(math.ceil(m*(float(a)/b)))
P = p = 2
primes = []
while P <= U:
p = nextprime(p)
primes.append(p)
P *= p
P //= p
b = 1
for p in primes[:-1]:
b *= p - 1
U = int(math.ceil(m*(float(P)/b)))
return L, U
from diofant.ntheory import isprime
left = right = {2, 3, 5, 7}
range10 = {"", "1", "2", "3", "4", "5", "6", "7", "8", "9"}
for _ in range(5):
left = {
int(f"{i}{j}")
for i in left for j in range10 if isprime(int(f"{i}{j}"))
}
right = {
int(f"{j}{i}")
for i in right for j in range10 if isprime(int(f"{j}{i}"))
}
def main() -> int:
return sum(left & right - {2, 3, 5, 7})
assert main() == 748317
def dup_zz_zassenhaus(f, K):
"""Factor primitive square-free polynomials in `Z[x]`. """
n = dup_degree(f)
if n == 1:
return [f]
fc = f[-1]
A = dup_max_norm(f, K)
b = dup_LC(f, K)
B = int(abs(K.sqrt(K(n + 1)) * 2**n * A * b))
C = int((n + 1)**(2 * n) * A**(2 * n - 1))
gamma = int(_ceil(2 * _log(C, 2)))
bound = int(2 * gamma * _log(gamma))
a = []
# choose a prime number `p` such that `f` be square free in Z_p
# if there are many factors in Z_p, choose among a few different `p`
# the one with fewer factors
for px in range(3, bound + 1):
if not isprime(px) or b % px == 0:
continue
px = K.convert(px)
F = gf_from_int_poly(f, px)
if not gf_sqf_p(F, px, K):
continue
fsqfx = gf_factor_sqf(F, px, K)[1]
a.append((px, fsqfx))
if len(fsqfx) < 15 or len(a) > 4:
break
p, fsqf = min(a, key=lambda x: len(x[1]))
l = int(_ceil(_log(2 * B + 1, p)))
modular = [gf_to_int_poly(ff, p) for ff in fsqf]
g = dup_zz_hensel_lift(p, f, modular, l, K)
sorted_T = range(len(g))
T = set(sorted_T)
factors, s = [], 1
pl = p**l
while 2 * s <= len(T):
for S in subsets(sorted_T, s):
# lift the constant coefficient of the product `G` of the factors
# in the subset `S`; if it is does not divide `fc`, `G` does
# not divide the input polynomial
if b == 1:
q = 1
for i in S:
q = q * g[i][-1]
q = q % pl
if not _test_pl(fc, q, pl):
continue
else:
G = [b]
for i in S:
G = dup_mul(G, g[i], K)
G = dup_trunc(G, pl, K)
G = dup_primitive(G, K)[1]
q = G[-1]
if q and fc % q != 0:
continue
H = [b]
S = set(S)
T_S = T - S
if b == 1:
G = [b]
for i in S:
G = dup_mul(G, g[i], K)
G = dup_trunc(G, pl, K)
for i in T_S:
H = dup_mul(H, g[i], K)
H = dup_trunc(H, pl, K)
G_norm = dup_l1_norm(G, K)
H_norm = dup_l1_norm(H, K)
if G_norm * H_norm <= B:
T = T_S
sorted_T = [i for i in sorted_T if i not in S]
G = dup_primitive(G, K)[1]
f = dup_primitive(H, K)[1]
factors.append(G)
b = dup_LC(f, K)
break
else:
s += 1
return factors + [f]