def hamming_n_count(limit, n):
exclusions = 0
prime = nextprime(n)
while prime <= limit:
print prime, limit // prime
exclusions += limit // prime
prime = nextprime(prime)
print limit, exclusions
return limit - exclusions
def euler134():
ret = 0
p1, p2 = 3, 5
while p2 < 1000000:
p1, p2 = p2, nextprime(p2)
ret += S(p1, p2)
return ret
def test_dmp_zz_wang():
R, x,y,z = ring("x,y,z", ZZ)
UV, _x = ring("x", ZZ)
p = ZZ(nextprime(R.dmp_zz_mignotte_bound(w_1)))
assert p == 6291469
t_1, k_1, e_1 = y, 1, ZZ(-14)
t_2, k_2, e_2 = z, 2, ZZ(3)
t_3, k_3, e_3 = y + z, 2, ZZ(-11)
t_4, k_4, e_4 = y - z, 1, ZZ(-17)
T = [t_1, t_2, t_3, t_4]
K = [k_1, k_2, k_3, k_4]
E = [e_1, e_2, e_3, e_4]
T = zip([ t.drop(x) for t in T ], K)
A = [ZZ(-14), ZZ(3)]
S = R.dmp_eval_tail(w_1, A)
cs, s = UV.dup_primitive(S)
assert cs == 1 and s == S == \
1036728*_x**6 + 915552*_x**5 + 55748*_x**4 + 105621*_x**3 - 17304*_x**2 - 26841*_x - 644
assert R.dmp_zz_wang_non_divisors(E, cs, ZZ(4)) == [7, 3, 11, 17]
assert UV.dup_sqf_p(s) and UV.dup_degree(s) == R.dmp_degree(w_1)
_, H = UV.dup_zz_factor_sqf(s)
h_1 = 44*_x**2 + 42*_x + 1
h_2 = 126*_x**2 - 9*_x + 28
h_3 = 187*_x**2 - 23
assert H == [h_1, h_2, h_3]
LC = [ lc.drop(x) for lc in [-4*y - 4*z, -y*z**2, y**2 - z**2] ]
assert R.dmp_zz_wang_lead_coeffs(w_1, T, cs, E, H, A) == (w_1, H, LC)
H_1 = [44*x**2 + 42*x + 1, 126*x**2 - 9*x + 28, 187*x**2 - 23]
H_2 = [-4*x**2*y - 12*x**2 - 3*x*y + 1, -9*x**2*y - 9*x - 2*y, x**2*y**2 - 9*x**2 + y - 9]
H_3 = [-4*x**2*y - 12*x**2 - 3*x*y + 1, -9*x**2*y - 9*x - 2*y, x**2*y**2 - 9*x**2 + y - 9]
c_1 = -70686*x**5 - 5863*x**4 - 17826*x**3 + 2009*x**2 + 5031*x + 74
c_2 = 9*x**5*y**4 + 12*x**5*y**3 - 45*x**5*y**2 - 108*x**5*y - 324*x**5 + 18*x**4*y**3 - 216*x**4*y**2 - 810*x**4*y + 2*x**3*y**4 + 9*x**3*y**3 - 252*x**3*y**2 - 288*x**3*y - 945*x**3 - 30*x**2*y**2 - 414*x**2*y + 2*x*y**3 - 54*x*y**2 - 3*x*y + 81*x + 12*y
c_3 = -36*x**4*y**2 - 108*x**4*y - 27*x**3*y**2 - 36*x**3*y - 108*x**3 - 8*x**2*y**2 - 42*x**2*y - 6*x*y**2 + 9*x + 2*y
# TODO
#assert R.dmp_zz_diophantine(H_1, c_1, [], 5, p) == [-3*x, -2, 1]
#assert R.dmp_zz_diophantine(H_2, c_2, [ZZ(-14)], 5, p) == [-x*y, -3*x, -6]
#assert R.dmp_zz_diophantine(H_3, c_3, [ZZ(-14)], 5, p) == [0, 0, -1]
factors = R.dmp_zz_wang_hensel_lifting(w_1, H, LC, A, p)
assert R.dmp_expand(factors) == w_1
def doProblem(): count = 0 sum = 0 i=10 while count < 11: i = nextprime(i) if TruncatableTest(i): # print(i) sum += i count += 1 return sum
def oeis_a037992(maxsize):
cnt = 0
todo = PriorityQueueSet([(2,2,0)])
done = set([2])
while maxsize != cnt:
out, p, k = todo.pop_smallest()
yield out
cnt += 1
np = sympy.nextprime(p)
if np not in done:
todo.add((np,np,0))
done.add(np)
todo.add((p**(2**(k+1)), p, k+1))
def dh_private_key(digit = 10):
"""
Return two number tuple as private key.
Diffie-Hellman key exchange is based on the mathematical problem
called the Discrete Logarithm Problem (see ElGamal).
Diffie-Hellman key exchange is divided into the following steps:
* Alice and Bob agree on a base that consist of a prime p and a
primitive root of p called g
* Alice choses a number a and Bob choses a number b where a
and b are random numbers with 1 < a, b < p. These are their
private keys.
* Alice then publicly sends Bob `g^{a} \pmod p` while Bob sends
Alice `g^{b} \pmod p`
* They both raise the received value to their secretly chose number
(a or b) and now have both as their shared key `g^{ab} \pmod p`
Parameters
==========
digit: Key length in binary
Returns
=======
(p, g, a) : p = prime number, g = primitive root of p,
a = random number in between 2 and p - 1
Examples
========
>>> from sympy.crypto.crypto import dh_private_key
>>> from sympy.ntheory import isprime, is_primitive_root
>>> p, g, _ = dh_private_key()
>>> isprime(p)
True
>>> is_primitive_root(g, p)
True
>>> p, g, _ = dh_private_key(5)
>>> isprime(p)
True
>>> is_primitive_root(g, p)
True
"""
p = nextprime(2 ** digit)
g = primitive_root(p)
a = randrange(2, p)
return p, g, a
def p134(limit):
t=time.clock()
ps=list(primesieve(limit)[2:])
ps.append(sp.nextprime(limit))
S=0
for i in range(len(ps)-1) :
b=ps[i+1]
c=ps[i]
a=-10**(int(math.log10(c))+1)
x1,y1=primeLD(a,b,c)
x=x1%b #b is prime, and a is a multiple of 100, so gcd(a,b)=1
S+=-a*x+c
print(S,time.clock()-t)
def truncatables_primes():
count =0
s = 0
last_prime = 7
while count<11:
prime = nextprime(last_prime)
prime_str = str(prime)
is_cool = True
for i in range(1,len(prime_str)):
p1 = int(prime_str[i:])
p2 = int(prime_str[:-i])
if not (isprime(p1) and isprime(p2)):
is_cool = False
if is_cool:
count = count +1
s = s+prime
last_prime = prime
return s
def primes_gen(*args):
"""
Prime number generator
Possible usages:
primes_gen(): infinite generator
primes_gen(a): generate all primes below a
primes_gen(a, b): generate all primes below a and b
"""
if len(args) == 1:
for x in primerange(2, args[0]):
yield x
elif len(args) == 2:
for x in primerange(*args):
yield x
elif len(args) == 0:
x = 1
while 1:
x = nextprime(x)
yield x
def test_gf_ddf():
f = gf_from_dict({15: 1, 0: -1}, 11, ZZ)
g = [([1, 0, 0, 0, 0, 10], 1),
([1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1], 2)]
assert gf_ddf_zassenhaus(f, 11, ZZ) == g
assert gf_ddf_shoup(f, 11, ZZ) == g
f = gf_from_dict({63: 1, 0: 1}, 2, ZZ)
g = [([1, 1], 1),
([1, 1, 1], 2),
([1, 1, 1, 1, 1, 1, 1], 3),
([1, 1, 0, 1, 1, 0, 1, 0, 1, 1, 0, 1, 1, 0, 0, 0, 0, 0, 0,
0, 0, 1, 1, 0, 1, 1, 0, 1, 0, 1, 1, 0, 1, 1, 0, 0, 0, 0,
0, 0, 0, 0, 1, 1, 0, 1, 1, 0, 1, 0, 1, 1, 0, 1, 1], 6)]
assert gf_ddf_zassenhaus(f, 2, ZZ) == g
assert gf_ddf_shoup(f, 2, ZZ) == g
f = gf_from_dict({6: 1, 5: -1, 4: 1, 3: 1, 1: -1}, 3, ZZ)
g = [([1, 1, 0], 1),
([1, 1, 0, 1, 2], 2)]
assert gf_ddf_zassenhaus(f, 3, ZZ) == g
assert gf_ddf_shoup(f, 3, ZZ) == g
f = [1, 2, 5, 26, 677, 436, 791, 325, 456, 24, 577]
g = [([1, 701], 1),
([1, 110, 559, 532, 694, 151, 110, 70, 735, 122], 9)]
assert gf_ddf_zassenhaus(f, 809, ZZ) == g
assert gf_ddf_shoup(f, 809, ZZ) == g
p = ZZ(nextprime(int((2**15 * pi).evalf())))
f = gf_from_dict({15: 1, 1: 1, 0: 1}, p, ZZ)
g = [([1, 22730, 68144], 2),
([1, 64876, 83977, 10787, 12561, 68608, 52650, 88001, 84356], 4),
([1, 15347, 95022, 84569, 94508, 92335], 5)]
assert gf_ddf_zassenhaus(f, p, ZZ) == g
assert gf_ddf_shoup(f, p, ZZ) == g
def elgamal_private_key(digit=10):
"""
Return three number tuple as private key.
Elgamal encryption is based on the mathmatical problem
called the Discrete Logarithm Problem (DLP). For example,
`a^{b} \equiv c \pmod p`
In general, if a and b are known, c is easily
calculated. If b is unknown, it is hard to use
a and c to get b.
Parameters
==========
digit : Key length in binary
Returns
=======
(p, r, d) : p = prime number, r = primitive root, d = random number
Examples
========
>>> from sympy.crypto.crypto import elgamal_private_key
>>> from sympy.ntheory import is_primitive_root, isprime
>>> a, b, _ = elgamal_private_key()
>>> isprime(a)
True
>>> is_primitive_root(b, a)
True
"""
p = nextprime(2**digit)
return p, primitive_root(p), randrange(2, p)
def test_dmp_zz_wang():
p = ZZ(nextprime(dmp_zz_mignotte_bound(w_1, 2, ZZ)))
assert p == ZZ(6291469)
t_1, k_1, e_1 = dmp_normal([[1],[]], 1, ZZ), 1, ZZ(-14)
t_2, k_2, e_2 = dmp_normal([[1, 0]], 1, ZZ), 2, ZZ(3)
t_3, k_3, e_3 = dmp_normal([[1],[ 1, 0]], 1, ZZ), 2, ZZ(-11)
t_4, k_4, e_4 = dmp_normal([[1],[-1, 0]], 1, ZZ), 1, ZZ(-17)
T = [t_1, t_2, t_3, t_4]
K = [k_1, k_2, k_3, k_4]
E = [e_1, e_2, e_3, e_4]
T = zip(T, K)
A = [ZZ(-14), ZZ(3)]
S = dmp_eval_tail(w_1, A, 2, ZZ)
cs, s = dup_primitive(S, ZZ)
assert cs == 1 and s == S == \
dup_normal([1036728, 915552, 55748, 105621, -17304, -26841, -644], ZZ)
assert dmp_zz_wang_non_divisors(E, cs, 4, ZZ) == [7, 3, 11, 17]
assert dup_sqf_p(s, ZZ) and dup_degree(s) == dmp_degree(w_1, 2)
_, H = dup_zz_factor_sqf(s, ZZ)
h_1 = dup_normal([44, 42, 1], ZZ)
h_2 = dup_normal([126, -9, 28], ZZ)
h_3 = dup_normal([187, 0, -23], ZZ)
assert H == [h_1, h_2, h_3]
lc_1 = dmp_normal([[-4], [-4,0]], 1, ZZ)
lc_2 = dmp_normal([[-1,0,0], []], 1, ZZ)
lc_3 = dmp_normal([[1], [], [-1,0,0]], 1, ZZ)
LC = [lc_1, lc_2, lc_3]
assert dmp_zz_wang_lead_coeffs(w_1, T, cs, E, H, A, 2, ZZ) == (w_1, H, LC)
H_1 = [ dmp_normal(t, 0, ZZ) for t in [[44L,42L,1L],[126L,-9L,28L],[187L,0L,-23L]] ]
H_2 = [ dmp_normal(t, 1, ZZ) for t in [[[-4,-12],[-3,0],[1]],[[-9,0],[-9],[-2,0]],[[1,0,-9],[],[1,-9]]] ]
H_3 = [ dmp_normal(t, 1, ZZ) for t in [[[-4,-12],[-3,0],[1]],[[-9,0],[-9],[-2,0]],[[1,0,-9],[],[1,-9]]] ]
c_1 = dmp_normal([-70686,-5863,-17826,2009,5031,74], 0, ZZ)
c_2 = dmp_normal([[9,12,-45,-108,-324],[18,-216,-810,0],[2,9,-252,-288,-945],[-30,-414,0],[2,-54,-3,81],[12,0]], 1, ZZ)
c_3 = dmp_normal([[-36,-108,0],[-27,-36,-108],[-8,-42,0],[-6,0,9],[2,0]], 1, ZZ)
T_1 = [ dmp_normal(t, 0, ZZ) for t in [[-3,0],[-2],[1]] ]
T_2 = [ dmp_normal(t, 1, ZZ) for t in [[[-1,0],[]],[[-3],[]],[[-6]]] ]
T_3 = [ dmp_normal(t, 1, ZZ) for t in [[[]],[[]],[[-1]]] ]
assert dmp_zz_diophantine(H_1, c_1, [], 5, p, 0, ZZ) == T_1
assert dmp_zz_diophantine(H_2, c_2, [ZZ(-14)], 5, p, 1, ZZ) == T_2
assert dmp_zz_diophantine(H_3, c_3, [ZZ(-14)], 5, p, 1, ZZ) == T_3
factors = dmp_zz_wang_hensel_lifting(w_1, H, LC, A, p, 2, ZZ)
assert dmp_expand(factors, 2, ZZ) == w_1
def test_powers_Integer():
"""Test Integer._eval_power"""
# check infinity
assert S(1)**S.Infinity == S.NaN
assert S(-1)**S.Infinity == S.NaN
assert S(2)**S.Infinity == S.Infinity
assert S(-2)**S.Infinity == S.Infinity + S.Infinity * S.ImaginaryUnit
assert S(0)**S.Infinity == 0
# check Nan
assert S(1)**S.NaN == S.NaN
assert S(-1)**S.NaN == S.NaN
# check for exact roots
assert S(-1)**Rational(6, 5) == -(-1)**(S(1) / 5)
assert sqrt(S(4)) == 2
assert sqrt(S(-4)) == I * 2
assert S(16)**Rational(1, 4) == 2
assert S(-16)**Rational(1, 4) == 2 * (-1)**Rational(1, 4)
assert S(9)**Rational(3, 2) == 27
assert S(-9)**Rational(3, 2) == -27 * I
assert S(27)**Rational(2, 3) == 9
assert S(-27)**Rational(2, 3) == 9 * (S(-1)**Rational(2, 3))
assert (-2)**Rational(-2, 1) == Rational(1, 4)
# not exact roots
assert sqrt(-3) == I * sqrt(3)
assert (3)**(S(3) / 2) == 3 * sqrt(3)
assert (-3)**(S(3) / 2) == -3 * sqrt(-3)
assert (-3)**(S(5) / 2) == 9 * I * sqrt(3)
assert (-3)**(S(7) / 2) == -I * 27 * sqrt(3)
assert (2)**(S(3) / 2) == 2 * sqrt(2)
assert (2)**(S(-3) / 2) == sqrt(2) / 4
assert (81)**(S(2) / 3) == 9 * (S(3)**(S(2) / 3))
assert (-81)**(S(2) / 3) == 9 * (S(-3)**(S(2) / 3))
assert (-3) ** Rational(-7, 3) == \
-(-1)**Rational(2, 3)*3**Rational(2, 3)/27
assert (-3) ** Rational(-2, 3) == \
-(-1)**Rational(1, 3)*3**Rational(1, 3)/3
# join roots
assert sqrt(6) + sqrt(24) == 3 * sqrt(6)
assert sqrt(2) * sqrt(3) == sqrt(6)
# separate symbols & constansts
x = Symbol("x")
assert sqrt(49 * x) == 7 * sqrt(x)
assert sqrt((3 - sqrt(pi))**2) == 3 - sqrt(pi)
# check that it is fast for big numbers
assert (2**64 + 1)**Rational(4, 3)
assert (2**64 + 1)**Rational(17, 25)
# negative rational power and negative base
assert (-3) ** Rational(-7, 3) == \
-(-1)**Rational(2, 3)*3**Rational(2, 3)/27
assert (-3) ** Rational(-2, 3) == \
-(-1)**Rational(1, 3)*3**Rational(1, 3)/3
assert S(1234).factors() == {617: 1, 2: 1}
assert Rational(2 * 3, 3 * 5 * 7).factors() == {2: 1, 5: -1, 7: -1}
# test that eval_power factors numbers bigger than
# the current limit in factor_trial_division (2**15)
from sympy import nextprime
n = nextprime(2**15)
assert sqrt(n**2) == n
assert sqrt(n**3) == n * sqrt(n)
assert sqrt(4 * n) == 2 * sqrt(n)
# check that factors of base with powers sharing gcd with power are removed
assert (2**4 * 3)**Rational(1, 6) == 2**Rational(2, 3) * 3**Rational(1, 6)
assert (2**4 * 3)**Rational(
5, 6) == 8 * 2**Rational(1, 3) * 3**Rational(5, 6)
# check that bases sharing a gcd are exptracted
assert 2**Rational(1, 3)*3**Rational(1, 4)*6**Rational(1, 5) == \
2**Rational(8, 15)*3**Rational(9, 20)
assert sqrt(8)*24**Rational(1, 3)*6**Rational(1, 5) == \
4*2**Rational(7, 10)*3**Rational(8, 15)
assert sqrt(8)*(-24)**Rational(1, 3)*(-6)**Rational(1, 5) == \
4*(-3)**Rational(8, 15)*2**Rational(7, 10)
assert 2**Rational(1, 3) * 2**Rational(8, 9) == 2 * 2**Rational(2, 9)
assert 2**Rational(2, 3) * 6**Rational(1, 3) == 2 * 3**Rational(1, 3)
assert 2**Rational(2, 3)*6**Rational(8, 9) == \
2*2**Rational(5, 9)*3**Rational(8, 9)
assert (-2)**Rational(2, S(3)) * (-4)**Rational(
1, S(3)) == -2 * 2**Rational(1, 3)
assert 3 * Pow(3, 2, evaluate=False) == 3**3
assert 3 * Pow(3, -1 / S(3), evaluate=False) == 3**(2 / S(3))
assert (-2)**(1/S(3))*(-3)**(1/S(4))*(-5)**(5/S(6)) == \
-(-1)**Rational(5, 12)*2**Rational(1, 3)*3**Rational(1, 4) * \
5**Rational(5, 6)
assert Integer(-2)**Symbol('', even=True) == \
Integer(2)**Symbol('', even=True)
assert (-1)**Float(.5) == 1.0 * I
import sympy
import time
start = time.time()
primes = []
current_prime = 11
while len(primes) < 11:
i = str(current_prime)
if all(
sympy.isprime(int(i[:num + 1])) and sympy.isprime(int(i[num:]))
for num in range(len(str(i)))):
primes.append(int(i))
current_prime = sympy.nextprime(current_prime)
print("Prime's list:", primes)
print("Sum:", sum(primes))
print("Elapsed Time:", time.time() - start)
# 10 sec solution
'''
import sympy
import time
start = time.time()
# generate primes in range from 8 to 1 000 000
def get_primes():
x = list(sympy.primerange(8, 1000000))
return x
def next_prime(num): # next prime after num (skip 2) return 3 if num < 3 else sympy.nextprime(num)
def test_dmp_zz_wang():
p = ZZ(nextprime(dmp_zz_mignotte_bound(w_1, 2, ZZ)))
assert p == ZZ(6291469)
t_1, k_1, e_1 = dmp_normal([[1], []], 1, ZZ), 1, ZZ(-14)
t_2, k_2, e_2 = dmp_normal([[1, 0]], 1, ZZ), 2, ZZ(3)
t_3, k_3, e_3 = dmp_normal([[1], [1, 0]], 1, ZZ), 2, ZZ(-11)
t_4, k_4, e_4 = dmp_normal([[1], [-1, 0]], 1, ZZ), 1, ZZ(-17)
T = [t_1, t_2, t_3, t_4]
K = [k_1, k_2, k_3, k_4]
E = [e_1, e_2, e_3, e_4]
T = zip(T, K)
A = [ZZ(-14), ZZ(3)]
S = dmp_eval_tail(w_1, A, 2, ZZ)
cs, s = dup_primitive(S, ZZ)
assert cs == 1 and s == S == \
dup_normal([1036728, 915552, 55748, 105621, -17304, -26841, -644], ZZ)
assert dmp_zz_wang_non_divisors(E, cs, 4, ZZ) == [7, 3, 11, 17]
assert dup_sqf_p(s, ZZ) and dup_degree(s) == dmp_degree(w_1, 2)
_, H = dup_zz_factor_sqf(s, ZZ)
h_1 = dup_normal([44, 42, 1], ZZ)
h_2 = dup_normal([126, -9, 28], ZZ)
h_3 = dup_normal([187, 0, -23], ZZ)
assert H == [h_1, h_2, h_3]
lc_1 = dmp_normal([[-4], [-4, 0]], 1, ZZ)
lc_2 = dmp_normal([[-1, 0, 0], []], 1, ZZ)
lc_3 = dmp_normal([[1], [], [-1, 0, 0]], 1, ZZ)
LC = [lc_1, lc_2, lc_3]
assert dmp_zz_wang_lead_coeffs(w_1, T, cs, E, H, A, 2, ZZ) == (w_1, H, LC)
H_1 = [
dmp_normal(t, 0, ZZ)
for t in [[44L, 42L, 1L], [126L, -9L, 28L], [187L, 0L, -23L]]
]
H_2 = [
dmp_normal(t, 1, ZZ)
for t in [[[-4, -12], [-3, 0], [1]], [[-9, 0], [-9], [-2, 0]],
[[1, 0, -9], [], [1, -9]]]
]
H_3 = [
dmp_normal(t, 1, ZZ)
for t in [[[-4, -12], [-3, 0], [1]], [[-9, 0], [-9], [-2, 0]],
[[1, 0, -9], [], [1, -9]]]
]
c_1 = dmp_normal([-70686, -5863, -17826, 2009, 5031, 74], 0, ZZ)
c_2 = dmp_normal(
[[9, 12, -45, -108, -324], [18, -216, -810, 0],
[2, 9, -252, -288, -945], [-30, -414, 0], [2, -54, -3, 81], [12, 0]],
1, ZZ)
c_3 = dmp_normal(
[[-36, -108, 0], [-27, -36, -108], [-8, -42, 0], [-6, 0, 9], [2, 0]],
1, ZZ)
T_1 = [dmp_normal(t, 0, ZZ) for t in [[-3, 0], [-2], [1]]]
T_2 = [dmp_normal(t, 1, ZZ) for t in [[[-1, 0], []], [[-3], []], [[-6]]]]
T_3 = [dmp_normal(t, 1, ZZ) for t in [[[]], [[]], [[-1]]]]
assert dmp_zz_diophantine(H_1, c_1, [], 5, p, 0, ZZ) == T_1
assert dmp_zz_diophantine(H_2, c_2, [ZZ(-14)], 5, p, 1, ZZ) == T_2
assert dmp_zz_diophantine(H_3, c_3, [ZZ(-14)], 5, p, 1, ZZ) == T_3
factors = dmp_zz_wang_hensel_lifting(w_1, H, LC, A, p, 2, ZZ)
assert dmp_expand(factors, 2, ZZ) == w_1
def test_dmp_zz_wang():
R, x, y, z = ring("x,y,z", ZZ)
UV, _x = ring("x", ZZ)
p = ZZ(nextprime(R.dmp_zz_mignotte_bound(w_1)))
assert p == 6291469
t_1, k_1, e_1 = y, 1, ZZ(-14)
t_2, k_2, e_2 = z, 2, ZZ(3)
t_3, k_3, e_3 = y + z, 2, ZZ(-11)
t_4, k_4, e_4 = y - z, 1, ZZ(-17)
T = [t_1, t_2, t_3, t_4]
K = [k_1, k_2, k_3, k_4]
E = [e_1, e_2, e_3, e_4]
T = zip([t.drop(x) for t in T], K)
A = [ZZ(-14), ZZ(3)]
S = R.dmp_eval_tail(w_1, A)
cs, s = UV.dup_primitive(S)
assert cs == 1 and s == S == \
1036728*_x**6 + 915552*_x**5 + 55748*_x**4 + 105621*_x**3 - 17304*_x**2 - 26841*_x - 644
assert R.dmp_zz_wang_non_divisors(E, cs, ZZ(4)) == [7, 3, 11, 17]
assert UV.dup_sqf_p(s) and UV.dup_degree(s) == R.dmp_degree(w_1)
_, H = UV.dup_zz_factor_sqf(s)
h_1 = 44 * _x**2 + 42 * _x + 1
h_2 = 126 * _x**2 - 9 * _x + 28
h_3 = 187 * _x**2 - 23
assert H == [h_1, h_2, h_3]
LC = [lc.drop(x) for lc in [-4 * y - 4 * z, -y * z**2, y**2 - z**2]]
assert R.dmp_zz_wang_lead_coeffs(w_1, T, cs, E, H, A) == (w_1, H, LC)
H_1 = [44 * x**2 + 42 * x + 1, 126 * x**2 - 9 * x + 28, 187 * x**2 - 23]
H_2 = [
-4 * x**2 * y - 12 * x**2 - 3 * x * y + 1,
-9 * x**2 * y - 9 * x - 2 * y, x**2 * y**2 - 9 * x**2 + y - 9
]
H_3 = [
-4 * x**2 * y - 12 * x**2 - 3 * x * y + 1,
-9 * x**2 * y - 9 * x - 2 * y, x**2 * y**2 - 9 * x**2 + y - 9
]
c_1 = -70686 * x**5 - 5863 * x**4 - 17826 * x**3 + 2009 * x**2 + 5031 * x + 74
c_2 = 9 * x**5 * y**4 + 12 * x**5 * y**3 - 45 * x**5 * y**2 - 108 * x**5 * y - 324 * x**5 + 18 * x**4 * y**3 - 216 * x**4 * y**2 - 810 * x**4 * y + 2 * x**3 * y**4 + 9 * x**3 * y**3 - 252 * x**3 * y**2 - 288 * x**3 * y - 945 * x**3 - 30 * x**2 * y**2 - 414 * x**2 * y + 2 * x * y**3 - 54 * x * y**2 - 3 * x * y + 81 * x + 12 * y
c_3 = -36 * x**4 * y**2 - 108 * x**4 * y - 27 * x**3 * y**2 - 36 * x**3 * y - 108 * x**3 - 8 * x**2 * y**2 - 42 * x**2 * y - 6 * x * y**2 + 9 * x + 2 * y
# TODO
#assert R.dmp_zz_diophantine(H_1, c_1, [], 5, p) == [-3*x, -2, 1]
#assert R.dmp_zz_diophantine(H_2, c_2, [ZZ(-14)], 5, p) == [-x*y, -3*x, -6]
#assert R.dmp_zz_diophantine(H_3, c_3, [ZZ(-14)], 5, p) == [0, 0, -1]
factors = R.dmp_zz_wang_hensel_lifting(w_1, H, LC, A, p)
assert R.dmp_expand(factors) == w_1
# T = 1
# MN = int(1e5)
# MA = int(1e9)
# L = 1
# R = MN
# # R = MN*(MN-1) + 1
# # L = R - (MN - 1)
#
# print(T)
# for test in range(T):
# print(MN, L, R)
# print(' '.join([str(MA - i) for i in range(MN)]))
from sympy import nextprime
D = 1
cur = 1
while D <= int(1e15):
cur = nextprime(cur)
D *= cur
print(D // cur)
def next_ends_by_1(p):
np = nextprime(p)
return (np % 10) == 1
def generate_random_M():
p = next_usable_prime(x)
q = next_usable_prime(y)
M = p * q
assert (1 < seed < M)
assert (seed % p != 0)
assert (seed % q != 0)
return M
def blum_blum_shub(xn, M):
return (xn**2) % M
if __name__ == "__main__":
rn = []
M = generate_random_M()
n = int(input("How many numbers do you want? "))
x = sympy.nextprime(random.randint(0, (int(time()))))
for _ in range(n):
x = blum_blum_shub(x, M)
rn.append(x)
max = max(rn)
print("integers:", rn)
norm = [a / max for a in rn]
print("normalised:", norm)
temp=mat[int(val[1])][int(val[0])]
#char = chr(int(temp))
print(temp,end ="")
def decryption(y):
s=find_s(y,w,m)
r=superIncressingAlgo(a_sort,s)
x=permutationPi(pi,r)
x=map(str,x)
res=str(int(''.join(x),2))
if(len(res)<14):
l=14-len(res)
str1 = "0"*l
res=str1+res
return res
if __name__ == "__main__":
m=nextprime(2036764117802210446778721319780021001)
w = nextprime(127552671440279916013001)
#m=2036764117802210446778721319780021357
#w=127552671440279916013021
y=readFile("y")
b=readFile("knapsack")
a=find_a(b,m,w)
a_sort = sorted(a)
pi=find_pi(a,a_sort)
res=[]
for val in y:
res.append(decryption(val))
for r in res:
intTochar(r)
def __init__(self, shared_secret_value):
public_key1 = sympy.nextprime(shared_secret_value)
public_key2 = sympy.nextprime(public_key1 // 2)
super.__init__(public_key1, public_key2)
self.flag_generated_key = False
import sympy
print('5 is a Prime Number',sympy.isprime(5))
print(list(sympy.primerange(0,100))
print(sympy.randprime(0,100)) #Output 83
print(sympy.randprime(0,100)) #Output 41
print(sympy.prime(3)) #Output 5
print(sympy.prevprime(50)) #Output 47
print(sympy.nextprime(50)) #Output 53
from sympy import nextprime
from math import gcd
import random
import math
#問題(1)1.公開鍵(n)を求める
#任意の相異なる素数p,qを問題文の条件のもと、選ぶ
a = 42 % 30 + 10 #問題(1)1.(a)より
b = 13 + 10 #問題(1)1.(b)より
p = nextprime(a) #p>aを満たす最小の素数
q = nextprime(b)
if q == p: #もしqがpと同値であれば次の素数を選ぶ
q = nextprime(q)
print("[pとqを出力して確認]")
print("p=", p, ",", "q=", q) #確認：pとqの値
n = p * q #nを求める
print("n=q*q =", n) #確認:n
#最小公倍数lcm(p-1,q-1)を求めて、Lと互いに素でLより小さな任意の整数eを選ぶ
def lcm(p, q): #最小公倍数を求める関数(Lの計算過程)
return (p * q) // gcd(p, q) #lcm(a,b) = ab / gcd(a,b)より
#問題(1)2.公開鍵(暗号鍵):eを生成する
L = lcm(p - 1, q - 1) #最小公倍数を求める
for i in range(2, L): #1<e<Lとなるiを生成
if numero%xifrat != 0: return False candidat = numero//xifrat return isprime(candidat) def right_harshad(numero): while numero > 0: if harshad(numero) == False: return False break numero = numero//10 else: return True fita = 10**14 primer = 11 suma = 0 while primer < fita: candidat = primer//10 if right_harshad(candidat) == False: pass elif strong_harshad(candidat) == False: pass else: suma += primer primer = nextprime(primer) print(suma)
# https://www.johndcook.com/blog/2018/12/19/rsa-with-one-known-prime/ # need Python 3 for secrets module # py -3 rsa-factoring.py from secrets import randbits from sympy import nextprime, gcd from timeit import default_timer as timer numbits = 2048 p = nextprime(randbits(numbits)) q = nextprime(randbits(numbits)) r = nextprime(randbits(numbits)) m = p*q n = p*r t0 = timer() g = gcd(m, n) assert(p == g) assert(q == m//p) assert(r == n//p) t1 = timer() # Print time in milliseconds print(1000*(t1 - t0)) print(p) print(q) print(r)
def getPrime(x):
p = sympy.nextprime(x)
while p % 4 != 3:
p = sympy.nextprime(p)
return p
import sympy as sp
n = 1
p = 2
while True:
r = 2 * n * p % p**2
if r > 10**10:
break
n += 2
p = sp.nextprime(sp.nextprime(p))
print(n)
def prime_gen():
n = 1
while True:
yield (n := nextprime(n))
import sympy as s
t=int(input())
for k in range(t):
z=int(input())
temp=1
p1=2
p2=3
run=True
while run:
if p1*p2<=z:
p1=s.nextprime(p1)
p2=s.nextprime(p2)
temp=s.nextprime(temp)
else:
run=False
print('Case #'+str(k+1)+': '+str(temp*p1))
for i in range(100):
for j in range(100):
try:
if f(i,j)==A:
print i,j
except Exception:
continue
x=2
y=83
#p=next_prime(z*x*y)
#q=next_prime(z)
n=117930806043507374325982291823027285148807239117987369609583515353889814856088099671454394340816761242974462268435911765045576377767711593100416932019831889059333166946263184861287975722954992219766493089630810876984781113645362450398009234556085330943125568377741065242183073882558834603430862598066786475299918395341014877416901185392905676043795425126968745185649565106322336954427505104906770493155723995382318346714944184577894150229037758434597242564815299174950147754426950251419204917376517360505024549691723683358170823416757973059354784142601436519500811159036795034676360028928301979780528294114933347127
z=842868045681390934539739959201847552284980179958879667933078453950968566151662147267006293571765463137270594151138695778986165111380428806545593588078365331313084230014618714412959584843421586674162688321942889369912392031882620994944241987153078156389470370195514285850736541078623854327959382156753457477
start=z#gmpy2.iroot(n/x/y,2)[0]
while True:
p=sympy.nextprime(start*x*y)
q=sympy.nextprime(start)
n_=p*q
#print n_
if n_==n:
print start
z=start
break
print n_-n
start -= 1
#z=842868045681390934539739959201847552284980179958879667933078453950968566151662147267006293571765463137270594151138695778986165111380428806545593588078365331313084230014618714412959584843421586674162688321942889369912392031882620994944241987153078156389470370195514285850736541078623854327959382156753457477
p=sympy.nextprime(z*x*y)
q=sympy.nextprime(z)
assert(p*q==n)
c=21637349983376240298781643137017527643006420437956416224341194119376827720906813134837051662331110800533726249916480511461651883091547167072238889133223663294552334442051093640989347891754536977519831106118699250847317797879843851255826918152121510988443260626502258826108177049550321258420523982587818065892584009185540294356508050406622208242619887465832495988412605479326659942708790414892644525081859890294893951328910205466191795443278975666713957632984396458370561861108820821299355143041504204271934655444506866309772339580031558766661592776211084022129268334078625159825028582506422947933620995804155386524
e=0x10001
a1 = 21856963452461630437348278434191434000066076750419027493852463513469865262064340836613831066602300959772632397773487317560339056658299954464169264467234407
b1 = 21856963452461630437348278434191434000066076750419027493852463513469865262064340836613831066602300959772632397773487317560339056658299954464169264467140596
a2 = 16466113115839228119767887899308820025749260933863446888224167169857612178664139545726340867406790754560227516013796269941438076818194617030304851858418927
b2 = 16466113115839228119767887899308820025749260933863446888224167169857612178664139545726340867406790754560227516013796269941438076818194617030304851858351026
n = 85492663786275292159831603391083876175149354309327673008716627650718160585639723100793347534649628330416631255660901307533909900431413447524262332232659153047067908693481947121069070451562822417357656432171870951184673132554213690123308042697361969986360375060954702920656364144154145812838558365334172935931441424096270206140691814662318562696925767991937369782627908408239087358033165410020690152067715711112732252038588432896758405898709010342467882264362733
c = 75700883021669577739329316795450706204502635802310731477156998834710820770245219468703245302009998932067080383977560299708060476222089630209972629755965140317526034680452483360917378812244365884527186056341888615564335560765053550155758362271622330017433403027261127561225585912484777829588501213961110690451987625502701331485141639684356427316905122995759825241133872734362716041819819948645662803292418802204430874521342108413623635150475963121220095236776428
p = 1
q = 1
i = 1
l = 0
for i in range(b1 + 1, a1 - 1):
p *= modinv(i, a1)
p %= a1
p = sympy.nextprime(p)
print "p="
print p
for i in range(b2 + 1, a2 - 1):
q *= modinv(i, a2)
q %= a2
q = sympy.nextprime(q)
print "q="
print q
r = n / q / p
print "r="
print r
fn = (p - 1) * (q - 1) * (r - 1)
print "fn="
print fn
e = 4097
def generate_tree_reversal_terms(residue, initial, base, modulus):
p = lambda n: backwards_base(nextprime(n), base)
c = lambda n: backwards_base(nextcompo(n), base)
return generate_tree_terms(residue, initial, base, p, c, modulus)
from sympy import primerange
from sympy import nextprime
# We're going to understand this algorithm by the exemple (151,157): we compute the first integer for which n*157[0] == 151[0] == 1
#which is 157*3 = 471, now we have found the first last integer, it is now unnecessary to try adding n*157, we can go for n*(1570) because the first last integer is already
#found, let's find the first integer for which n*1570 + 471 [1] == 151 [1] == 5, which is found for n==4 and the number is 6751, doing the same thing for the last integer 7
#we find 38151 :3
def s(p1,p2):
cmp = p2
s_p1 = str(p1)
l = len(s_p1)
inx = 1
while inx <= l:
if str(cmp)[len(str(cmp))-inx] == s_p1[l - inx]:
inx +=1
p2 = p2*10
else:
cmp += p2
return cmp
pr = [i for i in primerange(5,10**6 +1) ]
pr.append(nextprime(pr[len(pr)-1])) #the problem states p1<= 10**6 ;)
print sum([s(pr[i],pr[i+1]) for i in xrange(0,len(pr)-1)])
def base_b_terms(b):
p = lambda n: backwards_base(nextprime(n), b)
c = lambda n: backwards_base(nextcompo(n), b)
evens, odds = generate_terms(2, p, c)
return evens, odds
def genprime(n, K):
return K(nextprime(int((2**n * pi).evalf())))
def test_powers_Integer():
"""Test Integer._eval_power"""
# check infinity
assert S(1) ** S.Infinity == 1
assert S(-1)** S.Infinity == S.NaN
assert S(2) ** S.Infinity == S.Infinity
assert S(-2)** S.Infinity == S.Infinity + S.Infinity * S.ImaginaryUnit
assert S(0) ** S.Infinity == 0
# check Nan
assert S(1) ** S.NaN == S.NaN
assert S(-1) ** S.NaN == S.NaN
# check for exact roots
assert S(-1) ** Rational(6, 5) == - (-1)**(S(1)/5)
assert S(4) ** Rational(1, 2) == 2
assert S(-4) ** Rational(1, 2) == I * 2
assert S(16) ** Rational(1, 4) == 2
assert S(-16) ** Rational(1, 4) == 2 * (-1)**Rational(1,4)
assert S(9) ** Rational(3, 2) == 27
assert S(-9) ** Rational(3, 2) == -27*I
assert S(27) ** Rational(2, 3) == 9
assert S(-27) ** Rational(2, 3) == 9 * (S(-1) ** Rational(2, 3))
assert (-2) ** Rational(-2, 1) == Rational(1, 4)
# not exact roots
assert (-3) ** (S(1)/2) == sqrt(-3)
assert (3) ** (S(3)/2) == 3 * sqrt(3)
assert (-3) ** (S(3)/2) == - 3 * sqrt(-3)
assert (-3) ** (S(5)/2) == 9 * I * sqrt(3)
assert (-3) ** (S(7)/2) == - I * 27 * sqrt(3)
assert (2) ** (S(3)/2) == 2 * sqrt(2)
assert (2) ** (S(-3)/2) == sqrt(2) / 4
assert (81) ** (S(2)/3) == 9 * (S(3) ** (S(2)/3))
assert (-81) ** (S(2)/3) == 9 * (S(-3) ** (S(2)/3))
assert (-3) ** Rational(-7, 3) == -(-1)**Rational(2, 3)*3**Rational(2, 3)/27
assert (-3) ** Rational(-2, 3) == -(-1)**Rational(1, 3)*3**Rational(1, 3)/3
# join roots
assert sqrt(6) + sqrt(24) == 3*sqrt(6)
assert sqrt(2) * sqrt(3) == sqrt(6)
# separate sybols & constansts
x = Symbol("x")
assert sqrt(49 * x) == 7 * sqrt(x)
assert sqrt((3 - sqrt(pi)) ** 2) == 3 - sqrt(pi)
# check that it is fast for big numbers
assert (2**64+1) ** Rational(4, 3)
assert (2**64+1) ** Rational(17,25)
# negative rational power and negative base
assert (-3) ** Rational(-7, 3) == -(-1)**Rational(2, 3)*3**Rational(2, 3)/27
assert (-3) ** Rational(-2, 3) == -(-1)**Rational(1, 3)*3**Rational(1, 3)/3
assert S(1234).factors() == {617: 1, 2: 1}
assert Rational(2*3, 3*5*7).factors() == {2: 1, 5: -1, 7: -1}
# test that eval_power factors numbers bigger than limit (2**15)
from sympy import nextprime
n = nextprime(2**15) # bigger than the current limit in factor_trial_division
assert sqrt(n**2) == n
assert sqrt(n**3) == n*sqrt(n)
assert sqrt(4*n) == 2*sqrt(n)
def nth_prime(n):
prime_num = 0
for i in range(n):
prime_num = sympy.nextprime(prime_num)
return prime_num
def test_gf_factor():
assert gf_factor([], 11) == (0, [])
assert gf_factor([1], 11) == (1, [])
assert gf_factor([1,1], 11) == (1, [([1, 1], 1)])
f, p = [1,0,0,1,0], 2
g = (1, [([1, 0], 1),
([1, 1], 1),
([1, 1, 1], 1)])
assert gf_factor(f, p, method='zassenhaus') == g
assert gf_factor(f, p, method='shoup') == g
g = (1, [[1, 0],
[1, 1],
[1, 1, 1]])
assert gf_factor_sqf(f, p, method='zassenhaus') == g
assert gf_factor_sqf(f, p, method='shoup') == g
assert gf_factor([1, 5, 8, 4], 11) == \
(1, [([1, 1], 1), ([1, 2], 2)])
assert gf_factor([1, 1, 10, 1, 0, 10, 10, 10, 0, 0], 11) == \
(1, [([1, 0], 2), ([1, 9, 5], 1), ([1, 3, 0, 8, 5, 2], 1)])
assert gf_factor(gf_from_dict({32: 1, 0: 1}, 11), 11) == \
(1, [([1, 0, 0, 0, 0, 0, 0, 0, 3, 0, 0, 0, 0, 0, 0, 0, 10], 1),
([1, 0, 0, 0, 0, 0, 0, 0, 8, 0, 0, 0, 0, 0, 0, 0, 10], 1)])
assert gf_factor(gf_from_dict({32: 8, 0: 5}, 11), 11) == \
(8, [([1, 3], 1),
([1, 8], 1),
([1, 0, 9], 1),
([1, 2, 2], 1),
([1, 9, 2], 1),
([1, 0, 5, 0, 7], 1),
([1, 0, 6, 0, 7], 1),
([1, 0, 0, 0, 1, 0, 0, 0, 6], 1),
([1, 0, 0, 0, 10, 0, 0, 0, 6], 1)])
assert gf_factor(gf_from_dict({63: 8, 0: 5}, 11), 11) == \
(8, [([1, 7], 1),
([1, 4, 5], 1),
([1, 6, 8, 2], 1),
([1, 9, 9, 2], 1),
([1, 0, 0, 9, 0, 0, 4], 1),
([1, 2, 0, 8, 4, 6, 4], 1),
([1, 2, 3, 8, 0, 6, 4], 1),
([1, 2, 6, 0, 8, 4, 4], 1),
([1, 3, 3, 1, 6, 8, 4], 1),
([1, 5, 6, 0, 8, 6, 4], 1),
([1, 6, 2, 7, 9, 8, 4], 1),
([1, 10, 4, 7, 10, 7, 4], 1),
([1, 10, 10, 1, 4, 9, 4], 1)])
# Gathen polynomials: x**n + x + 1 (mod p > 2**n * pi)
p = nextprime(int((2**15 * pi).evalf()))
f = gf_from_dict({15: 1, 1: 1, 0: 1}, p)
assert gf_sqf_p(f, p) == True
g = (1, [([1, 22730, 68144], 1),
([1, 81553, 77449, 86810, 4724], 1),
([1, 86276, 56779, 14859, 31575], 1),
([1, 15347, 95022, 84569, 94508, 92335], 1)])
assert gf_factor(f, p, method='zassenhaus') == g
assert gf_factor(f, p, method='shoup') == g
g = (1, [[1, 22730, 68144],
[1, 81553, 77449, 86810, 4724],
[1, 86276, 56779, 14859, 31575],
[1, 15347, 95022, 84569, 94508, 92335]])
assert gf_factor_sqf(f, p, method='zassenhaus') == g
assert gf_factor_sqf(f, p, method='shoup') == g
# Shoup polynomials: f = a_0 x**n + a_1 x**(n-1) + ... + a_n
# (mod p > 2**(n-2) * pi), where a_n = a_{n-1}**2 + 1, a_0 = 1
p = nextprime(int((2**4 * pi).evalf()))
f = [1, 2, 5, 26, 41, 39, 38] # deg(f) = 6
assert gf_sqf_p(f, p) == True
g = (1, [([1, 44, 26], 1),
([1, 11, 25, 18, 30], 1)])
assert gf_factor(f, p, method='zassenhaus') == g
assert gf_factor(f, p, method='shoup') == g
g = (1, [[1, 44, 26],
[1, 11, 25, 18, 30]])
assert gf_factor_sqf(f, p, method='zassenhaus') == g
assert gf_factor_sqf(f, p, method='shoup') == g
def genprime(n, K):
return K(nextprime(int((2**n * pi).evalf())))
def next_prime(n):
return nextprime(n)
def test_zzX_wang():
f = zzX_from_poly(W_1)
p = nextprime(zzX_mignotte_bound(f))
assert p == 6291469
V_1, k_1, E_1 = [[1],[]], 1, -14
V_2, k_2, E_2 = [[1, 0]], 2, 3
V_3, k_3, E_3 = [[1],[ 1, 0]], 2, -11
V_4, k_4, E_4 = [[1],[-1, 0]], 1, -17
V = [V_1, V_2, V_3, V_4]
K = [k_1, k_2, k_3, k_4]
E = [E_1, E_2, E_3, E_4]
V = zip(V, K)
A = [-14, 3]
U = zzX_eval_list(f, A)
cu, u = zzx_primitive(U)
assert cu == 1 and u == U == \
[1036728, 915552, 55748, 105621, -17304, -26841, -644]
assert zzX_wang_non_divisors(E, cu, 4) == [7, 3, 11, 17]
assert zzx_sqf_p(u) and zzx_degree(u) == zzX_degree(f)
_, H = zzx_factor_sqf(u)
h_1 = [44, 42, 1]
h_2 = [126, -9, 28]
h_3 = [187, 0, -23]
assert H == [h_1, h_2, h_3]
LC_1 = [[-4], [-4,0]]
LC_2 = [[-1,0,0], []]
LC_3 = [[1], [], [-1,0,0]]
LC = [LC_1, LC_2, LC_3]
assert zzX_wang_lead_coeffs(f, V, cu, E, H, A) == (f, H, LC)
H_1 = [[44L, 42L, 1L], [126L, -9L, 28L], [187L, 0L, -23L]]
C_1 = [-70686, -5863, -17826, 2009, 5031, 74]
H_2 = [[[-4, -12], [-3, 0], [1]], [[-9, 0], [-9], [-2, 0]], [[1, 0, -9], [], [1, -9]]]
C_2 = [[9, 12, -45, -108, -324], [18, -216, -810, 0], [2, 9, -252, -288, -945], [-30, -414, 0], [2, -54, -3, 81], [12, 0]]
H_3 = [[[-4, -12], [-3, 0], [1]], [[-9, 0], [-9], [-2, 0]], [[1, 0, -9], [], [1, -9]]]
C_3 = [[-36, -108, 0], [-27, -36, -108], [-8, -42, 0], [-6, 0, 9], [2, 0]]
T_1 = [[-3, 0], [-2], [1]]
T_2 = [[[-1, 0], []], [[-3], []], [[-6]]]
T_3 = [[[]], [[]], [[-1]]]
assert zzX_diophantine(H_1, C_1, [], 5, p) == T_1
assert zzX_diophantine(H_2, C_2, [-14], 5, p) == T_2
assert zzX_diophantine(H_3, C_3, [-14], 5, p) == T_3
factors = zzX_wang_hensel_lifting(f, H, LC, A, p)
f_1 = zzX_to_poly(factors[0], x, y, z)
f_2 = zzX_to_poly(factors[1], x, y, z)
f_3 = zzX_to_poly(factors[2], x, y, z)
assert f_1 == -(4*(y + z)*x**2 + x*y*z - 1).as_poly(x, y, z)
assert f_2 == -(y*z**2*x**2 + 3*x*z + 2*y).as_poly(x, y, z)
assert f_3 == ((y**2 - z**2)*x**2 + y - z**2).as_poly(x, y, z)
assert f_1*f_2*f_3 == W_1
def get_prime_number(digits):
number = int(time.time() * 10 ** digits)
prime_number = sympy.nextprime(number)
while prime_number % 4 != 3:
prime_number = sympy.nextprime(prime_number)
return prime_number
def genprime(n):
return nextprime(int((2**n * pi).evalf()))
def random_prime(bits):
return sympy.nextprime(2 ** bits + random.randint(0, 2 ** bits))
def prime_numbers(digits):
seed = int(time.time() * 10**digits)
p = sympy.nextprime(seed)
q = sympy.nextprime(p)
return p, q
from sympy import prime
from sympy import nextprime
from sys import exit
""" by doing the math we find out that when n=2p , the remainder is 2, otherwise it's 2*n*Pn :D, alse, 2*n < Pn
for big n """
n = 7037
p = prime(7037)
while 1:
if (2*p*n) - 10**10 >0:
print n, 2*(prime(n-2))*(n-2)
exit(1)
p = nextprime(nextprime(p))
n+=2
def test_gf_factor():
assert gf_factor([], 11, ZZ) == (0, [])
assert gf_factor([1], 11, ZZ) == (1, [])
assert gf_factor([1,1], 11, ZZ) == (1, [([1, 1], 1)])
assert gf_factor_sqf([], 11, ZZ) == (0, [])
assert gf_factor_sqf([1], 11, ZZ) == (1, [])
assert gf_factor_sqf([1,1], 11, ZZ) == (1, [[1, 1]])
assert gf_factor_sqf([], 11, ZZ, method='berlekamp') == (0, [])
assert gf_factor_sqf([1], 11, ZZ, method='berlekamp') == (1, [])
assert gf_factor_sqf([1,1], 11, ZZ, method='berlekamp') == (1, [[1, 1]])
assert gf_factor_sqf([], 11, ZZ, method='zassenhaus') == (0, [])
assert gf_factor_sqf([1], 11, ZZ, method='zassenhaus') == (1, [])
assert gf_factor_sqf([1,1], 11, ZZ, method='zassenhaus') == (1, [[1, 1]])
assert gf_factor_sqf([], 11, ZZ, method='shoup') == (0, [])
assert gf_factor_sqf([1], 11, ZZ, method='shoup') == (1, [])
assert gf_factor_sqf([1,1], 11, ZZ, method='shoup') == (1, [[1, 1]])
f, p = [1,0,0,1,0], 2
g = (1, [([1, 0], 1),
([1, 1], 1),
([1, 1, 1], 1)])
assert gf_factor(f, p, ZZ, method='berlekamp') == g
assert gf_factor(f, p, ZZ, method='zassenhaus') == g
assert gf_factor(f, p, ZZ, method='shoup') == g
g = (1, [[1, 0],
[1, 1],
[1, 1, 1]])
assert gf_factor_sqf(f, p, ZZ, method='berlekamp') == g
assert gf_factor_sqf(f, p, ZZ, method='zassenhaus') == g
assert gf_factor_sqf(f, p, ZZ, method='shoup') == g
f, p = gf_from_int_poly([1,-3,1,-3,-1,-3,1], 11), 11
g = (1, [([1, 1], 1),
([1, 5, 3], 1),
([1, 2, 3, 4], 1)])
assert gf_factor(f, p, ZZ, method='berlekamp') == g
assert gf_factor(f, p, ZZ, method='zassenhaus') == g
assert gf_factor(f, p, ZZ, method='shoup') == g
f, p = [1, 5, 8, 4], 11
g = (1, [([1, 1], 1), ([1, 2], 2)])
assert gf_factor(f, p, ZZ, method='berlekamp') == g
assert gf_factor(f, p, ZZ, method='zassenhaus') == g
assert gf_factor(f, p, ZZ, method='shoup') == g
f, p = [1, 1, 10, 1, 0, 10, 10, 10, 0, 0], 11
g = (1, [([1, 0], 2), ([1, 9, 5], 1), ([1, 3, 0, 8, 5, 2], 1)])
assert gf_factor(f, p, ZZ, method='berlekamp') == g
assert gf_factor(f, p, ZZ, method='zassenhaus') == g
assert gf_factor(f, p, ZZ, method='shoup') == g
f, p = gf_from_dict({32: 1, 0: 1}, 11, ZZ), 11
g = (1, [([1, 0, 0, 0, 0, 0, 0, 0, 3, 0, 0, 0, 0, 0, 0, 0, 10], 1),
([1, 0, 0, 0, 0, 0, 0, 0, 8, 0, 0, 0, 0, 0, 0, 0, 10], 1)])
assert gf_factor(f, p, ZZ, method='berlekamp') == g
assert gf_factor(f, p, ZZ, method='zassenhaus') == g
assert gf_factor(f, p, ZZ, method='shoup') == g
f, p = gf_from_dict({32: 8, 0: 5}, 11, ZZ), 11
g = (8, [([1, 3], 1),
([1, 8], 1),
([1, 0, 9], 1),
([1, 2, 2], 1),
([1, 9, 2], 1),
([1, 0, 5, 0, 7], 1),
([1, 0, 6, 0, 7], 1),
([1, 0, 0, 0, 1, 0, 0, 0, 6], 1),
([1, 0, 0, 0, 10, 0, 0, 0, 6], 1)])
assert gf_factor(f, p, ZZ, method='berlekamp') == g
assert gf_factor(f, p, ZZ, method='zassenhaus') == g
assert gf_factor(f, p, ZZ, method='shoup') == g
f, p = gf_from_dict({63: 8, 0: 5}, 11, ZZ), 11
g = (8, [([1, 7], 1),
([1, 4, 5], 1),
([1, 6, 8, 2], 1),
([1, 9, 9, 2], 1),
([1, 0, 0, 9, 0, 0, 4], 1),
([1, 2, 0, 8, 4, 6, 4], 1),
([1, 2, 3, 8, 0, 6, 4], 1),
([1, 2, 6, 0, 8, 4, 4], 1),
([1, 3, 3, 1, 6, 8, 4], 1),
([1, 5, 6, 0, 8, 6, 4], 1),
([1, 6, 2, 7, 9, 8, 4], 1),
([1, 10, 4, 7, 10, 7, 4], 1),
([1, 10, 10, 1, 4, 9, 4], 1)])
assert gf_factor(f, p, ZZ, method='berlekamp') == g
assert gf_factor(f, p, ZZ, method='zassenhaus') == g
assert gf_factor(f, p, ZZ, method='shoup') == g
# Gathen polynomials: x**n + x + 1 (mod p > 2**n * pi)
p = ZZ(nextprime(int((2**15 * pi).evalf())))
f = gf_from_dict({15: 1, 1: 1, 0: 1}, p, ZZ)
assert gf_sqf_p(f, p, ZZ) == True
g = (1, [([1, 22730, 68144], 1),
([1, 81553, 77449, 86810, 4724], 1),
([1, 86276, 56779, 14859, 31575], 1),
([1, 15347, 95022, 84569, 94508, 92335], 1)])
assert gf_factor(f, p, ZZ, method='zassenhaus') == g
assert gf_factor(f, p, ZZ, method='shoup') == g
g = (1, [[1, 22730, 68144],
[1, 81553, 77449, 86810, 4724],
[1, 86276, 56779, 14859, 31575],
[1, 15347, 95022, 84569, 94508, 92335]])
assert gf_factor_sqf(f, p, ZZ, method='zassenhaus') == g
assert gf_factor_sqf(f, p, ZZ, method='shoup') == g
# Shoup polynomials: f = a_0 x**n + a_1 x**(n-1) + ... + a_n
# (mod p > 2**(n-2) * pi), where a_n = a_{n-1}**2 + 1, a_0 = 1
p = ZZ(nextprime(int((2**4 * pi).evalf())))
f = [1, 2, 5, 26, 41, 39, 38]
assert gf_sqf_p(f, p, ZZ) == True
g = (1, [([1, 44, 26], 1),
([1, 11, 25, 18, 30], 1)])
assert gf_factor(f, p, ZZ, method='zassenhaus') == g
assert gf_factor(f, p, ZZ, method='shoup') == g
g = (1, [[1, 44, 26],
[1, 11, 25, 18, 30]])
assert gf_factor_sqf(f, p, ZZ, method='zassenhaus') == g
assert gf_factor_sqf(f, p, ZZ, method='shoup') == g
import math
from collections import deque
from sympy import nextprime
start = 10 * 10 + 1
primes = deque([start] + [nextprime(start, x + 1) for x in range(5)])
def prime_pattern(arr):
if not ((arr[5] - arr[0] == 26) and \
(arr[4] - arr[0] == 12) and \
(arr[3] - arr[0] == 8) and \
(arr[2] - arr[0] == 6) and \
(arr[1] - arr[0] == 2)):
return 0
n2 = arr[0] - 1
n = int(math.sqrt(n2))
if n * n == n2:
return n
else:
return 0
limit = 10 ** 6
ret = 10
while 1:
primes.popleft()
primes.append(nextprime(primes[-1]))
if primes[-1] >= limit:
break
ret += prime_pattern(primes)
print ret
def test_gf_factor():
assert gf_factor([], 11, ZZ) == (0, [])
assert gf_factor([1], 11, ZZ) == (1, [])
assert gf_factor([1,1], 11, ZZ) == (1, [([1, 1], 1)])
assert gf_factor_sqf([], 11, ZZ) == (0, [])
assert gf_factor_sqf([1], 11, ZZ) == (1, [])
assert gf_factor_sqf([1,1], 11, ZZ) == (1, [[1, 1]])
config.setup('GF_FACTOR_METHOD', 'berlekamp')
assert gf_factor_sqf([], 11, ZZ) == (0, [])
assert gf_factor_sqf([1], 11, ZZ) == (1, [])
assert gf_factor_sqf([1,1], 11, ZZ) == (1, [[1, 1]])
config.setup('GF_FACTOR_METHOD', 'zassenhaus')
assert gf_factor_sqf([], 11, ZZ) == (0, [])
assert gf_factor_sqf([1], 11, ZZ) == (1, [])
assert gf_factor_sqf([1,1], 11, ZZ) == (1, [[1, 1]])
config.setup('GF_FACTOR_METHOD', 'shoup')
assert gf_factor_sqf([], 11, ZZ) == (0, [])
assert gf_factor_sqf([1], 11, ZZ) == (1, [])
assert gf_factor_sqf([1,1], 11, ZZ) == (1, [[1, 1]])
f, p = [1,0,0,1,0], 2
g = (1, [([1, 0], 1),
([1, 1], 1),
([1, 1, 1], 1)])
config.setup('GF_FACTOR_METHOD', 'berlekamp')
assert gf_factor(f, p, ZZ) == g
config.setup('GF_FACTOR_METHOD', 'zassenhaus')
assert gf_factor(f, p, ZZ) == g
config.setup('GF_FACTOR_METHOD', 'shoup')
assert gf_factor(f, p, ZZ) == g
g = (1, [[1, 0],
[1, 1],
[1, 1, 1]])
config.setup('GF_FACTOR_METHOD', 'berlekamp')
assert gf_factor_sqf(f, p, ZZ) == g
config.setup('GF_FACTOR_METHOD', 'zassenhaus')
assert gf_factor_sqf(f, p, ZZ) == g
config.setup('GF_FACTOR_METHOD', 'shoup')
assert gf_factor_sqf(f, p, ZZ) == g
f, p = gf_from_int_poly([1,-3,1,-3,-1,-3,1], 11), 11
g = (1, [([1, 1], 1),
([1, 5, 3], 1),
([1, 2, 3, 4], 1)])
config.setup('GF_FACTOR_METHOD', 'berlekamp')
assert gf_factor(f, p, ZZ) == g
config.setup('GF_FACTOR_METHOD', 'zassenhaus')
assert gf_factor(f, p, ZZ) == g
config.setup('GF_FACTOR_METHOD', 'shoup')
assert gf_factor(f, p, ZZ) == g
f, p = [1, 5, 8, 4], 11
g = (1, [([1, 1], 1), ([1, 2], 2)])
config.setup('GF_FACTOR_METHOD', 'berlekamp')
assert gf_factor(f, p, ZZ) == g
config.setup('GF_FACTOR_METHOD', 'zassenhaus')
assert gf_factor(f, p, ZZ) == g
config.setup('GF_FACTOR_METHOD', 'shoup')
assert gf_factor(f, p, ZZ) == g
f, p = [1, 1, 10, 1, 0, 10, 10, 10, 0, 0], 11
g = (1, [([1, 0], 2), ([1, 9, 5], 1), ([1, 3, 0, 8, 5, 2], 1)])
config.setup('GF_FACTOR_METHOD', 'berlekamp')
assert gf_factor(f, p, ZZ) == g
config.setup('GF_FACTOR_METHOD', 'zassenhaus')
assert gf_factor(f, p, ZZ) == g
config.setup('GF_FACTOR_METHOD', 'shoup')
assert gf_factor(f, p, ZZ) == g
f, p = gf_from_dict({32: 1, 0: 1}, 11, ZZ), 11
g = (1, [([1, 0, 0, 0, 0, 0, 0, 0, 3, 0, 0, 0, 0, 0, 0, 0, 10], 1),
([1, 0, 0, 0, 0, 0, 0, 0, 8, 0, 0, 0, 0, 0, 0, 0, 10], 1)])
config.setup('GF_FACTOR_METHOD', 'berlekamp')
assert gf_factor(f, p, ZZ) == g
config.setup('GF_FACTOR_METHOD', 'zassenhaus')
assert gf_factor(f, p, ZZ) == g
config.setup('GF_FACTOR_METHOD', 'shoup')
assert gf_factor(f, p, ZZ) == g
f, p = gf_from_dict({32: 8, 0: 5}, 11, ZZ), 11
g = (8, [([1, 3], 1),
([1, 8], 1),
([1, 0, 9], 1),
([1, 2, 2], 1),
([1, 9, 2], 1),
([1, 0, 5, 0, 7], 1),
([1, 0, 6, 0, 7], 1),
([1, 0, 0, 0, 1, 0, 0, 0, 6], 1),
([1, 0, 0, 0, 10, 0, 0, 0, 6], 1)])
config.setup('GF_FACTOR_METHOD', 'berlekamp')
assert gf_factor(f, p, ZZ) == g
config.setup('GF_FACTOR_METHOD', 'zassenhaus')
assert gf_factor(f, p, ZZ) == g
config.setup('GF_FACTOR_METHOD', 'shoup')
assert gf_factor(f, p, ZZ) == g
f, p = gf_from_dict({63: 8, 0: 5}, 11, ZZ), 11
g = (8, [([1, 7], 1),
([1, 4, 5], 1),
([1, 6, 8, 2], 1),
([1, 9, 9, 2], 1),
([1, 0, 0, 9, 0, 0, 4], 1),
([1, 2, 0, 8, 4, 6, 4], 1),
([1, 2, 3, 8, 0, 6, 4], 1),
([1, 2, 6, 0, 8, 4, 4], 1),
([1, 3, 3, 1, 6, 8, 4], 1),
([1, 5, 6, 0, 8, 6, 4], 1),
([1, 6, 2, 7, 9, 8, 4], 1),
([1, 10, 4, 7, 10, 7, 4], 1),
([1, 10, 10, 1, 4, 9, 4], 1)])
config.setup('GF_FACTOR_METHOD', 'berlekamp')
assert gf_factor(f, p, ZZ) == g
config.setup('GF_FACTOR_METHOD', 'zassenhaus')
assert gf_factor(f, p, ZZ) == g
config.setup('GF_FACTOR_METHOD', 'shoup')
assert gf_factor(f, p, ZZ) == g
# Gathen polynomials: x**n + x + 1 (mod p > 2**n * pi)
p = ZZ(nextprime(int((2**15 * pi).evalf())))
f = gf_from_dict({15: 1, 1: 1, 0: 1}, p, ZZ)
assert gf_sqf_p(f, p, ZZ) == True
g = (1, [([1, 22730, 68144], 1),
([1, 81553, 77449, 86810, 4724], 1),
([1, 86276, 56779, 14859, 31575], 1),
([1, 15347, 95022, 84569, 94508, 92335], 1)])
config.setup('GF_FACTOR_METHOD', 'zassenhaus')
assert gf_factor(f, p, ZZ) == g
config.setup('GF_FACTOR_METHOD', 'shoup')
assert gf_factor(f, p, ZZ) == g
g = (1, [[1, 22730, 68144],
[1, 81553, 77449, 86810, 4724],
[1, 86276, 56779, 14859, 31575],
[1, 15347, 95022, 84569, 94508, 92335]])
config.setup('GF_FACTOR_METHOD', 'zassenhaus')
assert gf_factor_sqf(f, p, ZZ) == g
config.setup('GF_FACTOR_METHOD', 'shoup')
assert gf_factor_sqf(f, p, ZZ) == g
# Shoup polynomials: f = a_0 x**n + a_1 x**(n-1) + ... + a_n
# (mod p > 2**(n-2) * pi), where a_n = a_{n-1}**2 + 1, a_0 = 1
p = ZZ(nextprime(int((2**4 * pi).evalf())))
f = [1, 2, 5, 26, 41, 39, 38]
assert gf_sqf_p(f, p, ZZ) == True
g = (1, [([1, 44, 26], 1),
([1, 11, 25, 18, 30], 1)])
config.setup('GF_FACTOR_METHOD', 'zassenhaus')
assert gf_factor(f, p, ZZ) == g
config.setup('GF_FACTOR_METHOD', 'shoup')
assert gf_factor(f, p, ZZ) == g
g = (1, [[1, 44, 26],
[1, 11, 25, 18, 30]])
config.setup('GF_FACTOR_METHOD', 'zassenhaus')
assert gf_factor_sqf(f, p, ZZ) == g
config.setup('GF_FACTOR_METHOD', 'shoup')
assert gf_factor_sqf(f, p, ZZ) == g
config.setup('GF_FACTOR_METHOD', 'other')
raises(KeyError, lambda: gf_factor([1,1], 11, ZZ))
config.setup('GF_FACTOR_METHOD')
def p007(n):
"""Finds the n-th prime number."""
p = 0
for i in range(n):
p = nextprime(p)
return p
def test_powers_Integer():
"""Test Integer._eval_power"""
# check infinity
assert S(1) ** S.Infinity == 1
assert S(-1)** S.Infinity == S.NaN
assert S(2) ** S.Infinity == S.Infinity
assert S(-2)** S.Infinity == S.Infinity + S.Infinity * S.ImaginaryUnit
assert S(0) ** S.Infinity == 0
# check Nan
assert S(1) ** S.NaN == S.NaN
assert S(-1) ** S.NaN == S.NaN
# check for exact roots
assert S(-1) ** Rational(6, 5) == - (-1)**(S(1)/5)
assert sqrt(S(4)) == 2
assert sqrt(S(-4)) == I * 2
assert S(16) ** Rational(1, 4) == 2
assert S(-16) ** Rational(1, 4) == 2 * (-1)**Rational(1,4)
assert S(9) ** Rational(3, 2) == 27
assert S(-9) ** Rational(3, 2) == -27*I
assert S(27) ** Rational(2, 3) == 9
assert S(-27) ** Rational(2, 3) == 9 * (S(-1) ** Rational(2, 3))
assert (-2) ** Rational(-2, 1) == Rational(1, 4)
# not exact roots
assert sqrt(-3) == I*sqrt(3)
assert (3) ** (S(3)/2) == 3 * sqrt(3)
assert (-3) ** (S(3)/2) == - 3 * sqrt(-3)
assert (-3) ** (S(5)/2) == 9 * I * sqrt(3)
assert (-3) ** (S(7)/2) == - I * 27 * sqrt(3)
assert (2) ** (S(3)/2) == 2 * sqrt(2)
assert (2) ** (S(-3)/2) == sqrt(2) / 4
assert (81) ** (S(2)/3) == 9 * (S(3) ** (S(2)/3))
assert (-81) ** (S(2)/3) == 9 * (S(-3) ** (S(2)/3))
assert (-3) ** Rational(-7, 3) == -(-1)**Rational(2, 3)*3**Rational(2, 3)/27
assert (-3) ** Rational(-2, 3) == -(-1)**Rational(1, 3)*3**Rational(1, 3)/3
# join roots
assert sqrt(6) + sqrt(24) == 3*sqrt(6)
assert sqrt(2) * sqrt(3) == sqrt(6)
# separate symbols & constansts
x = Symbol("x")
assert sqrt(49 * x) == 7 * sqrt(x)
assert sqrt((3 - sqrt(pi)) ** 2) == 3 - sqrt(pi)
# check that it is fast for big numbers
assert (2**64+1) ** Rational(4, 3)
assert (2**64+1) ** Rational(17,25)
# negative rational power and negative base
assert (-3) ** Rational(-7, 3) == -(-1)**Rational(2, 3)*3**Rational(2, 3)/27
assert (-3) ** Rational(-2, 3) == -(-1)**Rational(1, 3)*3**Rational(1, 3)/3
assert S(1234).factors() == {617: 1, 2: 1}
assert Rational(2*3, 3*5*7).factors() == {2: 1, 5: -1, 7: -1}
# test that eval_power factors numbers bigger than limit (2**15)
from sympy import nextprime
n = nextprime(2**15) # bigger than the current limit in factor_trial_division
assert sqrt(n**2) == n
assert sqrt(n**3) == n*sqrt(n)
assert sqrt(4*n) == 2*sqrt(n)
# check that factors of base with powers sharing gcd with power are removed
assert (2**4*3)**Rational(1, 6) == 2**Rational(2, 3)*3**Rational(1, 6)
assert (2**4*3)**Rational(5, 6) == 8*2**Rational(1, 3)*3**Rational(5, 6)
# check that bases sharing a gcd are exptracted
assert 2**Rational(1, 3)*3**Rational(1, 4)*6**Rational(1, 5) == \
2**Rational(8, 15)*3**Rational(9, 20)
assert sqrt(8)*24**Rational(1, 3)*6**Rational(1, 5) == \
4*2**Rational(7, 10)*3**Rational(8, 15)
assert sqrt(8)*(-24)**Rational(1, 3)*(-6)**Rational(1, 5) == \
4*(-3)**Rational(8, 15)*2**Rational(7, 10)
assert 2**Rational(1, 3)*2**Rational(8, 9) == 2*2**Rational(2, 9)
assert 2**Rational(2, 3)*6**Rational(1, 3) == 2*3**Rational(1, 3)
assert 2**Rational(2, 3)*6**Rational(8, 9) == 2*2**Rational(5, 9)*3**Rational(8, 9)
assert (-2)**Rational(2, S(3))*(-4)**Rational(1, S(3)) == -2*2**Rational(1, 3)
assert 3*Pow(3, 2, evaluate=False) == 3**3
assert 3*Pow(3, -1/S(3), evaluate=False) == 3**(2/S(3))
assert (-2)**(1/S(3))*(-3)**(1/S(4))*(-5)**(5/S(6)) == \
-(-1)**Rational(5, 12)*2**Rational(1, 3)*3**Rational(1, 4)*5**Rational(5, 6)
from sympy import nextprime
from pwnlib.tubes.remote import remote
r = remote('188.166.133.53', 11059)
while True:
lines = r.recvlines(2)
print '\n'.join(lines)
p = nextprime(lines[1].split()[8][:-1])
print p
r.send(str(p))
