def isirreducible(n):
if (n.real**2 == 1 and n.imag**2 == 1):
return True
elif (n.real % 4 == 3 and isprime(n.real) and n.imag == 0):
return True
elif(n*n.conjugate() % 4 == 1 and isprime(n*n.conjugate())):
return True
return False
def mySimulate():
a = 1
b = 0
c = 0
d = 0
e = 0
f = 0
g = 0
h = 0
b = 81
c = b
if a != 0:
b = 108100
c = b + 17000
else:
b = 81
c = b
while True:
f = 1
if not isprime(b):
h += 1
if c == b:
return h
b += 17
def find_prime_power(N):
for i in range(2, int(math.log(N, 2) + 1)):
[flag, b] = find_largest_b(N, i)
if flag == 1:
if isprime(b):
return [b, i]
return str(N) + " is not a prime power"
def test_prps():
oddcomposites = [n for n in range(1, 10**5) if
n % 2 and not isprime(n)]
# A checksum would be better.
assert sum(oddcomposites) == 2045603465
assert [n for n in oddcomposites if mr(n, [2])] == [
2047, 3277, 4033, 4681, 8321, 15841, 29341, 42799, 49141,
52633, 65281, 74665, 80581, 85489, 88357, 90751]
assert [n for n in oddcomposites if mr(n, [3])] == [
121, 703, 1891, 3281, 8401, 8911, 10585, 12403, 16531,
18721, 19345, 23521, 31621, 44287, 47197, 55969, 63139,
74593, 79003, 82513, 87913, 88573, 97567]
assert [n for n in oddcomposites if mr(n, [325])] == [
9, 25, 27, 49, 65, 81, 325, 341, 343, 697, 1141, 2059,
2149, 3097, 3537, 4033, 4681, 4941, 5833, 6517, 7987, 8911,
12403, 12913, 15043, 16021, 20017, 22261, 23221, 24649,
24929, 31841, 35371, 38503, 43213, 44173, 47197, 50041,
55909, 56033, 58969, 59089, 61337, 65441, 68823, 72641,
76793, 78409, 85879]
assert not any(mr(n, [9345883071009581737]) for n in oddcomposites)
assert [n for n in oddcomposites if is_lucas_prp(n)] == [
323, 377, 1159, 1829, 3827, 5459, 5777, 9071, 9179, 10877,
11419, 11663, 13919, 14839, 16109, 16211, 18407, 18971,
19043, 22499, 23407, 24569, 25199, 25877, 26069, 27323,
32759, 34943, 35207, 39059, 39203, 39689, 40309, 44099,
46979, 47879, 50183, 51983, 53663, 56279, 58519, 60377,
63881, 69509, 72389, 73919, 75077, 77219, 79547, 79799,
82983, 84419, 86063, 90287, 94667, 97019, 97439]
assert [n for n in oddcomposites if is_strong_lucas_prp(n)] == [
5459, 5777, 10877, 16109, 18971, 22499, 24569, 25199, 40309,
58519, 75077, 97439]
assert [n for n in oddcomposites if is_extra_strong_lucas_prp(n)
] == [
989, 3239, 5777, 10877, 27971, 29681, 30739, 31631, 39059,
72389, 73919, 75077]
def test_prps():
oddcomposites = [n for n in range(1, 10**5) if
n % 2 and not isprime(n)]
# A checksum would be better.
assert sum(oddcomposites) == 2045603465
assert [n for n in oddcomposites if mr(n, [2])] == [
2047, 3277, 4033, 4681, 8321, 15841, 29341, 42799, 49141,
52633, 65281, 74665, 80581, 85489, 88357, 90751]
assert [n for n in oddcomposites if mr(n, [3])] == [
121, 703, 1891, 3281, 8401, 8911, 10585, 12403, 16531,
18721, 19345, 23521, 31621, 44287, 47197, 55969, 63139,
74593, 79003, 82513, 87913, 88573, 97567]
assert [n for n in oddcomposites if mr(n, [325])] == [
9, 25, 27, 49, 65, 81, 325, 341, 343, 697, 1141, 2059,
2149, 3097, 3537, 4033, 4681, 4941, 5833, 6517, 7987, 8911,
12403, 12913, 15043, 16021, 20017, 22261, 23221, 24649,
24929, 31841, 35371, 38503, 43213, 44173, 47197, 50041,
55909, 56033, 58969, 59089, 61337, 65441, 68823, 72641,
76793, 78409, 85879]
assert not any(mr(n, [9345883071009581737]) for n in oddcomposites)
assert [n for n in oddcomposites if is_lucas_prp(n)] == [
323, 377, 1159, 1829, 3827, 5459, 5777, 9071, 9179, 10877,
11419, 11663, 13919, 14839, 16109, 16211, 18407, 18971,
19043, 22499, 23407, 24569, 25199, 25877, 26069, 27323,
32759, 34943, 35207, 39059, 39203, 39689, 40309, 44099,
46979, 47879, 50183, 51983, 53663, 56279, 58519, 60377,
63881, 69509, 72389, 73919, 75077, 77219, 79547, 79799,
82983, 84419, 86063, 90287, 94667, 97019, 97439]
assert [n for n in oddcomposites if is_strong_lucas_prp(n)] == [
5459, 5777, 10877, 16109, 18971, 22499, 24569, 25199, 40309,
58519, 75077, 97439]
assert [n for n in oddcomposites if is_extra_strong_lucas_prp(n)
] == [
989, 3239, 5777, 10877, 27971, 29681, 30739, 31631, 39059,
72389, 73919, 75077]
def prime_list():
"""Create prime number list below 1000000"""
L = []
for n in range(1, 1000000):
if isprime(n):
L.append(n)
return L
def shor(N):
if isprime(N):
print('{} is a prime'.format(N))
return -1
if N % 2 == 0:
print('{}: N is even'.format(N))
return 2
for i in range(3, int(np.log2(N))):
j = i
while (j <= N):
if j == N:
print(str(N) + ': trivially, N={}^a for some int a'.format(i))
return i
j *= i
while (1):
x = random.randint(2, int(np.sqrt(N)))
if (np.gcd(x, N) != 1):
print('{}: by randoming, got factor {}'.format(N, np.gcd(x, N)))
return (np.gcd(x, N))
#print('Attempting: N={}, x={}'.format(N,x))
r = quantum_find_r_sped(x, N)
#print('found r={}'.format(r))
if (r % 2 == 1): continue
#print('(N,x,r)=({},{},{})'.format(N,x,r))
a = np.gcd((x**(int(r / 2)) - 1) % N, N)
#print('a={}, ie {}'.format(a, int(a)))
if (a == 1 or a == N): continue
print('{}: Found (N,x,r)={},{},{} with factor {}'.format(
N, N, x, r, np.gcd(int(a), N)))
return np.gcd(int(a), N)
def __init__(self, p, g, y, name="Verifier"):
"""
Initializes the verifier. Checks the validity of the arguments, and sets the public parameters p, g, y.
This method has to initialize instance variables p, g and y (c and b will remain unset until choose_b method
is called).
:param p: integer, modulus
:param g: integer, base
:param y: integer, g^x mod p
:param name: optional string, name of the verifier (to be used when printing data to the console)
"""
self.p = None
self.g = None
self.y = None
self.c = None
self.b = None
self.name = name
try:
assert (isprime(p))
assert (y < p)
assert (is_primitive_root(g, p))
except:
raise InvalidParams
self.p = p
self.g = g
self.y = y
print_debug("{}:\t\tInitialized with p = {}, g = {} and y = {}".format(
self.name, self.p, self.g, self.y), LOG_INFO)
def LargestPrimeFactor(n="NAN"):
if n == "NAN":
n = int(input("LargestPrimeFactor(n), n = "))
num = str(n)
rng = int(np.sqrt(n))
primes = np.array(primeslist(1, rng))
val = n
while isprime(val) == False:
vals = np.linspace(val, val, len(primes))
print(vals)
divs = vals / primes
print(divs)
counter = 0
for i in range(len(primes)):
if int(divs[i]) == divs[i]:
print(divs[i], " for ", primes[i])
val = divs[i]
print(val)
if val == 1:
LPF = primes[i]
print("The largest prime factor of ", num, " is ", LPF)
return LPF
else:
break
else:
counter += 1
if counter == len(primes):
print("loop encountered")
LPF = val
print("The largest prime factor of ", num, " is ", LPF)
return LPF
def is_prime_as_int(n0: Union[int, float, complex]) -> int:
n = cint(n0)
if n != n0: return 0
try:
from sympy.ntheory.primetest import isprime
except ModuleNotFoundError:
raise Exception("Install sympy to use number-theoretic functions!")
# Note: If it ever matters, nonpositive integers are not prime.
return int(isprime(n))
def is_exatly_prime_64(num):
"""
Точная проверка того, простое ли (True) или составное (False) число,
работает только в диапазоне [0, 2*64)
"""
if num >= 2**64:
raise ValueError(
"Эта функция работает только в диапазоне [0, 2*64)")
return primetest.isprime(num)
def inv_fermat(self, a, n):
if isprime(n):
if a < 0:
a = a % n
b = a
for i in range(n - 3):
b = b * a % n
return b
else:
return "n is not prime"
def generatesMaxSequence(self):
'''
The length K of a multiplicative congruence generator
satisfies the following properties:
- If K = M-1, then M is prime
- K divides M -1
- K = M - 1 if and only if a is a primitive root of M and M is prime
'''
res = isprime(self.moduli) and isPrimitiveRoot(self.a, self.moduli)
return res
def primegenerating(n):
if is_square(n):
return 0
for div in divisors(
n
): #unfortunately a generator does not work, as it does not order the divisors, and we would break out of the loop too soon.
if div * div > n:
break
if not isprime(div + (n // div)):
return 0
return n
def quick(position):
"""Quick tests for whether the position status is known."""
# All enders/quiet-enders are N except {2, 3}.
# Note that [1] is irreducible (p) according to our definitions
if position.gcd == 1 and position.irreducible \
and position.generators != [2, 3]:
return "N"
# Single primes greater than 3 are P
if len(position.generators) == 1 and position.generators[0] > 3 \
and isprime(position.generators[0]):
return "P"
def N(a, b):
primes = []
for n in range(max(abs(a), abs(b))):
num = f(n, a, b)
# print(num)
if isprime(num):
primes.append(num)
else:
break
nprimes = len(primes)
return (nprimes)
def solve1(printflag, allflag, b):
sols = canned1.get(b)
if sols:
if printflag:
for sol in sols:
print("{}!".format(sol))
if not allflag:
break
return sols[0]
if isprime(b):
return 0
raise ValueError("Cannot solve {{{}}}".format(b))
def peel_right(i):
""""Remove digits right to left,
then check for prime number"""
x = list(str(i))
for n in range(1, len(x)):
y = x[:(-n)]
s = int(''.join(y))
if isprime(s):
continue
else:
return "False"
return "True"
def generate_prime_number(length=1024):
""" Generate a prime
Args:
length -- int -- length of the prime to generate, in bits
return a prime
"""
p = 4
# keep generating while the primality test fail
while not isprime(p):
p = generate_prime_candidate(length)
# print("prime generated: %d" %p)
return p
def generate_primes():
p = None
q = None
while (p == None) or (q == None):
odd_rand_num = random.randrange(100000, 999999)
if isprime(odd_rand_num):
if p == None:
p = odd_rand_num
else:
q = odd_rand_num
print("Your private key is: ", (p, q))
return (p, q)
def quadratic_primes(arange):
a, b, max_n = [0, 0, 0]
for i in prime_sieve(arange + 1): # b must be prime
for j in range(-i + 2, 0, 2):
n = 1
while isprime(
n * n + j * n + i
): # equation= n^2 + a*n + b, it has to generate prime numbers
n += 1
if n > max_n:
a, b, max_n = j, i, n
return a * b
def check_prime(self, n):
"""
Функция проверки простоты числа (используется класс Primes из библиотеки sage)
Parameters
----------
n : Union[int, gmpy2.mpz]
Число для проверки
Returns
-------
bool
Простое ли число
"""
return isprime(int(n))
def naive_search(M, N, J):
p = np.array([0] * N)
p[0], p[-1] = 1, 1
jamcoins = []
proofs = []
for i in range(0, 2**(N - 2)):
p_ = [int(a) for a in str(bin(i))[2:]]
p[-(len(p_) + 1):-1] = p_
c = [sum([a * b for a, b in zip(p, D)]) for D in M]
if all([not isprime(a) for a in c]):
jamcoins.append("".join([str(a) for a in p]))
proofs.append([min(list(factorint(a).keys())) for a in c])
if len(jamcoins) == J:
break
return jamcoins, proofs
def __init__(self, p, n=1):
p, n = int(p), int(n)
if not isprime(p):
raise ValueError("p must be a prime number, not %s" % p)
if n <= 0:
raise ValueError("n must be a positive integer, not %s" % n)
self.p = p
self.n = n
if n == 1:
self.reducing = [1, 0]
else:
for c in itertools.product(range(p), repeat=n):
poly = (1, *c)
if gf_irreducible_p(poly, p, ZZ):
self.reducing = poly
break
def add_primes(primes, switch):
"""Return the last number in the list (largest prime),
and the in-loop counter (consecutive numbers)"""
counter = switch # start with this prime number
in_counter = 0
sum = 0
L = []
while sum < 1000000:
sum = sum + primes[counter]
if isprime(sum):
L.append(sum)
in_counter += 1 # in loop counter
counter += 1
if L[-1] < 1000000:
return L[-1], in_counter
elif L[-1] > 1000000:
print("****")
return L[-2], in_counter
def main_code(x):
N, J = tuple(map(int, input().split()))
current = 1 + (2**(N - 1))
print("Case #" + str(x + 1) + ":")
for _ in range(J):
prime = True
while prime:
jamcoin = format(current, 'b')
current += 2
prime = False
proof = []
for base in range(2, 11):
interpretation = int(jamcoin, base)
if isprime(interpretation):
prime = True
break
else:
proof.append(str(primefactors(interpretation)[0]))
print(jamcoin, " ".join(proof))
def compute_min_length():
cur_value = 1
num_primes = 0
for spiral_radius in itertools.count(1):
gap_between_diagonals = 2 * spiral_radius
# For each of the first three corners
for _ in range(3):
cur_value += gap_between_diagonals
if isprime(cur_value):
num_primes += 1
# Ignore the bottom right corner
cur_value += gap_between_diagonals
# There are 4 corners in each spiral, and one center number
prime_ratio_of_diagonal = num_primes / (4 * spiral_radius + 1)
if prime_ratio_of_diagonal < 0.1:
return spiral_radius * 2 + 1
def number_replacement(combo, digit_dict, prime_set):
prime_list = []
not_prime = 0
for digit in range(0,10):
test_dict = dict(digit_dict)
for pos in combo:
test_dict[pos] = digit
new_dig = int(''.join([str(x) for x in test_dict.values()]))
#make sure the number hasn't been shortened by a leading zero
if len(str(new_dig)) == len(digit_dict.keys()):
if isprime(new_dig):
prime_list.append(new_dig)
else:
not_prime += 1
if not_prime > (10 - prime_set):
return []
return prime_list
def quadratic_primes(max_coefficient):
'''Calculate the product of the coefficients that generate the most primes
for a sequence of n starting with n=0 of the form:
n^2 + an + b
for |a| < max_coefficient and |b| < max_coefficient.
'''
b_lst = list(sieve.primerange(1, max_coefficient))
max_n = 0
for a in range(-max_coefficient + 1, max_coefficient):
for b in b_lst:
n = 0
while isprime(n**2 + a * n + b):
n += 1
if n > max_n:
max_n = n
max_product = a * b
return max_product
def __init__(self, p, g, y, x, name="HonestProver"):
"""
Initializes the prover. Checks the validity of the arguments, sets the public parameters p, g, y and sets the
secret x.
This method has to initialize instance variables p, g, y and x (r will remain unset until compute_c method
is called).
:param p: integer, modulus
:param g: integer, base
:param y: integer, g^x mod p
:param x: integer, secret
:param name: optional string, name of the prover (to be used when printing data to the console)
"""
self.p = None
self.g = None
self.y = None
self.x = None
self.r = None
self.name = name
try:
assert (isprime(p))
assert (y < p)
assert (is_primitive_root(g, p))
if x is not None:
assert (pow(g, x, p) == y)
except:
raise InvalidParams
self.p = p
self.g = g
self.y = y
self.x = x
print_debug("{}:\tInitialized with p = {}, g = {}, y = {} and x = {}".format(
self.name, self.p, self.g, self.y, self.x), LOG_INFO)
def __init__(self, p, n=None):
if n is None:
L = primeFact(p)
if len(set(L)) > 1:
raise ValueError('order must be a prime power')
p = L[0]
n = len(L)
p, n = int(p), int(n)
if not isprime(p):
raise ValueError("p must be a prime number, not %s" % p)
if n <= 0:
raise ValueError("n must be a positive integer, not %s" % n)
self.p = p
self.n = n
if n == 1:
self.reducing = [1, 0]
else:
for c in itertools.product(range(p), repeat=n):
poly = (1, *c)
if gf_irreducible_p(poly, p, ZZ):
self.reducing = poly
break
self.iterpos = 0
def solve_congruence(*remainder_modulus_pairs, **hint):
"""Compute the integer ``n`` that has the residual ``ai`` when it is
divided by ``mi`` where the ``ai`` and ``mi`` are given as pairs to
this function: ((a1, m1), (a2, m2), ...). If there is no solution,
return None. Otherwise return ``n`` and its modulus.
The ``mi`` values need not be co-prime. If it is known that the moduli are
not co-prime then the hint ``check`` can be set to False (default=True) and
the check for a quicker solution via crt() (valid when the moduli are
co-prime) will be skipped.
If the hint ``symmetric`` is True (default is False), the value of ``n``
will be within 1/2 of the modulus, possibly negative.
Examples
========
>>> from sympy.ntheory.modular import solve_congruence
What number is 2 mod 3, 3 mod 5 and 2 mod 7?
>>> solve_congruence((2, 3), (3, 5), (2, 7))
(23, 105)
>>> [23 % m for m in [3, 5, 7]]
[2, 3, 2]
If you prefer to work with all remainder in one list and
all moduli in another, send the arguments like this:
>>> solve_congruence(*zip((2, 3, 2), (3, 5, 7)))
(23, 105)
The moduli need not be co-prime; in this case there may or
may not be a solution:
>>> solve_congruence((2, 3), (4, 6)) is None
True
>>> solve_congruence((2, 3), (5, 6))
(5, 6)
The symmetric flag will make the result be within 1/2 of the modulus:
>>> solve_congruence((2, 3), (5, 6), symmetric=True)
(-1, 6)
See Also
========
crt : high level routine implementing the Chinese Remainder Theorem
"""
def combine(c1, c2):
"""Return the tuple (a, m) which satisfies the requirement
that n = a + i*m satisfy n = a1 + j*m1 and n = a2 = k*m2.
References
==========
- http://en.wikipedia.org/wiki/Method_of_successive_substitution
"""
a1, m1 = c1
a2, m2 = c2
a, b, c = m1, a2 - a1, m2
g = reduce(igcd, [a, b, c])
a, b, c = [i//g for i in [a, b, c]]
if a != 1:
inv_a, _, g = igcdex(a, c)
if g != 1:
return None
b *= inv_a
a, m = a1 + m1*b, m1*c
return a, m
rm = remainder_modulus_pairs
symmetric = hint.get('symmetric', False)
if hint.get('check', True):
rm = [(as_int(r), as_int(m)) for r, m in rm]
# ignore redundant pairs but raise an error otherwise; also
# make sure that a unique set of bases is sent to gf_crt if
# they are all prime.
#
# The routine will work out less-trivial violations and
# return None, e.g. for the pairs (1,3) and (14,42) there
# is no answer because 14 mod 42 (having a gcd of 14) implies
# (14/2) mod (42/2), (14/7) mod (42/7) and (14/14) mod (42/14)
# which, being 0 mod 3, is inconsistent with 1 mod 3. But to
# preprocess the input beyond checking of another pair with 42
# or 3 as the modulus (for this example) is not necessary.
uniq = {}
for r, m in rm:
r %= m
if m in uniq:
if r != uniq[m]:
return None
continue
uniq[m] = r
rm = [(r, m) for m, r in uniq.iteritems()]
del uniq
# if the moduli are co-prime, the crt will be significantly faster;
# checking all pairs for being co-prime gets to be slow but a prime
# test is a good trade-off
if all(isprime(m) for r, m in rm):
r, m = zip(*rm)
return crt(m, r, symmetric=symmetric, check=False)
rv = (0, 1)
for rmi in rm:
rv = combine(rv, rmi)
if rv is None:
break
n, m = rv
n = n % m
else:
if symmetric:
return symmetric_residue(n, m), m
return n, m
def test_isprime():
s = Sieve()
s.extend(100000)
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)
assert isprime(20678048681)
assert isprime(1968188556461)
assert isprime(2614941710599)
assert isprime(65635624165761929287)
assert isprime(1162566711635022452267983)
assert isprime(77123077103005189615466924501)
assert isprime(3991617775553178702574451996736229)
assert isprime(273952953553395851092382714516720001799)
assert isprime(int('''
531137992816767098689588206552468627329593117727031923199444138200403\
559860852242739162502265229285668889329486246501015346579337652707239\
409519978766587351943831270835393219031728127'''))
# Some Mersenne primes
assert isprime(2**61 - 1)
assert isprime(2**89 - 1)
assert isprime(2**607 - 1)
# (but not all Mersenne's are primes
assert not isprime(2**601 - 1)
# pseudoprimes
#-------------
# to some small bases
assert not isprime(2152302898747)
assert not isprime(3474749660383)
assert not isprime(341550071728321)
assert not isprime(3825123056546413051)
# passes the base set [2, 3, 7, 61, 24251]
assert not isprime(9188353522314541)
# large examples
assert not isprime(877777777777777777777777)
# conjectured psi_12 given at http://mathworld.wolfram.com/StrongPseudoprime.html
assert not isprime(318665857834031151167461)
# conjectured psi_17 given at http://mathworld.wolfram.com/StrongPseudoprime.html
assert not isprime(564132928021909221014087501701)
# Arnault's 1993 number; a factor of it is
# 400958216639499605418306452084546853005188166041132508774506\
# 204738003217070119624271622319159721973358216316508535816696\
# 9145233813917169287527980445796800452592031836601
assert not isprime(int('''
803837457453639491257079614341942108138837688287558145837488917522297\
427376533365218650233616396004545791504202360320876656996676098728404\
396540823292873879185086916685732826776177102938969773947016708230428\
687109997439976544144845341155872450633409279022275296229414984230688\
1685404326457534018329786111298960644845216191652872597534901'''))
# Arnault's 1995 number; can be factored as
# p1*(313*(p1 - 1) + 1)*(353*(p1 - 1) + 1) where p1 is
# 296744956686855105501541746429053327307719917998530433509950\
# 755312768387531717701995942385964281211880336647542183455624\
# 93168782883
assert not isprime(int('''
288714823805077121267142959713039399197760945927972270092651602419743\
230379915273311632898314463922594197780311092934965557841894944174093\
380561511397999942154241693397290542371100275104208013496673175515285\
922696291677532547504444585610194940420003990443211677661994962953925\
045269871932907037356403227370127845389912612030924484149472897688540\
6024976768122077071687938121709811322297802059565867'''))
sieve.extend(3000)
assert isprime(2819)
assert not isprime(2931)
assert not isprime(2.0)