def verify_all_params(self) -> bool:
"""
verify all parameters including p, q, r, g
:return: True if all parameters are verified to fit in designated equations or have specific values,
False otherwise
"""
error = self.initialize_error()
# check if p and q are the expected values
if not number.equals(self.large_prime, self.LARGE_PRIME_EXPECTED):
# if not, use Miller-Rabin algorithm to check the primality of p and q, 5 iterations by default
if not number.is_prime(self.large_prime):
error = self.set_error()
print("Large prime value error. ")
if not number.equals(self.small_prime, self.SMALL_PRIME_EXPECTED):
if not number.is_prime(self.small_prime):
error = self.set_error()
print("Small prime value error. ")
# get basic parameters
cofactor = self.param_g.get_cofactor()
# check equation p - 1 = qr
if not number.equals(self.large_prime - 1, self.small_prime * cofactor):
error = self.set_error()
print("p - 1 does not equals to r * q.")
# check q is not a divisor of r
if number.is_divisor(self.small_prime, cofactor):
error = self.set_error()
print("q is a divisor of r.")
# check 1 < g < p
if not number.is_within_range(self.generator, 1, self.large_prime):
error = self.set_error()
print("g is not in the range of 1 to p. ")
# check g^q mod p = 1
if not number.equals(pow(self.generator, self.small_prime, self.large_prime), 1):
error = self.set_error()
print("g^q mod p does not equal to 1. ")
# print out message
output = "Baseline parameter check"
if error:
output += " failure. "
else:
output += " success. "
print(output)
return not error
def F_valid(factors):
ps = powerset(factors, 1, len(factors))
for x in ps:
f = product(x)+1
if not f in (2, 3, 5, 23, 29) and is_prime(f, trials=10):
return False
return True
def test_is_prime():
unittest.TestCase.assertEqual(number.is_prime(0), True)
unittest.TestCase.assertEqual(number.is_prime(1), True)
unittest.TestCase.assertEqual(number.is_prime(2), True)
unittest.TestCase.assertEqual(number.is_prime(3), False)
unittest.TestCase.assertEqual(number.is_prime(19), False)
unittest.TestCase.assertEqual(number.is_prime(-6), False)
def S(n, d):
s = 0
# Test first factor (p^2-1)/(p-1) = p+1, so d|p+1, p < 10^11
for f in range(2*d, n, 2*d): # Test even f
if is_prime(f-1):
if (f-1)%1000 == 727: print(f-1, 1)
# Multiples of f-1
pass
# Test rest of factors
primes = sieve(int(n**0.5)+1)
for a_1 in range(2, 40):
for p_1 in primes:
if p_1 < n**(1/a_1):
if (p_1**(a_1+1)-1)//(p_1-1) % d == 0:
print(p_1, a_1)
def S(n, d):
s = 0
found_M = False
M = n
other_digits = [r for r in range(10) if r != d]
while not found_M:
for non_rep in rep_combo(other_digits, n-M):
digits = non_rep + (d,)*M
for perm in unique_permutations(digits):
if perm[0] == 0: continue # Skip nums with leading 0s
p = 0
for x in perm: p = 10*p + x
if is_prime(p):
print(p)
s += p
found_M = True
M -= 1
return s
def test_is_prime_4(self):
self.assertFalse(number.is_prime(4))
def test_is_prime_2(self):
self.assertTrue(number.is_prime(2))
def test_is_prime_7(self):
self.assertTrue(number.is_prime(7))
def test_is_prime():
assert number.is_prime(2)
assert number.is_prime(7)
assert number.is_prime(11)
assert not number.is_prime(10)
assert not number.is_prime(9)
from number import is_prime
s, n = 2, 1
a = 0
for i in range(10**5):
if is_prime(n): a += 1
if i%1000 == 0: print(s, a/(i+1))
if i%4 == 0 and a/(i+1) < 0.10 and s > 7:
print(s+1)
break
if i%4 == 0 and i != 0: s += 2
n += s
import itertools, number
l = 0
for i in range(7, 1, -1): # 9 and 8 digit can't be prime
for perm in itertools.permutations(range(1, i+1), i):
n = int(''.join(map(str, perm)))
if number.is_prime(n):
l = max(l, n)
print(l)
def test_case(self):
self.prime = number.is_prime(4)
self.assertTrue(self.prime,msg="Is not prime!")
# Derivations (mathblog.dk) # n^3 + n^2p = k^3 <=> n cbrt(p + n) / cbrt(n) = k # n = x^3, p + n = y^3; p = y^3 - x^3 = (y-x)(y^2 + yx + x^2) # Since p has 2 factors, y-x = 1 # (i+1)^3 - i^3 < 1000000, i < 577 from number import is_prime print(sum(is_prime((i+1)**3 - i**3) for i in range(1, 577)))
# Brute force, combine together all primes with unique digits
from itertools import permutations
from number import is_prime
primes = []
prime_digits = []
for l in range(1, 10):
for perm in permutations(range(1, 10), l):
p = 0
for x in perm: p = 10*p + x
if is_prime(p):
primes.append(p)
prime_digits.append(perm)
prime_sets = 0
# pool is an array of bools, with pool[d] representing if d has been used
def create_set(i, pool):
if all(pool):
global prime_sets
prime_sets += 1
digits_left = pool.count(False)
for j in range(i+1, len(primes)):
if len(prime_digits[j]) > digits_left: break
if not any(pool[d] for d in prime_digits[j]):
for d in prime_digits[j]: pool[d] = True
create_set(j, pool)
# Cyclic number (10^(p-1)-1)/p = 00000000137...56789
# p*56789 = ...99999, p = ...09891
from number import is_prime
def starts137(p):
return str((10**30 - 1) / p)[:3] == "137" # Approximate 999... / p
for p in range(9891, 10**10, 100000):
if(is_prime(p)) and starts137(p):
print(p)