def search(goal):
"""
>>> search(6)
13
>>> search(7)
56003
"""
already_searched = set()
for prime in primes():
if prime in already_searched:
continue
positions_list = list(position_combinations(prime))
for positions in positions_list:
digits = list(digits_of(prime))
family = set()
for digit in range(10):
if 0 in positions and digit == 0:
continue
for pos in positions:
digits[pos] = digit
number = from_digits(digits)
if is_prime(number):
already_searched.add(number)
family.add(number)
if len(family) >= goal:
return min(family)
def odd_composites():
"""
>>> list(up_to(33, odd_composites()))
[9, 15, 21, 25, 27, 33]
"""
for odd in islice(count(3), None, None, 2):
if not is_prime(odd):
yield odd
def prime_sqrt(n):
"""
>>> prime_sqrt(4)
(2, 2)
>>> prime_sqrt(1000)
(31, 37)
"""
if n < 4:
raise ValueError(n)
s = math.sqrt(n)
lps = math.floor(s)
while not is_prime(lps):
lps -= 1
ups = math.ceil(s)
while not is_prime(ups):
ups += 1
return (lps, ups)
def prime_sequence_length(coefficients):
"""
Find the length of the prime sequence generated by a polynomial.
>>> prime_sequence_length((1, 1, 41))
40
>>> prime_sequence_length((1, -79, 1601))
80
"""
for n in itertools.count():
if not is_prime(eval_polynomial(coefficients, n)):
return n
def prime_sequence_sums(limit):
prime_list = []
for p in primes():
prime_list.append(p)
if sum(prime_list) >= limit:
return
for i in range(len(prime_list)):
sequence = prime_list[i:]
if len(sequence) <= 1:
continue
s = sum(sequence)
if is_prime(s):
yield sequence, s
def RSAKeyGeneration(socket, option):
while(True):
p = secrets.randbits(256)
if(util.is_prime(p)):
break
while(True):
q = secrets.randbits(256)
if(util.is_prime(q) and p!=q):
break
N = p*q
carmichael = util.lcm(p-1,q-1)
e = 65537
d = float(1+carmichael)/e
k = 1
while(not d.is_integer()):
d = float(1+k*carmichael)/e
k += 1
d = int(d)
socket.send(str(e).encode())
e = int(socket.recv(1024).decode())
socket.send(str(N).encode())
n = int(socket.recv(1024).decode())
return d, n, e, N
from utility import is_prime
n = abs(int(input('Enter a number: ')))
print('The number is prime!' if is_prime(n) else 'The number is not prime!')
import itertools
from utility import digits_of, from_digits, is_prime
# We only need to search 4 and 7 digit pandigital numbers for primality. For
# other sizes of numbers, their digital roots show that they are always
# divisible by 3 (http://en.wikipedia.org/wiki/Digital_root).
# 1+2+3+4+5+6+7+8+9 = 45, 4+5 = 9
# 1+2+3+4+5+6+7+8 = 36, 3+6 = 9
# etc.
def digit_permutations(n):
for digits in itertools.permutations(digits_of(n)):
yield from_digits(digits)
print(max(p for p in digit_permutations(1234567) if is_prime(p)))
41 20 7 8 9 10 27
42 21 22 23 24 25 26
43 44 45 46 47 48 49
>>> import itertools
>>> list(itertools.islice(spiral_corners(), 4))
[(1,), (3, 5, 7, 9), (13, 17, 21, 25), (31, 37, 43, 49)]
"""
size = 0
n = 1
yield (n,)
while True:
size += 2
nums = []
for _ in range(4):
n += size
nums.append(n)
yield tuple(nums)
if __name__ == '__main__':
total = 0
prime_count = 0
for i, nums in enumerate(spiral_corners()):
for n in nums:
total += 1
if is_prime(n):
prime_count += 1
if i > 0 and prime_count * 10 < total:
print(2 * i + 1)
break
