def is_truncatable_prime(number):
if not is_prime(number):
return False
number_string = str(number)
number_length = len(number_string)
if number_length < 2:
return False
for i in range(1, number_length):
if (not is_prime(int(number_string[i:]))
or not is_prime(int(number_string[:i]))):
return False
return True
def nth_prime_number(number):
count = 0
n = 1
while count < number:
n += 1
if is_prime(n):
count += 1
return n
def spiral_primes_at_ratio(ratio):
length = 1
diagonal_number = 1
diagonal_primes = 0
while length == 1 or diagonal_primes / ((2 * length) - 1) > ratio:
length += 2
for _ in range(4):
diagonal_number += length - 1
if is_prime(diagonal_number):
diagonal_primes += 1
return length
def maximum_quadratic_primes(maximum):
minimum = -maximum
result = (0, 0)
maximum_consecutive_primes = 0
for a in range(minimum + 1, maximum):
for b in range(max(0, 1 - a), maximum + 1):
n = 0
while is_prime(n**2 + (a * n) + b):
n += 1
if n > maximum_consecutive_primes:
result = (a, b)
maximum_consecutive_primes = n
return result
def smallest_non_goldbach_number():
primes = [2]
n = 1
while True:
n += 2
if is_prime(n):
primes.append(n)
else:
is_goldbach = False
for p in primes:
if sqrt((n - p) / 2).is_integer():
is_goldbach = True
break
if not is_goldbach:
return n
def prime_digit_replacement_family(family_size):
primality_map = {}
number = 0
while True:
number_string = str(number)
digit_counts = sorted(
[(int(digit), len(list(group)))
for (digit, group) in groupby(number_string)],
key=itemgetter(1),
reverse=True,
)
largest_count = digit_counts[0][1]
digits_to_replace = (digit_count[0] for digit_count in digit_counts
if digit_count[1] == largest_count)
for digit in digits_to_replace:
family = (int(''.join(
[str(n) if int(d) == digit else d for d in number_string]))
for n in range(10))
family_primality_pre_check = (
p for p in family
if p not in primality_map or primality_map[p])
family_primality = [(n, is_prime(n))
for n in family_primality_pre_check
if len(str(n)) == len(number_string)]
prime_family = [n[0] for n in family_primality if n[1]]
for (n, primality) in family_primality:
primality_map[n] = primality
if len(prime_family) >= family_size:
return prime_family
number += 1