def solve():
total = 0
# Filter out any primes that contain even numbers, as at least one rotation
# will be divisible by 2
prime_list = set(filter(lambda i: not set(str(i)).difference(["1", "3", "5", "7", "9"]),
sieve_of_atkin(10 ** 6)))
for prime in prime_list:
if are_rotations_prime(prime, prime_list):
total += 1
# +1 for 2, because it was excluded while removing numbers containing
# even digits
return total + 1
def solve():
longest_seq = 0
max_coefficients = (0, 0)
prime_list = sieve_of_atkin(87400)
# The range in which the condition |a| < 1000 is true
for a in range(-999, 0, 2):
# The range in which the condition |b| < 1000 is true
for b in range(-a, 1000, 2):
n = 1
while is_prime(n * (n + a) + b, prime_list):
n += 1
if n > longest_seq:
longest_seq = n
max_coefficients = (a, b)
return max_coefficients[0] * max_coefficients[1]
def solve():
def is_twice_square(n):
return sqrt(n / 2).is_integer()
answer = 1
found = False
primes = sieve_of_atkin(10000)
while not found:
index = 0
found = True
answer += 2
while answer >= primes[index]:
if is_twice_square(answer - primes[index]):
found = False
break
index += 1
return answer
def solve():
return sieve_of_atkin(105000)[10000]
def longest_cycle(ceiling):
# Only bother testing prime numbers, as they produce recurring cycles
return max(primes.sieve_of_atkin(ceiling), key=lambda n: len_recurring_cycle(n))