def solve():
primes = 0
for i, n in enumerate(spiral(100000)):
if is_prime(n):
primes += 1
if primes / (i + 1) < 0.1 and i % 4 == 0 and primes > 0:
return i // 2 + 1
def pandigitals():
# pandigitals of length 8 or 9 are divisible by 3
for length in range(1, 8):
for perm in permutations('1234567'[:length]):
n = int(''.join(perm))
if is_prime(n):
yield n
def is_goldbach(n):
return any(is_prime(n - 2 * i**2) for i in range(1, int(n**0.5)))
def solve():
return next(n for n in count(3, 2) if not is_prime(n)
if not is_goldbach(n))
def is_goldbach(n):
return any(is_prime(n - 2 * i ** 2) for i in range(1, int(n ** 0.5)))
def solve():
return next(n for n in count(3, 2)
if not is_prime(n)
if not is_goldbach(n))
