def largest_pandigital_prime():
maximum = 7654321 # 8- and 9-digit pandigital primes are divisible by 3
primes = sieve_of_eratosthenes(maximum)
for number in sorted(primes, reverse=True):
number_string = ''.join(sorted(str(number)))
number_length = len(number_string)
if ''.join(sorted(str(number))) == '1234567'[:number_length]:
return number
def circular_primes(maximum):
primes = sieve_of_eratosthenes(maximum)
result = set([2])
for number in range(3, maximum, 2):
if number not in primes or number in result:
continue
number_string = str(number)
rotations = [
int(number_string[i:] + number_string[:i])
for i in range(len(number_string))
]
if len([p for p in rotations if p in primes]) == len(rotations):
result.update([p for p in rotations if p < maximum])
return result
def prime_permutations(number_of_digits):
minimum = 10**(number_of_digits - 1)
maximum = 10 * minimum
primes = sieve_of_eratosthenes(maximum)
permutation_list = []
permutations_accounted_for = set()
for prime in [p for p in range(minimum, maximum) if p in primes]:
if prime in permutations_accounted_for:
continue
digit_permutations = set([
int(''.join(permutation))
for permutation in permutations(str(prime), number_of_digits)
])
prime_digit_permutations = [
p for p in digit_permutations
if p in primes and prime <= p < maximum
]
permutations_accounted_for.update(prime_digit_permutations)
permutation_list.append(sorted(prime_digit_permutations))
return [
permutation for permutation in permutation_list if len(permutation) > 2
]
def most_consecutive_prime_sum(maximum):
primes = sieve_of_eratosthenes(maximum)
prime_list = list(primes)
prime_set = set(primes)
number_of_primes_in_sum = result_sum_prime = total = 0
for (starting_index, starting_prime) in enumerate(prime_list):
next_prime_index = starting_index + number_of_primes_in_sum
total = sum(prime_list[starting_index:next_prime_index])
if total > maximum:
break
for (index, prime) in list(enumerate(prime_list))[next_prime_index:]:
if total > maximum:
break
if total in prime_set:
result_sum_prime = total
number_of_primes_in_sum = index - starting_index
total += prime
return result_sum_prime
def summation_of_primes(maximum):
return sum(sieve_of_eratosthenes(maximum))