problem 49

"""
The arithmetic sequence, 1487, 4817, 8147, in which each of the terms increases by 3330, is unusual in two ways: (i) each of the three terms are prime, and, (ii) each of the 4-digit numbers are permutations of one another.

There are no arithmetic sequences made up of three 1-, 2-, or 3-digit primes, exhibiting this property, but there is one other 4-digit increasing sequence.

What 12-digit number do you form by concatenating the three terms in this sequence?
"""
from itertools import permutations

def all_primes(upper_bound):
  primes  = [2]
  for i in range(3, upper_bound):
    j = 0
    while j < len(primes) and primes[j] <= int(i ** 0.5):
      if i % primes[j] == 0:
        break
      j += 1
    else:
      primes.append(i)
  return primes

def is_prime(x, primes):
  sqrt = int(x ** 0.5) + 1
  for prime in primes:
    if x % prime == 0:
      return False
    if prime == primes[-1]:
      return False
    if prime > sqrt:
      return True
  return True

primes = []
primes = [x for x in all_primes(10000) if x > 1487]

def all_priems(b):
  for i in range(len(b)):
    for j in range(i + 1, len(b)):
        last_num = b[j] - b[i]
        if b[j] + last_num in b:
          return str(b[i]) + str(b[j]) + str(b[j] + last_num)
  return False

for i in primes:
  perm = permutations(str(i))
  all_num = [int(''.join(x)) for x in perm]
  all_num = list(set([x for x in all_num if x in primes]))
  if all_priems(all_num):
    print (all_priems(all_num))
    break