def is_right_truncatable(l):
is_truncatable = 1
for size in xrange(0, len(l)):
n = num(l[size:])
prime._refresh(int(math.sqrt(n)))
if not prime._isprime(n):
is_truncatable = 0
break
return is_truncatable
def is_left_truncatable(l):
is_truncatable = 1
for size in xrange(1, len(l)+1):
n = num(l[:size])
prime._refresh(int(math.sqrt(n)))
if not prime._isprime(n):
is_truncatable = 0
break
return is_truncatable
'''
We shall say that an n-digit number is pandigital if it makes use of all the digits 1 to n exactly once. For example, 2143 is a 4-digit pandigital and is also prime.
What is the largest n-digit pandigital prime that exists?
'''
import prime
from combinatorics import permutations
# Pan-digital primes are 4 or 7 digits. Others divisible by 3
prime._refresh(2766) # sqrt(7654321)
for perm in permutations(range(7, 0, -1)):
num = 0
for n in perm: num = num * 10 + n
if prime._isprime(num):
print num
break
import prime print prime.prime(11) print prime._isprime(32) print prime.isprime(11)
'''
Starting with 1 and spiralling anticlockwise in the following way, a square spiral with side length 7 is formed.
37 36 35 34 33 32 31
38 17 16 15 14 13 30
39 18 5 4 3 12 29
40 19 6 1 2 11 28
41 20 7 8 9 10 27
42 21 22 23 24 25 26
43 44 45 46 47 48 49
It is interesting to note that the odd squares lie along the bottom right diagonal, but what is more interesting is that 8 out of the 13 numbers lying along both diagonals are prime; that is, a ratio of 8/13 ~ 62%.
If one complete new layer is wrapped around the spiral above, a square spiral with side length 9 will be formed. If this process is continued, what is the side length of the square spiral for which the ratio of primes along both diagonals first falls below 10%?
'''
import prime
prime._refresh(50000)
width, diagonal, base, primes = 1, 1, 1, 0
while True:
width = width + 2
increment = width - 1
for i in xrange(0, 4):
diagonal = diagonal + increment
if i < 3 and prime._isprime(diagonal): primes += 1
base = base + 4
if primes * 10 < base:
print width
break
def concatenations_are_prime(primes):
for pair in combinations(primes, 2):
if not prime._isprime(cat(pair[0], pair[1])): return False
return True