[Python / Project Euler]问题41

Question

I am trying to solve problem 41 , but i don't know why my code stops at 987654319:

def IsPandigital(No):
 a = sorted(str(No))
 for i in range(1,10):
  if a[i - 1] != str(i):
   return False
 return True
def IsPrime(No):
 i = 2
 while i < No:
  if No % i == 0:
   return False
  i += 1
 return True
i = 987654321
while i > 0:
 print i
 raw_input()
 if IsPrime(i) == True and IsPandigital(i) == True:
  print i
  break
 i -= 1
print i
print "EOP"
raw_input()

PS: I know that i should start from 799999999 because:

GergS: Every 9 digit and 8 digit pandigital number is divisible by 3.

Answer 1

Your IsPrime function is very slow. It quickly calculated that 987654321 is dividable by 3 and 987654320 is dividable by 2. However, 987654319 is prime and it takes very very long to check it against all divisors so it seems as if it stopped.

The problem requires more than just simple calculation, as you did. Calculating prime numbers is a slow process, so you should optimize it, for example:

• Test IsPandigital before IsPrime ,
• Even better, create a list of only pandigital numbers and make the prime test. Hint: the pandigital numbers are: [987654321,987654312,987654231,987654213,...]
• You dont have to make loop while i < No - it is enough to go up to sqrt(No) .
• Numbers ending with digit 0, 2, 4, 5, 6 or 8 are never prime
• Another option might be to calculate all n-digit prime numbers and then pick the largest pandigital number.

Your other problems:

• You are calculating the largest 9-digit pandigital prime. The problem says "the largest n-digit pandigital prime", so you should be able to calculate any length depending on a parameter
• You should not start from 799999999, as you wrote. You should simply skip calculations for n=3, n=5, n=6, n=8 and n=9 because none of those is prime.

Answer 2

The routine to find primes takes much longer than the pandigital, so switch the order that they are run. So it should be:

if IsPandigital(i) == True and IsPrime(i) == True:

Answer 3

Try using itertools.permutations to generate all pan-digital numbers then checked if they were prime. It generates all the permutations of the string passed.

Eg

itertools.permutations('9876543210')

You can narrow your search as explain above, 9876543210 is just an example.