Problem Description

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?

Analysis

What is the value of n? We know it’s between 4 and 9 but that leaves a huge set of prime number candidates to search through. So let’s eliminate some values of n by using the rule that any integer is divisible by 3 or 9 whose sum of digits is divisible by 3 or 9; therefore composite and not prime.

9+8+7+6+5+4+3+2+1 = 45,
8+7+6+5+4+3+2+1 = 36,
6+5+4+3+2+1 = 21, and
5+4+3+2+1 = 15,
all of which are divisible by 3 and therefore could not yield a 1 to {5, 6, 8, 9} pandigital prime. So that leaves 4 and 7 digit prime numbers less than 4321 and 7654321 respectively. Since we want the largest pandigital prime we’ll check the 7 digit group first.

Solution

Runs < 1 second in Python.

from Euler import is_prime, is_pandigital
 
n = 7654321
while not(is_prime(n) and is_pandigital(n, 7)):
  n-=2
print "Answer to PE41 = ", n

Comments

• BTW, the largest 4 digit pandigital prime is 4231.
• More information on the Euler module can be found on the tools page.