Pandigital number

A number whose representation contains every numerical digit from a specified alphabet. The convention must specify the base, whether zero is required, and whether repetitions are allowed. In many counting problems, “0-to-9 pandigital” means that all ten decimal numerical digits occur exactly once; other problems require only that each occur at least once.

A pandigital prime is a prime number that is also a pandigital number for a specified radix, digit alphabet and repetition convention. For example, $2143$ uses each of $1,2,3,4$ exactly once and is prime; trial division by prime numbers at most its square root verifies the latter condition.

The digit sum supplies a useful obstruction. A decimal number using $1,\dots ,n$ exactly once has digit sum $n(n+1)/2$. If this sum is divisible by $3$ and the number exceeds $3$, the number is composite. In particular, a number using all ten decimal digits exactly once has digit sum $45$ and cannot be prime. The filter follows from the digit sum integer congruence, independently of the order in which the digits occur.