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# Pandigital primes

A base ${\displaystyle \scriptstyle b\,}$ pandigital prime ${\displaystyle \scriptstyle p\,}$ is a prime number that is also a pandigital number in the given base. The smallest base ${\displaystyle \scriptstyle b\,}$ pandigital prime usually starts with the string "1012" rather than "102" for the smallest pandigital composite number, after which follows a concatenation of the other base ${\displaystyle \scriptstyle b\,}$ digits more or less in order (since having exactly one instance of each digit probably means the number is divisible by ${\displaystyle \scriptstyle b-1\,}$, at least when ${\displaystyle \scriptstyle b\,}$ is even.)

Sequences of pandigital primes base ${\displaystyle \scriptstyle b\,}$
${\displaystyle b\,}$ A-number Sequence
2 A138837 {2, 5, 11, 13, 17, 19, 23, 29, 37, 41, 43, 47, 53, 59, 61, ...}
3   Have not looked yet
4   Have not looked yet
5 A175277 {3319, 3323, 3347, 3469, 3491, 3539, 3547, 3559, 3571, 3607, 3613, 3617, 3691, 3823, 3847, 3863, 4019, ...}
6   Have not looked yet
7   Have not looked yet
8 A175271 {17119607, 17120573, 17121077, 17127839, 17128931, 17132347, 17135413, ...}
9   Have not looked yet
10 A050288 {10123457689, 10123465789, 10123465897, 10123485679, 10123485769, ...}

The base doesn't have to get too large for pandigital primes to get quite large. By most standards, 36 is not a terribly large number, yet the smallest pandigital prime in base 36 is already larger than the largest known Mersenne prime prior to 1952.[1] I have searched in vain for a compact way to express these numbers in terms of addition of powers of smaller numbers. However, the nature of these numbers provides a very efficient shortcut: for example, the smallest base 36 pandigital prime is 10123...STUXYZWV (the ellipsis omitting the predictable digits 4, 5, 6, 7, etc., up to P, Q, R).

## Notes

1. The base 2 logarithm of the number in question is approx. 186.11847728172409366, whereas the base 2 logarithm of the mentioned Mersenne prime is almost 127.0.