This site is supported by donations to The OEIS Foundation.

# Pandigital primes

A base **pandigital prime** is a prime number that is also a pandigital number in the given base. The smallest base 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 digits more or less in order (since having exactly one instance of each digit probably means the number is divisible by , at least when is even.)

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

- ↑ 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.