Primeval number
In mathematics, a primeval number is a natural number n for which the number of prime numbers which can be obtained by permuting some or all of its digits is larger than the number of primes obtainable in the same way for any smaller natural number. Primeval numbers were first described by Mike Keith.
The first few primeval numbers are
The number of primes that can be obtained from the primeval numbers is
The largest number of primes that can be obtained from a primeval number with n digits is
The smallest n-digit number to achieve this number of primes is
Primeval numbers can be composite. The first is 1037 = 17×61. A Primeval prime is a primeval number which is also a prime number:
The following table shows the first seven primeval numbers with the obtainable primes and the number of them.
| Primeval number | Primes obtained | Number of primes |
| 1 | 0 | |
| 2 | 2 | 1 |
| 13 | 3, 13, 31 | 3 |
| 37 | 3, 7, 37, 73 | 4 |
| 107 | 7, 17, 71, 107, 701 | 5 |
| 113 | 3, 11, 13, 31, 113, 131, 311 | 7 |
| 137 | 3, 7, 13, 17, 31, 37, 71, 73, 137, 173, 317 | 11 |
In base 12, the primeval numbers are:
The number of primes that can be obtained from the primeval numbers is:
| Primeval number | Primes obtained | Number of primes |
| 1 | 0 | |
| 2 | 2 | 1 |
| 13 | 3, 31 | 2 |
| 15 | 5, 15, 51 | 3 |
| 57 | 5, 7, 57, 75 | 4 |
| 115 | 5, 11, 15, 51, 511 | 5 |
| 117 | 7, 11, 17, 117, 171, 711 | 6 |
| 125 | 2, 5, 15, 25, 51, 125, 251 | 7 |
| 135 | 3, 5, 15, 31, 35, 51, 315, 531 | 8 |
| 157 | 5, 7, 15, 17, 51, 57, 75, 157, 175, 517, 751 | 11 |
Note that 13, 115 and 135 are composite: 13 = 3×5, 115 = 7×1Ɛ, and 135 = 5×31.