In
recreational number theory, 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
digit
Digit may refer to:
Mathematics and science
* Numerical digit, as used in mathematics or computer science
** Hindu-Arabic numerals, the most common modern representation of numerical digits
* Digit (anatomy), the most distal part of a limb, such ...
s (in
base 10) 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
:1, 2, 13, 37, 107, 113, 137, 1013, 1037, 1079, 1237, 1367, 1379, 10079, 10123, 10136, 10139, 10237, 10279, 10367, 10379, 12379, 13679, ...
The number of primes that can be obtained from the primeval numbers is
:0, 1, 3, 4, 5, 7, 11, 14, 19, 21, 26, 29, 31, 33, 35, 41, 53, 55, 60, 64, 89, 96, 106, ...
The largest number of primes that can be obtained from a primeval number with ''n'' digits is
:1, 4, 11, 31, 106, 402, 1953, 10542, 64905, 362451, 2970505, ...
The smallest ''n''-digit number to achieve this number of primes is
:2, 37, 137, 1379, 13679, 123479, 1234679, 12345679, 102345679, 1123456789, 10123456789, ...
Primeval numbers can be
composite. The first is 1037 = 17×61. A Primeval prime is a primeval number which is also a prime number:
:2, 13, 37, 107, 113, 137, 1013, 1237, 1367, 10079, 10139, 12379, 13679, 100279, 100379, 123479, 1001237, 1002347, 1003679, 1012379, ...
The following table shows the first seven primeval numbers with the obtainable primes and the number of them.
Base 12
In
base 12, the primeval numbers are: (using inverted two and three for ten and eleven, respectively)
:1, 2, 13, 15, 57, 115, 117, 125, 135, 157, 1017, 1057, 1157, 1257, 125Ɛ, 157Ɛ, 167Ɛ, ...
The number of primes that can be obtained from the primeval numbers is: (written in base 10)
:0, 1, 2, 3, 4, 5, 6, 7, 8, 11, 12, 20, 23, 27, 29, 33, 35, ...
Note that 13, 115 and 135 are composite: 13 = 3×5, 115 = 7×1Ɛ, and 135 = 5×31.
See also
*
Permutable prime
*
Truncatable prime
In number theory, a left-truncatable prime is a prime number which, in a given base, contains no 0, and if the leading ("left") digit is successively removed, then all resulting numbers are prime. For example, 9137, since 9137, 137, 37 and 7 are ...
External links
* Chris Caldwell
The Prime Glossary: Primeval numberat The
Prime Pages
*
Mike Keith''Integers Containing Many Embedded Primes''
{{Prime number classes
Base-dependent integer sequences
Prime numbers