Editing Primeval number

{{Short description|Type of natural number in recreational number theory}}
In [[recreational mathematics|recreational]] [[number theory]], a '''primeval number''' is a [[natural number]] ''n'' for which the number of [[prime number]]s which can be obtained by [[permutation|permuting]] some or all of its [[Numerical digit|digit]]s (in [[Decimal|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 (mathematician)|Mike Keith]].

The first few primeval numbers are
:[[1]], [[2]], [[13 (number)|13]], [[37 (number)|37]], [[107 (number)|107]], [[113 (number)|113]], [[137 (number)|137]], 1013, 1037, 1079, 1237, 1367, 1379, 10079, 10123, 10136, 10139, 10237, 10279, 10367, 10379, 12379, 13679, ... {{OEIS|id=A072857}}
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, ... {{OEIS|id=A076497}}
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, ... {{OEIS|id=A076730}}
The smallest ''n''-digit number to achieve this number of primes is
:2, 37, 137, 1379, 13679, 123479, 1234679, 12345679, 102345679, 1123456789, 10123456789, ... {{OEIS|id=A134596}}

Primeval numbers can be [[Composite number|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, ... {{OEIS|id=A119535}}

The following table shows the first seven primeval numbers with the obtainable primes and the number of them.

{|class="wikitable"
! 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
|}

==Base 12==

In [[duodecimal|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, ...

{|class="wikitable"
! Primeval number !! Primes obtained !! Number of primes (written in base 10)
|-
| 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.

== See also ==
*[[Permutable prime]]
*[[Truncatable prime]]

== External links ==
* Chris Caldwell, [http://primes.utm.edu/glossary/page.php?sort=Primeval The Prime Glossary: Primeval number] at The [[Prime Pages]]
* [[Mike Keith (mathematician)|Mike Keith]], [http://www.cadaeic.net/primeval.htm ''Integers Containing Many Embedded Primes'']

{{Classes of natural numbers}}
{{Prime number classes}}

[[Category:Base-dependent integer sequences]]
[[Category:Prime numbers]]