recreational Recreation is an activity of leisure, leisure being discretionary time. The "need to do something for recreation" is an essential element of human biology and psychology. Recreational activities are often done for enjoyment, amusement, or pleasu ...

number theory Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integers and integer-valued functions. German mathematician Carl Friedrich Gauss (1777–1855) said, "Math ...

, a primeval number is a

natural number In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country"). Numbers used for counting are called '' cardinal ...

''n'' for which the number of

prime number A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only way ...

s 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 The decimal numeral system (also called the base-ten positional numeral system and denary or decanary) is the standard system for denoting integer and non-integer numbers. It is the extension to non-integer numbers of the Hindu–Arabic numeral ...

) 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

base 12 The duodecimal system (also known as base 12, dozenal, or, rarely, uncial) is a positional notation numeral system using twelve as its base. The number twelve (that is, the number written as "12" in the decimal numerical system) is instead wr ...

, 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.

External links

* Chris Caldwell
The Prime Glossary: Primeval number
at The

Prime Pages The PrimePages is a website about prime numbers maintained by Chris Caldwell at the University of Tennessee at Martin. The site maintains the list of the "5,000 largest known primes", selected smaller primes of special forms, and many "top twenty" ...

* Mike Keith
''Integers Containing Many Embedded Primes''
{{Prime number classes Base-dependent integer sequences Prime numbers

Base 12

See also

External links