In mathematics

Mathematics

Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, a primeval number is a natural number

Natural number

In mathematics, the natural numbers are the ordinary whole numbers used for counting and ordering . These purposes are related to the linguistic notions of cardinal and ordinal numbers, respectively...

 n for which the number of prime number

Prime number

A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. A natural number greater than 1 that is not a prime number is called a composite number. For example 5 is prime, as only 1 and 5 divide it, whereas 6 is composite, since it has the divisors 2...

s which can be obtained by permuting

Permutation

In mathematics, the notion of permutation is used with several slightly different meanings, all related to the act of permuting objects or values. Informally, a permutation of a set of objects is an arrangement of those objects into a particular order...

 some or all of its digit

Numerical digit

A digit is a symbol used in combinations to represent numbers in positional numeral systems. The name "digit" comes from the fact that the 10 digits of the hands correspond to the 10 symbols of the common base 10 number system, i.e...

s (in base 10

Decimal

The decimal numeral system has ten as its base. It is the numerical base most widely used by modern civilizations....

) 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

Mike Keith (mathematician)

Mike Keith is an American mathematician, software engineer, and author of works of constrained writing....

.

The first few primeval numbers are

1, 2, 13, 37, 107, 113, 137, 1013, 1037, 1079, 1237, 1367, ...

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, ...

The largest number of primes that can be obtained from a primeval number with n digits is

1, 4, 11, 31, 106, ...

The smallest n-digit prime to achieve this number of primes is

2, 37, 137, 1379, 13679, ...

Primeval numbers can be composite

Composite number

A composite number is a positive integer which has a positive divisor other than one or itself. In other words a composite number is any positive integer greater than one that is not a prime number....

. 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, ...

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

Primeval number	Primes obtained	Number of primes
1	none	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

External links

• Chris Caldwell, The Prime Glossary: Primeval number at The Prime Pages
  Prime pages
  The Prime Pages 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" lists for primes of various forms...
  
  
• Mike Keith
  Mike Keith (mathematician)
  Mike Keith is an American mathematician, software engineer, and author of works of constrained writing....
  
  , Integers Containing Many Embedded Primes