External links
- Chris Caldwell, The Prime Glossary: Primeval number at The Prime Pages
- Mike Keith, Integers Containing Many Embedded Primes
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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 digits (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
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: (using inverted two and three for ten and eleven, respectively)
The number of primes that can be obtained from the primeval numbers is: (written in base 10)
| 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.
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79 (seventy-nine) is the natural number following 78 and preceding 80.
73 (seventy-three) is the natural number following 72 and preceding 74. In English, it is the smallest natural number with twelve letters in its spelled out name.
31 (thirty-one) is the natural number following 30 and preceding 32. It is a prime number.
37 (thirty-seven) is the natural number following 36 and preceding 38.
92 (ninety-two) is the natural number following 91 and preceding 93
113 is the natural number following 112 and preceding 114.
A permutable prime, also known as anagrammatic prime, is a prime number which, in a given base, can have its digits' positions switched through any permutation and still be a prime number. H. E. Richert, who is supposedly the first to study these primes, called them permutable primes, but later they were also called absolute primes.
1000 or one thousand is the natural number following 999 and preceding 1001. In most English-speaking countries, it can be written with or without a comma or sometimes a period separating the thousands digit: 1,000.
300 is the natural number following 299 and preceding 301.
700 is the natural number following 699 and preceding 701.
2000 is a natural number following 1999 and preceding 2001.
1,000,000, or one thousand thousand, is the natural number following 999,999 and preceding 1,000,001. The word is derived from the early Italian millione, from mille, "thousand", plus the augmentative suffix -one.
100,000 (one hundred thousand) is the natural number following 99,999 and preceding 100,001. In scientific notation, it is written as 105.
A cyclic number is an integer for which cyclic permutations of the digits are successive integer multiples of the number. The most widely known is the six-digit number 142857, whose first six integer multiples are
223 is the natural number following 222 and preceding 224.
10,000,000 is the natural number following 9,999,999 and preceding 10,000,001.
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 all prime. Decimal representation is often assumed and always used in this article.
In recreational number theory, a minimal prime is a prime number for which there is no shorter subsequence of its digits in a given base that form a prime. In base 10 there are exactly 26 minimal primes:
A circular prime is a prime number with the property that the number generated at each intermediate step when cyclically permuting its (base 10) digits will be prime. For example, 1193 is a circular prime, since 1931, 9311 and 3119 all are also prime. A circular prime with at least two digits can only consist of combinations of the digits 1, 3, 7 or 9, because having 0, 2, 4, 6 or 8 as the last digit makes the number divisible by 2, and having 0 or 5 as the last digit makes it divisible by 5. The complete listing of the smallest representative prime from all known cycles of circular primes (The single-digit primes and repunits are the only members of their respective cycles) is 2, 3, 5, 7, R2, 13, 17, 37, 79, 113, 197, 199, 337, 1193, 3779, 11939, 19937, 193939, 199933, R19, R23, R317, R1031, R49081, R86453, R109297, R270343, R5794777 and R8177207, where Rn is a repunit prime with n digits. There are no other circular primes up to 1023. A type of prime related to the circular primes are the permutable primes, which are a subset of the circular primes (every permutable prime is also a circular prime, but not necessarily vice versa).
