Type of natural number in recreational number theory
Inrecreationalnumber theory, aprimeval number is anatural numbern for which the number ofprime numbers which can be obtained bypermuting some or all of itsdigits (inbase 10) is larger than the number of primes obtainable in the same way for any smaller natural number. Primeval numbers were first described byMike 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, ... (sequenceA072857 in theOEIS)
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, ... (sequenceA076497 in theOEIS)
The largest number of primes that can be obtained from a primeval number withn digits is
- 1, 4, 11, 31, 106, 402, 1953, 10542, 64905, 362451, 2970505, ... (sequenceA076730 in theOEIS)
The smallestn-digit number to achieve this number of primes is
- 2, 37, 137, 1379, 13679, 123479, 1234679, 12345679, 102345679, 1123456789, 10123456789, ... (sequenceA134596 in theOEIS)
Primeval numbers can becomposite. The first is 1037 = 17×61. APrimeval 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, ... (sequenceA119535 in theOEIS)
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 |
Inbase 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, ...
| 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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Possessing a specific set of other numbers |
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Expressible via specific sums |
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| By formula | |
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| By integer sequence | |
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| By property | |
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| Base-dependent | |
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| Patterns | | k-tuples | - Twin (p,p + 2)
- Triplet (p,p + 2 orp + 4,p + 6)
- Quadruplet (p,p + 2,p + 6,p + 8)
- Cousin (p,p + 4)
- Sexy (p,p + 6)
- Arithmetic progression (p + a·n,n = 0, 1, 2, 3, ...)
- Balanced (consecutivep − n,p,p + n)
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| By size | |
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| Complex numbers | |
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| Composite numbers | |
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| Related topics | |
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| First 60 primes | |
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