Primeval number

In the recreational mathematics is a Primeval number (from the English Primeval Number ) is a natural number for which the number of primes that are obtained by permutation may receive some or all of its digits (ie by exchanging or omission of their digits) is greater than the number of prime numbers that can be obtained in the same way for all smaller natural numbers . ${\displaystyle n\in \mathbb {N} }$ ${\displaystyle m<n}$

The mathematician Mike Keith ( en ) was the first to deal with these numbers in 1998.

Examples

• The following prime numbers can be generated from the number by interchanging and omitting the digits: ${\displaystyle n=1379}$

3, 7, 13, 17, 19, 31, 37, 71, 73, 79, 97, 137, 139, 173, 179, 193, 197, 317, 379, 397, 719, 739, 937, 971, 1973, 3719, 3917, 7193, 9137, 9173, 9371

There are a total of 31 prime numbers that can be generated. This does not yet make the number a primeval number. You have to prove beforehand that you can not generate so many, i.e. fewer than 31 prime numbers, from all smaller natural numbers . However, this is actually the case: there is not a single number from which one can generate 31 or more prime numbers by interchanging and omitting the digits. So is a primeval number. ${\displaystyle n=1379}$ ${\displaystyle 0<m<n=1379}$ ${\displaystyle m<n=1379}$ ${\displaystyle n=1379}$

• The first 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, 100279, 100379, 101237, 102347, 102379, 103679, 123479 , 1001237, 1002347, 1002379, 1003679, 1012349, 1012379… (sequence A072857 in OEIS )

• The number of prime numbers that can be made from the above-mentioned Primeval numbers are:

0, 1, 3, 4, 5, 7, 11, 14, 19, 21, 26, 29, 31, 33, 35 , 41, 53, 55, 60, 64, 89, 96, 106, 122, 153, 188, 248, 311, 349, 402 , 421, 547, 705, 812, 906, 1098 ... (sequence A076497 in OEIS )

Example:

At the 15th position of the above two lists you can read the numbers and . This means that one can make exactly different prime numbers from the number and that there is not a single number from which one or more different prime numbers can be made. ${\displaystyle n=10123}$ ${\displaystyle 35}$ ${\displaystyle n=10123}$ ${\displaystyle 35}$ ${\displaystyle m<n=10123}$ ${\displaystyle 35}$

• The largest number of prime numbers that can be made from a -digit (primeval) number are the following: ${\displaystyle k}$ ${\displaystyle k=1,2,3,\ldots }$

1, 4, 11, 31, 106, 402 , 1953, 10542, 75447, 398100, 3605464… (sequence A076730 in OEIS )

• The corresponding -digit Primeval numbers with the largest number of prime numbers are the following (they are also the largest -digit Primeval numbers): ${\displaystyle k}$ ${\displaystyle k}$

2, 37, 137, 1379, 13679, 123479 , 1234679, 12345679, 102345679, 1123456789, 10123456789… ( continuation A134596 in OEIS )

Example:

In the 6th position of the two lists above is the number and . This means that you can make a maximum of prime numbers out of a 6-digit (primeval) number . In this case it would be the Primeval number from which different prime numbers can be made (can also be found in the 30th position of the two top lists). ${\displaystyle 402}$ ${\displaystyle 123479}$ ${\displaystyle 402}$ ${\displaystyle n=123479}$ ${\displaystyle 402}$

• The following table shows the first seven Primeval numbers and which prime numbers can be made from them:

Primeval number	prime numbers obtained therefrom	Number of prime numbers obtained in this way
1		0
2	2	1
13	3, 13, 31	3
37	3, 7, 37, 73	4th
107	7, 17, 71, 107, 701	5
113	3, 11, 13, 31, 113, 131, 311	7th
137	3, 7, 13, 17, 31, 37, 71, 73, 137, 173, 317	11

Primeval prime numbers

A primeval number , which is a prime number, is called a primeval prime number . ${\displaystyle n\in \mathbb {N} }$

properties

• The maximum number of numbers that can be obtained by permutation are (for one, two, three or more-digit numbers):

1, 4, 15, 64, 325 ...

The largest primeval number with the property that all permutations of its digits are prime numbers is the number . ${\displaystyle n=37}$

Example:

Be . This number is a three-digit Primeval number. Overall, if you permute all three digits, there are possibilities. From the table above, however, one can see that only 5 prime numbers are obtained, namely and through interchanges. There are still 10 more options missing, namely and , all of which are either not prime or have the unusual spelling with zeros in front. According to the above statement, there are no primeval numbers which are greater than for which all permutations result in prime numbers. ${\displaystyle n=107}$ ${\displaystyle {\binom {3}{1}}\cdot 1!+{\binom {3}{2}}\cdot 2!+{\binom {3}{3}}\cdot 3!=3\cdot 1+3\cdot 2+1\cdot 6=15}$ ${\displaystyle 7,17,71,107}$ ${\displaystyle 701}$ ${\displaystyle 0,1,01,07,10,70,017,071,170}$ ${\displaystyle 710}$ ${\displaystyle n=37}$

Examples

• The smallest Primeval prime is . You can only make the prime number out of it. ${\displaystyle p=2}$ ${\displaystyle p=2}$
• The smallest Primeval number that is not also prime is is . The following prime numbers can be made from it: ${\displaystyle n=1037=17\cdot 61}$

3, 7, 13, 17, 31, 37, 71, 73, 103, 107, 137, 173, 307, 317, 701, 1307, 3701, 7013, 7103

There is no smaller natural number from which one can make 19 or more prime numbers. ${\displaystyle m<n=1037}$

• The smallest Primeval primes are the following:

2, 13, 37, 107, 113, 137, 1013, 1237, 1367, 10079, 10139, 12379, 13679, 100279, 100379, 123479, 1001237, 1002347, 1003679, 1012379… ( continuation A119535 in OEIS )

Primeval numbers to other bases

So far, only Primeval numbers in the decimal system , i.e. the basis, have been dealt with. For example, the primeval number would be the base number and you could make completely different prime numbers from it as the base (in this case only the two prime numbers and ). Therefore, the respective base plays a major role in Primeval numbers. ${\displaystyle b=10}$ ${\displaystyle n=37}$ ${\displaystyle b=8}$ ${\displaystyle n=37={\underline {4}}\cdot 8^{1}+{\underline {5}}\cdot 8^{0}=45_{8}}$ ${\displaystyle b=8}$ ${\displaystyle 5_{8}=5}$ ${\displaystyle 45_{8}=37}$

A primeval number for the base ${\displaystyle b}$ is a natural number for which the number of prime numbers that can be obtained by permutation (i.e. by interchanging or omitting) some or all of its base digits is greater than the number of prime numbers that can be obtained can be obtained in the same way for all smaller natural numbers . If this number is a prime number, it is called the primeval prime number for the base . ${\displaystyle n\in \mathbb {N} }$ ${\displaystyle b}$ ${\displaystyle m<n}$ ${\displaystyle b}$

Examples

• Be . The following prime numbers can be generated from it in the decimal system, i.e. for the base , by interchanging and omitting the digits: ${\displaystyle n=117}$ ${\displaystyle b=10}$

7, 11, 17, 71

There are only four of them. But you can already generate 5 prime numbers (namely 7, 17, 71, 107 and 701). So there is no primeval number in the decimal system . ${\displaystyle m=107<n=117}$ ${\displaystyle n=117}$

But now be a number in the duodecimal system , i.e. for the base (this number would be the number in the decimal system ). This number is also not a primeval number in the decimal system. The following prime numbers can be generated from this as a basis : ${\displaystyle n=117_{12}}$ ${\displaystyle b=12}$ ${\displaystyle n=117_{12}={\underline {1}}\cdot 12^{2}+{\underline {1}}\cdot 12^{1}+{\underline {7}}\cdot 12^{0}=144+12+7=163}$ ${\displaystyle b=12}$

${\displaystyle {\begin{aligned}7_{12}&=&&&&&{\underline {7}}\cdot 12^{0}&=&&&7\in \mathbb {P} \\11_{12}&=&&&{\underline {1}}\cdot 12^{1}&+&{\underline {1}}\cdot 12^{0}&=&12+1&=&13\in \mathbb {P} \\17_{12}&=&&&{\underline {1}}\cdot 12^{1}&+&{\underline {7}}\cdot 12^{0}&=&12+7&=&19\in \mathbb {P} \\117_{12}&=&{\underline {1}}\cdot 12^{2}&+&{\underline {1}}\cdot 12^{1}&+&{\underline {7}}\cdot 12^{0}&=&144+12+7&=&163\in \mathbb {P} \\171_{12}&=&{\underline {1}}\cdot 12^{2}&+&{\underline {7}}\cdot 12^{1}&+&{\underline {1}}\cdot 12^{0}&=&144+84+1&=&229\in \mathbb {P} \\711_{12}&=&{\underline {7}}\cdot 12^{2}&+&{\underline {1}}\cdot 12^{1}&+&{\underline {1}}\cdot 12^{0}&=&1008+12+1&=&1021\in \mathbb {P} \end{aligned}}}$

So you can generate a total of 6 different prime numbers from the individual digits in the duodecimal system. Because one cannot generate 6 or more prime numbers from any smaller number in the duodecimal system, a primeval number is the base . ${\displaystyle n=117_{12}}$ ${\displaystyle m<n=117_{12}}$ ${\displaystyle n=117_{12}}$ ${\displaystyle b=12}$

• The first Primeval numbers for the base are the following (for lack of further digits and ): ${\displaystyle b=12}$ ${\displaystyle A:=10}$ ${\displaystyle B:=11}$

1, 2, 13, 15, 57, 115, 117, 125, 135, 157 , 1017, 1057, 1157, 1257, 125B, 157B, 167B ...

• The number of prime numbers that can be made from the above-mentioned Primeval numbers as a base are: ${\displaystyle b=12}$

0, 1, 2, 3, 4, 5, 6, 7, 8, 11 , 12, 20, 23, 27, 29, 33, 35 ...

Example:

At the 10th position of the above two lists you can read the numbers and . This means that you can make exactly different prime numbers from the number and that there is not a single base number from which you can make different prime numbers or more. ${\displaystyle n=157_{12}}$ ${\displaystyle 11}$ ${\displaystyle n=157_{12}}$ ${\displaystyle 11}$ ${\displaystyle m<n=157_{12}}$ ${\displaystyle b=12}$ ${\displaystyle 11}$

• The following table shows the first 10 Primeval numbers for the base and which prime numbers can be made from them: ${\displaystyle b=12}$

Primeval number to base ${\displaystyle b=12}$	corresponds in the decimal system	primes obtained therefrom to the base ${\displaystyle b=12}$	prime numbers obtained therefrom in the decimal system	Number of prime numbers obtained in this way
${\displaystyle 1_{12}}$	${\displaystyle 1\not \in \mathbb {P} }$			0
${\displaystyle 2_{12}}$	${\displaystyle 2\in \mathbb {P} }$	${\displaystyle 2_{12}}$	2	1
${\displaystyle 13_{12}}$	${\displaystyle 15\not \in \mathbb {P} }$	${\displaystyle 3_{12},31_{12}}$	3, 37	2
${\displaystyle 15_{12}}$	${\displaystyle 17\in \mathbb {P} }$	${\displaystyle 5_{12},15_{12},51_{12}}$	5, 17, 61	3
${\displaystyle 57_{12}}$	${\displaystyle 67\in \mathbb {P} }$	${\displaystyle 5_{12},7_{12},57_{12},75_{12}}$	5, 7, 67, 89	4th
${\displaystyle 115_{12}}$	${\displaystyle 161\not \in \mathbb {P} }$	${\displaystyle 5_{12},11_{12},15_{12},51_{12},511_{12}}$	5, 13, 17, 61, 733	5
${\displaystyle 117_{12}}$	${\displaystyle 163\in \mathbb {P} }$	${\displaystyle 7_{12},11_{12},17_{12},117_{12},171_{12},711_{12}}$	7, 13, 19, 163, 229, 1021	6th
${\displaystyle 125_{12}}$	${\displaystyle 173\in \mathbb {P} }$	${\displaystyle 2_{12},5_{12},15_{12},25_{12},51_{12},125_{12},251_{12}}$	2, 5, 17, 29, 61, 173, 349	7th
${\displaystyle 135_{12}}$	${\displaystyle 185\not \in \mathbb {P} }$	${\displaystyle 3_{12},5_{12},15_{12},31_{12},35_{12},51_{12},315_{12},531_{12}}$	3, 5, 17, 37, 41, 61, 449, 757	8th
${\displaystyle 157_{12}}$	${\displaystyle 211\in \mathbb {P} }$	${\displaystyle 5_{12},7_{12},15_{12},17_{12},51_{12},57_{12},75_{12},157_{12},175_{12},517_{12},751_{12}}$	5, 7, 17, 19, 61, 67, 89, 211, 233, 739, 1069	11

• The table above shows that the numbers and are not prime numbers. They are the smallest composite primeval numbers to the base (because it is neither a prime nor a composite number). All other non-composite Primeval numbers (except ) are Primeval primes as the base . ${\displaystyle 13_{12}=3_{12}\cdot 5_{12},115_{12}=7_{12}\cdot 1B_{12}}$ ${\displaystyle 135_{12}=5_{12}\cdot 31_{12}}$ ${\displaystyle b=12}$ ${\displaystyle 1_{12}=1}$ ${\displaystyle 1_{12}=1}$ ${\displaystyle b=12}$

k-primeval numbers

There are numbers with which one can obtain all -digit prime numbers by permutation (i.e. by interchanging or omitting) some or all of their digits . These numbers are called k-primeval numbers . ${\displaystyle n\in \mathbb {N} }$ ${\displaystyle k}$

Examples

• If you are looking for the smallest number , which contains all single-digit prime numbers and , it can only be the number . This number is therefore a 1-primeval number. All other numbers that contain the digits and are of course 1-primeval numbers (such as ). ${\displaystyle n\in \mathbb {N} }$ ${\displaystyle 2,3,5}$ ${\displaystyle 7}$ ${\displaystyle n=2357}$ ${\displaystyle 2,3,5}$ ${\displaystyle 7}$ ${\displaystyle n=3018253975}$
• The smallest k-primeval numbers are the following (for ascending ): ${\displaystyle k=1,2,3,\ldots }$

2357, 1123456789, 1012233445566778899, 10011222333444555666777888999, 1000111222233334444555666777788889999, 100001111222233333444445555566666777778888899999, 100000111112222233333344444555556666677777788888999999 ... (sequence A134649 in OEIS )

k-primeval primes

A k-primeval number that is also prime is called a k-primeval prime number .

Examples

• The smallest k-primeval prime numbers are the following (for ascending ): ${\displaystyle k=1,2,3,\ldots }$

2357, 1123465789, 10112233445566788997, 100111222333444555666777998889, 1000111222233334444555666777798889899, ​​100001111222233333444445555566666777778888999989… (episode A135377 in OEIS )

Because these numbers get very high very quickly, the following notation has been established:

First all numbers begin with (1), which always occurs in such numbers, then the number of all subsequent zeros, then the number of all subsequent ones etc., towards the end the number of all subsequent eights follows, and finally there is a group of eights and nines that completes the number you are looking for.

Example:

The smallest 4-primeval prime number is and is in this notation: (1) 2 3 3 3 3 3 3 3 0 998889 . It starts with a one, followed by two zeros, three ones, three twos, etc., towards the end there are three sevens, zero eights and finally the sequence of digits 998889. ${\displaystyle n=100111222333444555666777998889}$

• With the above notation one can specify the further smallest k-Primeval prime numbers without taking up much space:

${\displaystyle k}$ smallest k-primeval prime number in the above notation 1 2357 ( spelling not suitable in this case ) 2 1123465789 ( spelling not suitable in this case ) 3 (1) 1 2 2 2 2 2 2 1 2 997 4th (1) 2 3 3 3 3 3 3 3 0 998889 5 (1) 3 3 4 4 4 3 3 4 0 98889899 6th (1) 4 4 4 5 5 5 5 5 4 999989 7th (1) 5 5 5 6 5 5 5 6 3 98899999 8th (1) 5 6 7 7 6 7 7 7 6 98999999 9 (1) 7 7 8 8 8 7 8 8 6 9999989899 10 (1) 8 8 8 9 9 9 9 9 7 9999899999

${\displaystyle k}$ smallest k-primeval prime number in the above notation 11 (1) 8 9 10 10 10 9 10 10 6 9889989999999 12 (1) 10 10 10 11 11 11 10 11 9 9998999999899 13 (1) 10 11 11 12 11 12 11 12 9 99899999999899 14th (1) 11 13 13 13 12 12 12 13 11 989999989999999 15th (1) 12 13 14 14 13 14 13 14 12 9999999988999999 16 (1) 13 14 14 15 14 14 14 15 12 99999999999999889 17th (1) 14 15 15 16 15 15 15 16 14 998999999999998999 18th (1) 16 17 17 17 16 17 17 17 14 9989999999999899999 19th (1) 17 18 17 18 17 17 17 18 15 988999999899999999999 20th (1) 17 19 18 19 19 18 19 19 16 999999998999999999989

${\displaystyle k}$ smallest k-primeval prime number in the above notation 21st (1) 18 19 19 20 19 19 20 20 17 9899999999999999998999 22nd (1) 18 20 20 21 20 21 21 21 18 99998999999999999998999 23 (1) 21 23 21 22 21 21 22 22 19 999999889999999999999999 24 (1) 20 22 22 23 22 22 22 23 21 999999999999999989999999 25th (1) 23 23 23 24 23 23 23 24 22 9999999999999999998999999 26th (1) 23 24 24 25 25 25 24 25 22 999999999999999999899999989 27 (1) 24 25 25 26 25 25 25 26 23 9999999998999999999999998999 28 (1) 25 26 27 27 27 26 27 27 25 9999899999999999999999999999 29 (1) 25 27 27 28 27 27 27 28 25 999999989999999999999999999989 30th (1) 26 29 28 29 29 28 28 29 27 999999999999998999999999999999

The smallest 30-primeval prime number already has digits. ${\displaystyle 1+26+29+28+29+29+28+28+29+27+30=284}$

See also

• Permutable prime number
• Truncatable prime number

Individual evidence

1. ↑ ^{a b} Mike Keith: Primeval Numbers. Retrieved December 30, 2018 . 

Web links

• Chris K. Caldwell: primeval number. Prime Pages, accessed December 30, 2018 .