Minimal prime number

In entertainment mathematics , a minimal prime number is a prime number in which no partial sequence of its digits in a given base is a prime number, as long as they are not interchanged. ${\displaystyle p\in \mathbb {P} }$

Examples in the decimal system

• The number is not a minimal prime number because one can make the prime number out of its digits . The single digits of the partial sequences thus have the original number not be contiguous. ${\displaystyle 109}$ ${\displaystyle 19}$
• From the number one following partial sequences can make their numbers: . Neither of these numbers is prime, so is minimal prime. ${\displaystyle 409}$ ${\displaystyle 0,4,9,09,40,49}$ ${\displaystyle 409}$
• The number is a minimal prime number because only the numbers and can be made from its digits and none of these numbers are prime. The individual digits of the original number must not be interchanged (otherwise the partial sequence would very well be a prime number in this case ). ${\displaystyle 991}$ ${\displaystyle 1,9,91}$ ${\displaystyle 99}$ ${\displaystyle 19}$
• The only minimal prime numbers for base 10 (i.e. in the decimal system ) are the following 26 prime numbers:

2, 3, 5, 7, 11, 19, 41, 61, 89, 409, 449, 499, 881, 991, 6469, 6949, 9001, 9049, 9649, 9949, 60649, 666649, 946669, 60000049, 66000049, 66600049 (Follow A071062 in OEIS )

Examples with base b

• This is followed by a table from which the minimal prime numbers in the base can be taken (whereby A = 10 and B = 11 are set for lack of further digits). It can be shown that there can no longer be minimal prime numbers for the respective base: ${\displaystyle b}$

Base ${\displaystyle b}$	minimal prime numbers to the base , written to the base ${\displaystyle b}$ ${\displaystyle b}$
1	11
2	10, 11
3	2, 10, 111
4th	2, 3, 11
5	2, 3, 10, 111, 401, 414, 14444, 44441 (a total of 8 minimal prime numbers)
6th	2, 3, 5, 11, 4401, 4441, 40041 (a total of 7 minimal prime numbers)
7th	2, 3, 5, 10, 14, 16, 41, 61, 11111 (a total of 9 minimum prime numbers)
8th	2, 3, 5, 7, 111, 141, 161, 401, 661, 4611, 6101, 6441, 60411, 444641, 444444441 (a total of 15 minimum prime numbers)
9	2, 3, 5, 7, 14, 18, 41, 81, 601, 661, 1011, 1101 (a total of 12 minimal prime numbers)
10	2, 3, 5, 7, 11, 19, 41, 61, 89, 409, 449, 499, 881, 991, 6469, 6949, 9001, 9049, 9649, 9949, 60649, 666649, 946669, 60000049, 66000049, 66600049 (a total of 26 minimal prime numbers)
11	2, 3, 5, 7, 10, 16, 18, 49, 61, 81, 89, 94, 98, 9A, 199, 1AA, 414, 919, A1A, AA1, 11A9, 66A9, A119, A911, AAA9, 11144, 11191, 1141A, 114A1, 1411A, 144A4, 14A11, 1A114, 1A411, 4041A, 40441, 404A1, 4111A, 411A1, 44401, 444A1, 44A01, 6A609, 6A669, 6A696, 6A906, 6A111, 90111, A0669, A0966, A0999, A0A09, A4401, A6096, A6966, A6999, A9091, A9699, A9969, 401A11, 404001, 404111, 440A41, 4A0401, 4A4041, 60A069, 6A0096, A0000909, 6A00A96, 6A909999, 6A009190, 6A9099999 A60609, A66069, A66906, A69006, A90099, A90996, A96006, A96666, 111114A, 1111A14, 1111A41, 1144441, 14A4444, 1A44444, 4000111, 4011111, 41A1111, 44110001A, 6000A1169, 44110001A, 6000A4111, 6000A116, 4A110001A, 4A4116, 4A110001A, 4A1140001A, 4A4111 9990091, A000696, A000991, A006906, A040041, A141111, A600A69, A906606, A909009, A990009, 40A00041, 60A99999, 99000001, A0004041, A9909006, A9990006, A9990606, A9999966, 40000A401, 44A444441, 900,000,091, A00990001, A44444111, A66666669, A90000606, A99999006, A99999099, 600000A999, A000144444, A900000066, A0 000000001, A0014444444, 40000000A0041, A000000014444, A044444444441, A144444444411, 40000000000401, A0000044444441, A00000000444441, 11111111111111111, 14444444444441111, 44444444444444111, A1444444444444444, A9999999999999996, 1444444444444444444, 4000000000000000A041, A999999999999999999999, A44444444444444444444444441, 40000000000000000000000000041, 440000000000000000000000000001, 999999999999999999999999999999991, 444444444444444444444444444444444444444444441 (a total of 152 minimum prime numbers)
12	2, 3, 5, 7, B, 11, 61, 81, 91, 401, A41, 4441, A0A1, AAAA1, 44AAA1, AAA0001, AA000001 (a total of 17 minimum prime numbers)

• The only minimal prime numbers for base 12 (i.e. in the duodecimal system ) are the 17 prime numbers above. In the decimal system, they are:

2, 3, 5, 7, 11, 13, 73, 97, 109, 577, 1489, 7537, 17401, 226201, 1097113, 32555521, 388177921 (series A110600 in OEIS )

Example:

The minimum prime number in the decimal system is the number . You can make the non-prime numbers and from it. ${\displaystyle A41_{12}}$ ${\displaystyle {\underline {10}}\cdot 12^{2}+{\underline {4}}\cdot 12^{1}+{\underline {1}}\cdot 12^{0}=1440+48+1=1489_{10}}$ ${\displaystyle 1_{12}=1_{10},4_{12}=4_{10},A_{12}=10_{10},41_{12}=49_{10},A1_{12}=121_{10}}$ ${\displaystyle A4_{12}=124_{10}}$

• The number of minimal (partly PRP ) prime numbers given a given base are the following: ${\displaystyle b=1,2,\ldots }$

1, 2, 3, 3, 8, 7, 9, 15, 12, 26, 152, 17, 228, 240, 100, 483, ≥1279, 50, ≥3462, 651, ≥2600, 1242, 6021, 306 , ≥17597, ≥5662, ≥17210, ≥5783, ≥57283, 220, ...

Example:

The number is in the 14th position in the list above . So there are minimal prime numbers for the base . ${\displaystyle 240}$ ${\displaystyle 240}$ ${\displaystyle b=14}$

• The number of digits of the largest minimal (partly PRP) prime numbers given a given base are the following: ${\displaystyle b=1,2,\ldots }$

2, 2, 3, 2, 5, 5, 5, 9, 4, 8, 45, 8, 32021, 86, 107, 3545, ≥111334, 33, ≥110986, 449, ≥479150, 764, 800874, 100 , ≥136967, ≥8773, ≥109006, ≥94538, ≥174240, 1024, ...

Example 1:

In the 13th position of the list above is the number . The largest minimal (PRP) prime number for the base has places. ${\displaystyle 32021}$ ${\displaystyle b=13}$ ${\displaystyle 32021}$

Example 2:

The entry is in the 26th position of the list above . The largest minimal (PRP) prime number for the base therefore has places, but there are still unsolved cases that have more places. ${\displaystyle \geq 8773}$ ${\displaystyle b=26}$ ${\displaystyle 8773}$

• The largest minimal (partly PRP) prime numbers for a given base are the following when written in the decimal system: ${\displaystyle b=1,2,\ldots }$

2, 3, 13, 5, 3121, 5209, 2801, 76695841, 811, 66600049, 29156193474041220857161146715104735751776055777, 388177921, ... (the next prime number has 35670 digits)

Example:

The number is at the 12th position in the list above . In fact, the largest minimum base prime is number . ${\displaystyle 388177921}$ ${\displaystyle 12}$ ${\displaystyle AA000001_{12}={\underline {10}}\cdot 12^{7}+{\underline {10}}\cdot 12^{6}+{\underline {1}}\cdot 12^{0}=388177921_{10}}$

Generalizations

• There are exactly 32 composite numbers in the decimal system, which consist of digits whose partial sequences do not result in any further composite numbers:

4, 6, 8, 9, 10, 12, 15, 20, 21, 22, 25, 27, 30, 32, 33, 35, 50, 51, 52, 55, 57, 70, 72, 75, 77, 111, 117, 171, 371, 711, 713, 731 (sequence A071070 in OEIS )

Example:

From the number one can make the numbers and , which are all prime numbers and therefore not composed. These numbers are thus the exact opposite of the minimal prime numbers. ${\displaystyle 731}$ ${\displaystyle 1,3,7,31,71}$ ${\displaystyle 73}$

• In the decimal system there are exactly 146 prime numbers (i.e. of the form with ), which consist of digits whose partial sequences in the decimal system do not result in any further prime numbers of the form : ${\displaystyle p\equiv 1{\pmod {4}}}$ ${\displaystyle p=4k+1}$ ${\displaystyle k\in \mathbb {N} }$ ${\displaystyle p\equiv 1{\pmod {4}}}$

5, 13, 17, 29, 37, 41, 61, 73, 89, 97, 101, 109, 149, 181, 233, 277, 281, 349, 409, 433, 449, 677, 701, 709, 769, 821, 877, 881, 1669, 2221, 3001, 3121, 3169, 3221, 3301, 3833, 4969, 4993, 6469, 6833, 6949, 7121, 7477, 7949, 9001, 9049, 9221, 9649, 9833, ... ( Follow A111055 in OEIS )

Example:

From the prime number one can make the numbers and , none of which are prime numbers of the form . ${\displaystyle 3169}$ ${\displaystyle 1,3,6,9,16,19,31,36,39,69,169,316,319}$ ${\displaystyle 369}$ ${\displaystyle p\equiv 1{\pmod {4}}}$

• In the decimal system there are exactly 112 prime numbers (i.e. of the form with ), which consist of digits whose partial sequences in the decimal system do not result in any further prime numbers of the form : ${\displaystyle p\equiv 3{\pmod {4}}}$ ${\displaystyle p=4k+3}$ ${\displaystyle k\in \mathbb {N} }$ ${\displaystyle p\equiv 3{\pmod {4}}}$

3, 7, 11, 19, 59, 251, 491, 499, 691, 991, 2099, 2699, 2999, 4051, 4451, 4651, 5051, 5651, 5851, 6299, 6451, 6551, 6899, 8291, 8699, 8951, 8999, 9551, 9851, 22091, 22291, 66851, 80051, 80651, 84551, 85451, 86851, 88651, 92899, 98299, 98899, ... (series A111056 in OEIS )

Example:

From the prime number one can make the numbers and , none of which are prime numbers of the form . ${\displaystyle 4051}$ ${\displaystyle 0,1,4,5,01,05,40,41,45,51,051,401,405}$ ${\displaystyle 451}$ ${\displaystyle p\equiv 3{\pmod {4}}}$

• The number of minimum composite numbers given the base are as follows: ${\displaystyle b=1,2,\ldots }$

1, 3, 4, 9, 10, 19, 18, 26, 28, 32, 32, 46, 43, 52, 54, 60, 60, 95, 77, 87, 90, 94, 97, 137, 117, 111, 115, 131, 123, 207, ...

• Given the base , the number of digits of the largest minimum composite numbers are as follows: ${\displaystyle b=1,2,\ldots }$

4, 4, 3, 3, 3, 4, 3, 3, 2, 3, 3, 4, 3, 3, 2, 3, 3, 4, 3, 3, 2, 3, 3, 4, 2, 3, 2, 3, 3, 4, ...

Individual evidence

1. ↑ ^{a b} Minimal prime numbers and unsolved cases ("families") with bases from 2 to 30 (English)
2. ^ ^{A b} Curtis Bright, Raymond Devillers, Jeffrey Shallit: Minimal Elements for the Prime Numbers. University of Waterloo , 2015, p. 15 , accessed July 4, 2018 . 
3. ↑ For the base b = 17 there are 1279 known minimal (PRP) prime numbers and one unsolved case
4. ↑ For the base b = 19 there are 3462 known minimal (PRP) prime numbers and one unsolved case
5. ↑ For the base b = 21 there are 2600 known minimal (PRP) prime numbers and one unsolved case
6. ↑ For the base b = 25 there are 17597 known minimal (PRP) prime numbers and 12 unsolved cases
7. ↑ For the base b = 26 there are 5662 known minimal (PRP) prime numbers and two unsolved cases
8. ↑ For the base b = 27 there are 17210 known minimal (PRP) prime numbers and 5 unsolved cases
9. ↑ For the base b = 28 there are 5783 known minimal (PRP) prime numbers and one unsolved case
10. ↑ For the base b = 29 there are 57,283 known minimal (PRP) prime numbers and 14 unsolved cases
11. ^ ^{A b} Curtis Bright, Raymond Devillers, Jeffrey Shallit: Minimal Elements for the Prime Numbers. University of Waterloo , 2015, p. 20 , accessed July 4, 2018 . 

Web links

• Chris K. Caldwell: minimal prime. Prime Pages - The Prime Glossary, accessed July 4, 2018 . 

• Minimal primes and unsolved families with bases from 2 to 30
• Minimal primes and unsolved families with bases from 28 to 50