極小質數

極小質數（minimal prime）是娛樂數學中的一個名詞。即是，若一質數在數字順序不變的前題下，其所有子序列都不是質數，那麼該質數就是極小質數。

## 概要

以類似的概念來看，以下的32個合數在數字順序不變下，所有子序列都不是合數：

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

若只考慮除以4會餘1的質數，以下146個質數在數字順序不變下，其子序列都沒有除以4會餘1的質數：

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, 9901, 9949, ...

若只考慮除以4會餘3的質數，以下113個質數在數字順序不變下，其子序列都沒有除以4會餘3的質數：

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, ...

## 十進位的例子

在十進位下，極小質數共有以下26個：

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

以409為例，其子序列有4,0,9,40,49,09，都不是質數，因此409為極小質數。子序列不一定要在原質數中連續的位子上。例如109不是極小質數，因為子序列中的19是質數。子序列的數字順序需和原來相同，不能將兩數字的順序對調。例如991，雖然19是質數，但因為位置對調，不在考慮範圍內，而其他子序列都不是質數，因此991是極小質數。

## 其他進制

極小質數也可以擴展到其他的進制。可以證明在每一個進制下，極小質數的個數都是有限個。換句話說，每一個足夠大的質數都至少會有一個子序列是質數。

b｜b下的極小質數（以b進制表示，其中的字母A, B, C, ... 表示數值10, 11, 12, ...）｜b進制下極小質數的個數

1｜11｜1

2｜10, 11｜2

3｜2, 10, 111｜3

4｜2, 3, 11｜3

5｜2, 3, 10, 111, 401, 414, 14444, 44441｜8

6｜2, 3, 5, 11, 4401, 4441, 40041｜7

7｜2, 3, 5, 10, 14, 16, 41, 61, 11111｜9

8｜2, 3, 5, 7, 111, 141, 161, 401, 661, 4611, 6101, 6441, 60411, 444641, 444444441｜15

9｜2, 3, 5, 7, 14, 18, 41, 81, 601, 661, 1011, 1101｜12

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｜26

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, 6A966, 90901, 99111, A0111, A0669, A0966, A0999, A0A09, A4401, A6096, A6966, A6999, A9091, A9699, A9969, 401A11, 404001, 404111, 440A41, 4A0401, 4A4041, 60A069, 6A0096, 6A0A96, 6A9099, 6A9909, 909991, 999901, A00009, A60609, A66069, A66906, A69006, A90099, A90996, A96006, A96666, 111114A, 1111A14, 1111A41, 1144441, 14A4444, 1A44444, 4000111, 4011111, 41A1111, 4411111, 444441A, 4A11111, 4A40001, 6000A69, 6000A96, 6A00069, 9900991, 9990091, A000696, A000991, A006906, A040041, A141111, A600A69, A906606, A909009, A990009, 40A00041, 60A99999, 99000001, A0004041, A9909006, A9990006, A9990606, A9999966, 40000A401, 44A444441, 900000091, A00990001, A44444111, A66666669, A90000606, A99999006, A99999099, 600000A999, A000144444, A900000066, A0000000001, A0014444444, 40000000A0041, A000000014444, A044444444441, A144444444411, 40000000000401, A0000044444441, A00000000444441, 11111111111111111, 14444444444441111, 44444444444444111, A1444444444444444, A9999999999999996, 1444444444444444444, 4000000000000000A041, A999999999999999999999, A44444444444444444444444441, 40000000000000000000000000041, 440000000000000000000000000001, 999999999999999999999999999999991, 444444444444444444444444444444444444444444441｜152

12｜2, 3, 5, 7, B, 11, 61, 81, 91, 401, A41, 4441, A0A1, AAAA1, 44AAA1, AAA0001, AA000001｜17

十進制下的12個極小質數列在。

## 參考資料

- Chris Caldwell, The Prime Glossary: minimal prime , from the Prime Pages
- 2到30進制的極小質數
- Minimal primes and unsolved families in bases 2 to 30
- Minimal primes and unsolved families in bases 28 to 50
- J. Shallit, Minimal primes , Journal of Recreational Mathematics, 30:2, pp. 113–117, 1999-2000.
- PRP records, search by form 8*13^n+183 (primes of the form 8{0}111 in base 13), n=32020
- PRP records, search by form (51*21^n-1243)/4 (primes of the form C{F}0K in base 21), n=479149
- PRP records, search by form (106*23^n-7)/11 (primes of the form 9{E} in base 23), n=800873