极小质数

极小质数（minimal prime）是娱乐数学中的一个名词。即是，若一质数在数字顺序不变的前题下，其所有子序列都不是质数，那么该质数就是极小质数。

## 概要

以类似的概念来看，以下的32个合数在数字顺序不变下，所有子序列都不是合数：

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

若只考虑除以4会余1的质数，以下146个质数在数字顺序不变下，其子序列都没有除以4会余1的质数：

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, ...

若只考虑除以4会余3的质数，以下113个质数在数字顺序不变下，其子序列都没有除以4会余3的质数：

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, ...

## 十进制的例子

在十进制下，极小质数共有以下26个：

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为例，其子序列有4,0,9,40,49,09，都不是质数，因此409为极小质数。子序列不一定要在原质数中连续的位子上。例如109不是极小质数，因为子序列中的19是质数。子序列的数字顺序需和原来相同，不能将两数字的顺序对调。例如991，虽然19是质数，但因为位置对调，不在考虑范围内，而其他子序列都不是质数，因此991是极小质数。

## 其他进制

极小质数也可以扩展到其他的进制。可以证明在每一个进制下，极小质数的个数都是有限个。换句话说，每一个足够大的质数都至少会有一个子序列是质数。

b｜b下的极小质数（以b进制表示，其中的字母A, B, C, ... 表示数值10, 11, 12, ...）｜b进制下极小质数的个数

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

十进制下的12个极小质数列在。

## 参考资料

- Chris Caldwell, The Prime Glossary: minimal prime , from the Prime Pages
- 2到30进制的极小质数
- 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