Minimal prime (recreational mathematics)


In recreational number theory, a minimal prime is a prime number for which there is no shorter subsequence of its digits in a given base that form a prime. In base 10 there are exactly 26 minimal primes:
For example, 409 is a minimal prime because there is no prime among the shorter subsequences of the digits: 4, 0, 9, 40, 49, 09. The subsequence does not have to consist of consecutive digits, so 109 is not a minimal prime. But it does have to be in the same order; so, for example, 991 is still a minimal prime even though a subset of the digits can form the shorter prime 19 by changing the order.
Similarly, there are exactly 32 composite numbers which have no shorter composite subsequence:
There are 146 primes congruent to 1 mod 4 which have no shorter prime congruent to 1 mod 4 subsequence:
There are 113 primes congruent to 3 mod 4 which have no shorter prime congruent to 3 mod 4 subsequence:

Other bases

Minimal primes can be generalized to other bases. It can be shown that there are only a finite number of minimal primes in every base. Equivalently, every sufficiently large prime contains a shorter subsequence that forms a prime.
bminimal primes in base b
1111
210, 112
32, 10, 1113
42, 3, 113
52, 3, 10, 111, 401, 414, 14444, 444418
62, 3, 5, 11, 4401, 4441, 400417
72, 3, 5, 10, 14, 16, 41, 61, 111119
82, 3, 5, 7, 111, 141, 161, 401, 661, 4611, 6101, 6441, 60411, 444641, 44444444115
92, 3, 5, 7, 14, 18, 41, 81, 601, 661, 1011, 110112
102, 3, 5, 7, 11, 19, 41, 61, 89, 409, 449, 499, 881, 991, 6469, 6949, 9001, 9049, 9649, 9949, 60649, 666649, 946669, 60000049, 66000049, 6660004926
112, 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, 444444444444444444444444444444444444444444441152
122, 3, 5, 7, B, 11, 61, 81, 91, 401, A41, 4441, A0A1, AAAA1, 44AAA1, AAA0001, AA00000117

The base 12 minimal primes written in base 10 are listed in.
Number of minimal primes in base n are
The length of the largest minimal prime in base n are
Largest minimal prime in base n are
Number of minimal composites in base n are
The length of the largest minimal composite in base n are