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 15 minimal primes:
2, 3, 5, 7, E, 11, 61, 81, 91, 401, X41, 4441, X0X1, XXXX1, 44XXX1, XXX0001, XX000001
Similarly, there are exactly 3X composite numbers which have no shorter composite subsequence:
4, 6, 8, 9, X, 10, 12, 13, 20, 21, 22, 23, 2E, 30, 32, 33, 50, 52, 53, 55, 70, 71, 72, 73, 77, 7E, E0, E1, E2, E3, EE, 115, 151, 15E, 257, 275, 311, 317, 31E, 351, E57, E75, 1111, 1117, 111E, 5111
All minimal primes are found for all bases up to 20 except bases 15, 17 and 19. All these three bases have only one unsolved family:
Base | The only one unsolved family | Formula (“n” is the number of digits in “{}”) |
---|---|---|
15 | 13:1:{9} | (2461*15^n−9)/14 |
17 | 12:12:1:{6} | (92X4*17^n−1)/3 |
19 | 14:{0}:13:18 | 4100*19^n+23E |
Besides, there are large probable prime found for these bases, e.g. in base 1E, the largest minimal (probable) prime is 9:{12}_{327575} (=(8X*1E^327575−7)/E), and in base 19, a large minimal (probable) prime is 10:{13}_{1E134E}:0:18 (=(43*19^1E134E−877)/4), this prime is likely the second-largest “base 19 minimal prime”.
However, this problem seems to be hard for bases > 20, e.g. in base 21, there are 10 unsolved families, and although this problem is solved for all bases <= 20 except bases 15, 17 and 19, but this problem is unsolved for all bases > 20 and <= 100 except bases 26, 36 and 50.
The bases > 20 and <= 100 with only few unsolved families are
Base | Unsolved families |
---|---|
22 | {X}:6:13 {16}:14:19 |
23 | 8:{0}:9:X 9:9:9:{14} 10:{19}:12 12:{16}:13:8 {13}:9:13:1X |
24 | 20:{X}:13 |
26 | (none) |
30 | 20:{19}:2E {21}:24:2E |
34 | 24:{22}:33 |
36 | (none) |
40 | X:{0}:24:21 10:{34}:2E {18}:16:21 30:{0}:1 34:19:{0}:2E 39:32:{34}:2E |
50 | (none) |
Even in base 11, the largest minimal (probable) prime is also large, it is 8:{0}_{16641}:1:1:1, which is equal to X208*11^16641+133.
In bases 11, 15, 17, 19 and 1E (also all bases > 20 and <= 100 except 26, 36 and 50), some minimal primes found are only probable primes, i.e. not proved primes. (For all bases <= 20 except 11, 15, 17, 19 and 1E, also for bases 26, 36 and 50, all minimal primes found are proven primes)
For bases up to 10, the largest minimal primes are:
Base | Largest minimal prime | Largest minimal prime (written in base 10) | Length | Number of minimal (probable) prime |
---|---|---|---|---|
2 | 11 | 3 | 2 | 2 |
3 | 111 | 11 | 3 | 3 |
4 | 11 | 5 | 2 | 3 |
5 | 44441 | 1981 | 5 | 8 |
6 | 40041 | 3021 | 5 | 7 |
7 | 11111 | 1755 | 5 | 9 |
8 | 444444441 | 21828201 | 9 | 13 |
9 | 1101 | 577 | 4 | 10 |
X | 66600049 | 1X379841 | 8 | 22 |
E | 444444444444444444444444444444444444444444441 | 1193880E7XE914EX9X696EX922319503040849110881 | 39 | 108 |
10 | XX000001 | XX000001 | 8 | 15 |
The known largest minimal (probable) primes for bases b > 10 and <= 100 are:
Base | Largest minimal (probable) prime | Length | Dozenal length | Number of minimal (probable) prime |
---|---|---|---|---|
11 | 8:{0}_{16641}:1:1:1 | 16645 | 17165 | 170 |
12 | 4:{0}_{6E}:4:9 | 72 | 77 | 180 |
13 | 9:{6}_{88}:0:8 | 8E | 99 | 84 |
14 | 9:{0}_{2072}:9:1 | 2075 | 2358 | 343 |
16 | 14:14:{0}_{26}:1 | 29 | 33 | 42 |
18 | {14}_{313}:9:9 | 315 | 392 | 463 |
1X | 18:{0}_{534}:12:10:1 | 538 | 673 | 876 |
1E | 9:{12}_{327575} | 327576 | 408990 | 3599 |
20 | 1X:{6}_{82}:1 | 84 | X8 | 216 |
26 | 10:{0}_{712}:1 | 714 | 98X | 164 |
36 | {23}_{346}:1 | 347 | 511 | 2773 |
50 | {34}_{1155}:1 | 1156 | 1X22 | (?) |
For these (probable) primes, only the largest minimal (probable) prime in bases 11, 14 and 1E are titanic primes (primes with at least 1000 dozenal digits), that in bases 11 and 1E are also gigantic primes (primes with at least 10000 dozenal digits), however, none of them are megaprimes (primes with at least 10^{6} dozenal digits).
The minimal primes in bases up to 10 are
b | minimal primes in base b (written in base b) | number of minimal primes in base b |
---|---|---|
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 | 13 |
9 | 2, 3, 5, 7, 14, 18, 41, 81, 601, 661, 1011, 1101 | 10 |
X | 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 | 22 |
E | 2, 3, 5, 7, 10, 16, 18, 49, 61, 81, 89, 94, 98, 9X, 199, 1XX, 414, 919, X1X, XX1, 11X9, 66X9, X119, X911, XXX9, 11144, 11191, 1141X, 114X1, 1411X, 144X4, 14X11, 1X114, 1X411, 4041X, 40441, 404X1, 4111X, 411X1, 44401, 444X1, 44X01, 6X609, 6X669, 6X696, 6X906, 6X966, 90901, 99111, X0111, X0669, X0966, X0999, X0X09, X4401, X6096, X6966, X6999, X9091, X9699, X9969, 401X11, 404001, 404111, 440X41, 4X0401, 4X4041, 60X069, 6X0096, 6X0X96, 6X9099, 6X9909, 909991, 999901, X00009, X60609, X66069, X66906, X69006, X90099, X90996, X96006, X96666, 111114X, 1111X14, 1111X41, 1144441, 14X4444, 1X44444, 4000111, 4011111, 41X1111, 4411111, 444441X, 4X11111, 4X40001, 6000X69, 6000X96, 6X00069, 9900991, 9990091, X000696, X000991, X006906, X040041, X141111, X600X69, X906606, X909009, X990009, 40X00041, 60X99999, 99000001, X0004041, X9909006, X9990006, X9990606, X9999966, 40000X401, 44X444441, 900000091, X00990001, X44444111, X66666669, X90000606, X99999006, X99999099, 600000X999, X000144444, X900000066, X0000000001, X0014444444, 40000000X0041, X000000014444, X044444444441, X144444444411, 40000000000401, X0000044444441, X00000000444441, 11111111111111111, 14444444444441111, 44444444444444111, X1444444444444444, X9999999999999996, 1444444444444444444, 4000000000000000X041, X999999999999999999999, X44444444444444444444444441, 40000000000000000000000000041, 440000000000000000000000000001, 999999999999999999999999999999991, 444444444444444444444444444444444444444444441 | 108 |
10 | 2, 3, 5, 7, E, 11, 61, 81, 91, 401, X41, 4441, X0X1, XXXX1, 44XXX1, XXX0001, XX000001 | 15 |