Minimal set of the strings for primes with at least two digits

[sweety439]

Originally Posted by sweety439

https://primes.utm.edu/glossary/page...t=MinimalPrime

In 1996, Jeffrey Shallit [Shallit96] suggested that we view prime numbers as strings of digits. He then used concepts from formal language theory to define an interesting set of primes called the minimal primes:

For example, if our set is the set of prime numbers (written in radix 10), then we get the set {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}, and if our set is the set of composite numbers (written in radix 10), then we get the set {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}

Besides, if our set is the set of prime numbers written in radix b, then we get these sets:

Now, let's consider: if our set is the set of prime numbers >= b written in radix b (i.e. the prime numbers with at least two digits in radix b), then we get the sets:

This puzzle is an extension of the original minimal prime base b puzzle, to include CRUS Sierpinski/Riesel conjectures base b with k-values < b

The original minimal prime base b puzzle does not cover CRUS Sierpinski/Riesel conjectures base b with CK < b (such Riesel bases are 14, 20, 29, 32, 34, 38, 41, 44, 47, 50, 54, 56, 59, 62, 64, 65, 68, 69, 74, 77, 81, 83, 84, 86, 89, 90, 92, 94, 98, 104, 110, 113, 114, 116, 118, 119, 122, 125, 128, 129, 131, 132, 134, 137, 139, 140, 142, 144, 146, 149, 152, 153, 155, 158, 164, 167, 170, 173, 174, 176, 178, 179, 182, 184, 185, 186, 188, 189, 194, 197, 200, 202, 203, 204, 206, 208, 209, 212, 214, 216, 218, 219, 221, 224, 227, 229, 230, 233, 234, 236, 237, 239, 242, 244, 245, 246, 248, 251, 252, 254, 257, 258, 259, 260, 263, 264, 265, 266, 269, 272, 274, 275, 278, 279, 284, 285, 286, 289, 290, 293, 294, 296, 298, 299, 300, ..., and such Sierpinski bases are 14, 20, 29, 32, 34, 38, 41, 44, 47, 50, 54, 56, 59, 62, 64, 65, 68, 69, 74, 76, 77, 83, 84, 86, 89, 90, 92, 94, 98, 101, 104, 109, 110, 113, 114, 116, 118, 119, 122, 125, 128, 129, 131, 132, 134, 137, 139, 140, 142, 144, 146, 149, 152, 153, 154, 155, 158, 159, 160, 164, 167, 169, 170, 172, 173, 174, 176, 179, 181, 182, 184, 185, 186, 188, 189, 194, 197, 200, 202, 203, 204, 206, 208, 209, 212, 214, 216, 218, 219, 220, 221, 224, 227, 229, 230, 233, 234, 236, 237, 239, 242, 244, 245, 246, 248, 251, 252, 254, 257, 258, 259, 260, 263, 264, 265, 266, 269, 272, 274, 275, 278, 279, 281, 284, 285, 289, 290, 293, 294, 296, 298, 299, 300, ...), since in Riesel side, the prime is not minimal prime if either k-1 or b-1 (or both) is prime, and in Sierpinski side, the prime is not minimal prime if k is prime (e.g. 25*30^34205-1 is not minimal prime in base 30, since it is OT₃₄₂₀₅ in base 30, and T (= 29 in decimal) is prime, but it is minimal prime in base 30 if single-digit primes are not counted), but this extended version of minimal prime base b problem does, this requires a restriction of prime >= b, i.e. primes should have >=2 digits, and the single-digit primes (including the k-1, b-1, k) are not allowed.

[sweety439]

CRUS requires exponent n>=1 for these primes, n=0 is not acceptable to avoid the trivial primes (e.g. 2*b^n+1, 4*b^n+1, 6*b^n+1, 10*b^n+1, 12*b^n+1, 3*b^n-1, 4*b^n-1, 6*b^n-1, 8*b^n-1, 12*b^n-1, ... cannot be quickly eliminated with n=0, or the conjectures become much more easy and uninteresting)

For the same reason, this minimal prime puzzle requires >=base (i.e. >=2 digits) for these primes, single-digit primes are not acceptable to avoid the trivial primes (e.g. simple families containing digit 2, 3, 5, 7, B, D, H, J, N, ... cannot be quickly eliminated with the single-digit prime, or the conjectures become much more easy and uninteresting)

[sweety439]

Originally Posted by sweety439

Base b minimal primes (start with 2 digits) includes:

Some known minimal primes (start with b+1) and unsolved families for large bases b:

* For the repunit case (family {(1)}), see https://mersenneforum.org/attachment...1&d=1597771406 and https://raw.githubusercontent.com/xa...iesel%20k1.txt

* Unsolved family {(1)} in bases b = 185, 269, 281, 380, 384, 385, 394, 396, 452, 465, 511, 574, 598, 601, 629, 631, 632, 636, 711, 713, 759, 771, 795, 861, 866, 881, 938, 948, 951, 956, 963, 1005, 1015 (less than 1024)

* Unsolved family (40):{(121)} in base 243

* For the GFN case (family (1){(0)}(1)), see http://jeppesn.dk/generalized-fermat.html and http://www.noprimeleftbehind.net/crus/GFN-primes.htm

* Unsolved family (1){(0)}(1) in bases b = 38, 50, 62, 68, 86, 92, 98, 104, 122, 144, 168, 182, 186, 200, 202, 212, 214, 218, 244, 246, 252, 258, 286, 294, 298, 302, 304, 308, 322, 324, 338, 344, 354, 356, 362, 368, 380, 390, 394, 398, 402, 404, 410, 416, 422, 424, 446, 450, 454, 458, 468, 480, 482, 484, 500, 514, 518, 524, 528, 530, 534, 538, 552, 558, 564, 572, 574, 578, 580, 590, 602, 604, 608, 620, 622, 626, 632, 638, 648, 650, 662, 666, 668, 670, 678, 684, 692, 694, 698, 706, 712, 720, 722, 724, 734, 744, 746, 752, 754, 762, 766, 770, 792, 794, 802, 806, 812, 814, 818, 836, 840, 842, 844, 848, 854, 868, 870, 872, 878, 888, 896, 902, 904, 908, 922, 924, 926, 932, 938, 942, 944, 948, 954, 958, 964, 968, 974, 978, 980, 988, 994, 998, 1002, 1006, 1014, 1016 (less than 1024)

* Unsolved families (4){(0)}(1) and (16){(0)}(1) in base 32, (16){(0)}(1) in base 128, (36){(0)}(1) in base 216, (2){(0)}(1), (4){(0)}(1), (16){(0)}(1), (32){(0)}(1), (256){(0)}(1) in base 512, (10){(0)}(1) and (100){(0)}(1) in base 1000, (4){(0)}(1) and (16){(0)}(1) in base 1024

* Unsolved families {((b-1)/2)}((b+1)/2) in base b = 31, 37, 55, 63, 67, 77, 83, 89, 91, 93, 97, 99, 107, 109, 117, 123, 127, 133, 135, 137, 143, 147, 149, 151, 155, 161, 177, 179, 183, 189, 193, 197, 207, 211, 213, 215, 217, 223, 225, 227, 233, 235, 241, 247, 249, 255, 257, 263, 265, 269, 273, 277, 281, 283, 285, 287, 291, 293, 297, 303, 307, 311, 319, 327, 347, 351, 355, 357, 359, 361, 367, 369, 377, 381, 383, 385, 387, 389, 393, 397, 401, 407, 411, 413, 417, 421, 423, 437, 439, 443, 447, 457, 465, 467, 469, 473, 475, 481, 483, 489, 493, 495, 497, 509, 511, 515, 533, 541, 547, 549, 555, 563, 591, 593, 597, 601, 603, 611, 615, 619, 621, 625, 627, 629, 633, 635, 637, 645, 647, 651, 653, 655, 659, 663, 667, 671, 673, 675, 679, 683, 687, 691, 693, 697, 707, 709, 717, 731, 733, 735, 737, 741, 743, 749, 753, 755, 757, 759, 765, 767, 771, 773, 775, 777, 783, 785, 787, 793, 797, 801, 807, 809, 813, 817, 823, 825, 849, 851, 853, 865, 867, 873, 877, 887, 889, 893, 897, 899, 903, 907, 911, 915, 923, 927, 933, 937, 939, 941, 943, 945, 947, 953, 957, 961, 967, 975, 977, 983, 987, 993, 999, 1003, 1005, 1009, 1017 (less than 1024)

* Unsolved family (12):{(62)}:(63) in base 125, (24):{(171)}:(172) in base 343

* For the Williams 1st case (family (b-2){(b-1)}), see https://harvey563.tripod.com/wills.txt and https://www.rieselprime.de/ziki/Williams_prime_MM_least

* Unsolved family (b-2){(b-1)} in bases b = 128, 233, 268, 293, 383, 478, 488, 533, 554, 665, 698, 779, 863, 878, 932, 941, 1010 (less than 1024)

* For the Williams 2nd case (family (b-1){(0)}1), see https://www.rieselprime.de/ziki/Williams_prime_MP_least

* Unsolved family (b-1){(0)}1 in bases b = 123, 342, 362, 422, 438, 479, 487, 512, 542, 602, 757, 767, 817, 830, 872, 893, 932, 992, 997, 1005, 1007 (less than 1024)

* For the Williams 4th case (family (1)(1){(0)}(1)), see https://www.rieselprime.de/ziki/Williams_prime_PP_least

* Unsolved family (1)(1){(0)}(1) in bases 813, 863, 1017 (not base 962, since in base 962, (1)(0)(0)(0)(1) is prime) (less than 1024)

* Minimal primes (70)₃₀₁₈(1) in base 71, (81)₁₆₈(1) in base 82, (82)₉₆₄(1) in base 83, (87)₂₈₄₇(1) in base 88, (113)₉₉₀(1) in base 114, (127)₄₀₀(1) in base 128, (142)₂₈₁(1) in base 143, (144)₂₅₄(1) in base 145

* Unsolved family {(92)}(1) in base 93 and {(112)}(1) in base 113, {(151)}(1) in base 152, {(157)}(1) in base 158

* Minimal primes (1)(0)₁₉₃(79) in base 80, (1)(0)₁₃₉₉(106) in base 107, (1)(0)₂₀₀₈₇(112) in base 113, (1)(0)₆₄₃₆₉(122) in base 123, (1)(0)₅₀₃(127) in base 128, (1)(0)₁₀₃(160) in base 161

* For the (2){(0)}(1), (3){(0)}(1), (4){(0)}(1), ..., (12){(0)}(1) case, see https://www.rieselprime.de/ziki/Prot..._bases_least_n

* Unsolved family (2){(0)}(1) in bases 365, 383, 461, 512 (GFN), 542, 647, 773, 801, 836, 878, 908, 914, 917, 947, 1004

* Unsolved family (3){(0)}(1) in bases 718, 912

etc.

* For the (1){(b-1)}, (2){(b-1)}, (3){(b-1)}, ..., (11){(b-1)} case, see https://www.rieselprime.de/ziki/Ries..._bases_least_n

* Unsolved family (1){(580)} in base 581, (1){(991)} in base 992, (1){(1018)} in base 1019

* Unsolved family (2){(587)} in base 588, (2){(971)} in base 972

etc.

* Minimal primes (1)(0)₁₁₂(2) in base 47, (1)(0)₂₅₄(2) in base 89, (1)(0)₁₃₅(2) in base 159

* Unsolved family (1){(0)}(2) in base 167

* Minimal primes (80)₁₂₉(79) in base 81, (96)₇₄₆(95) in base 97, (196)₁₆₃(195) in base 197, (208)₁₂₅(207) in base 209, (214)₁₃₃(213) in base 215, (220)₅₅₁(219) in base 221, (286)₃₄₀₉(285) in base 287

* Unsolved family {(304)}(303) in base 305

* For k*b^n+1, see http://www.noprimeleftbehind.net/cru...onjectures.htm, all 1<=k<=b-1 are minimal primes or unsolved families

** Also, all two-digit (when written in base b) k-values while both digits d of k cannot have prime of the form d*b^n+1 are minimal primes or unsolved families

* For k*b^n-1, see http://www.noprimeleftbehind.net/cru...onjectures.htm, all 1<=k<=b-1 are minimal primes or unsolved families

** Also, all two-digit (when written in base b) k-values while both digits d of k-1 cannot have prime of the form (d+1)*b^n-1 are minimal primes or unsolved families

[sweety439]

Examples of families which can be proven to contain no primes > base (and no subsequences of these families can be primes > base):

(using 0-9 for digit values 0-9, A-Z for digit values 10-35, a-z for digit values 36-61, "+" for digit value 62, "/" for digit value 63, for bases <=64; and using decimal to represent individual digits and using ":" as separating mark, for bases >64)

* all families with ending number not coprime to base (all such numbers are not coprime to base, thus are either composite or factors of base (thus <= base), thus cannot be primes > base)
* all families with gcd of the digits > 1 (all such numbers are divisible by the gcd, and since the gcd is < base, all such numbers are either composite or equal to the gcd (thus also < base), thus cannot be primes > base)
* 1{0}1 in base 3 (all such numbers are divisible by 2)
* 2{0}1 in base 4 (all such numbers are divisible by 3)
* 1{0}1 in base 5 (all such numbers are divisible by 2)
* 1{0}3 in base 5 (all such numbers are divisible by 2)
* 3{0}1 in base 5 (all such numbers are divisible by 2)
* 4{0}1 in base 6 (all such numbers are divisible by 5)
* 1{0}1 in base 8 (all such numbers factored as sum of cubes)
* {1} in base 9 (all such numbers factored as difference of squares)
* {1}5 in base 9 (all such numbers are divisible either by 2 or by 5)
* 2{7} in base 9 (all such numbers are divisible either by 2 or by 5)
* 3{1} in base 9 (all such numbers factored as difference of squares)
* {3}5 in base 9 (all such numbers are divisible either by 2 or by 5)
* {3}8 in base 9 (all such numbers are divisible either by 2 or by 5)
* 3{8} in base 9 (all such numbers factored as difference of squares)
* 5{1} in base 9 (all such numbers are divisible either by 2 or by 5)
* 5{7} in base 9 (all such numbers are divisible either by 2 or by 5)
* 6{1} in base 9 (all such numbers are divisible either by 2 or by 5)
* {7}2 in base 9 (all such numbers are divisible either by 2 or by 5)
* {7}5 in base 9 (all such numbers are divisible either by 2 or by 5)
* 8{3} in base 9 (all such numbers are divisible either by 2 or by 5)
* {8}5 in base 9 (all such numbers factored as difference of squares)
* 4{6}9 in base 10 (all such numbers are divisible by 7)
* 2{5} in base 11 (all such numbers are divisible either by 2 or by 3)
* 3{5} in base 11 (all such numbers are divisible either by 2 or by 3)
* 3{7} in base 11 (all such numbers are divisible either by 2 or by 3)
* 4{7} in base 11 (all such numbers are divisible either by 2 or by 3)
* {B}9B in base 12 (all such numbers with even length factored as difference of squares, and all such numbers with odd length are divisible by 13)
* 1{0}B in base 14 (all such numbers are divisible either by 3 or by 5)
* 3{D} in base 14 (all such numbers are divisible either by 3 or by 5)
* 4{0}1 in base 14 (all such numbers are divisible either by 3 or by 5)
* 8{D} (all such numbers with odd length factored as difference of squares, and all such numbers with even length are divisible by 5)
* A{D} in base 14 (all such numbers are divisible either by 3 or by 5)
* B{0}1 in base 14 (all such numbers are divisible either by 3 or by 5)
* {D}3 in base 14 (all such numbers are divisible either by 3 or by 5)
* {D}5 in base 14 (all such numbers with even length factored as difference of squares, and all such numbers with odd length are divisible by 5)
* 9{6}8 in base 15 (all such numbers are divisible by 11)
* 1{5} in base 16 (all such numbers factored as difference of squares)
* {4}1 in base 16 (all such numbers factored as difference of squares)
* {4}D in base 16 (all such numbers are divisible by 3, 7, or 13)
* {8}F in base 16 (all such numbers are divisible by 3, 7, or 13)
* 8{F} in base 16 (all such numbers factored as difference of squares)
* B{4}1 in base 16 (all such numbers factored as difference of squares)
* {C}B in base 16 (all such numbers factored as difference of squares)
* {C}D in base 16 (all such numbers factored as x^4+4*y^4)
* {C}DD in base 16 (all such numbers factored as x^4+4*y^4)
* 1{9} in base 17 (all such numbers with odd length factored as difference of squares, and all such numbers with even length are divisible by 2)
* 1{6} in base 19 (all such numbers with odd length factored as difference of squares, and all such numbers with even length are divisible by 5)
* 1{0}D in base 20 (all such numbers are divisible either by 3 or by 7)
* 7{J} in base 20 (all such numbers are divisible either by 3 or by 7)
* 8{0}1 in base 20 (all such numbers are divisible either by 3 or by 7)
* C{J} in base 20 (all such numbers are divisible either by 3 or by 7)
* D{0}1 in base 20 (all such numbers are divisible either by 3 or by 7)
* {J}7 in base 20 (all such numbers are divisible either by 3 or by 7)
* 3{N} in base 24 (all such numbers with odd length factored as difference of squares, and all such numbers with even length are divisible by 5)
* 5{N} in base 24 (all such numbers with even length factored as difference of squares, and all such numbers with odd length are divisible by 5)
* 8{N} in base 24 (all such numbers with odd length factored as difference of squares, and all such numbers with even length are divisible by 5)
* {1} in base 25 (all such numbers factored as difference of squares)
* 2{1} in base 25 (all such numbers factored as difference of squares)
* 1{3} in base 25 (all such numbers factored as difference of squares)
* 1{8} in base 25 (all such numbers factored as difference of squares)
* 5{1} in base 25 (all such numbers factored as difference of squares)
* 5{8} in base 25 (all such numbers factored as difference of squares)
* 7{1} in base 25 (all such numbers factored as difference of squares)
* A{3} in base 25 (all such numbers factored as difference of squares)
* L{8} in base 25 (all such numbers factored as difference of squares)
* 1{0}8 in base 27 (all such numbers factored as sum of cubes)
* 7{Q} in base 27 (all such numbers factored as difference of cubes)
* 8{0}1 in base 27 (all such numbers factored as sum of cubes)
* 9{G} in base 27 (all such numbers factored as sum of cubes)
* {D}E in base 27 (all such numbers factored as sum of cubes)
* {Q}J in base 27 (all such numbers factored as difference of cubes)
* 1{0}1 in base 32 (all such numbers factored as sum of 5th powers)
* {1} in base 32 (all such numbers factored as difference of 5th powers)
* F{W} in base 33 (all such numbers with odd length factored as difference of squares, and all such numbers with even length are divisible by 17)
* 1{B} in base 34 (all such numbers with odd length factored as difference of squares, and all such numbers with even length are divisible by 5)
* 8{X} in base 34 (all such numbers with odd length factored as difference of squares, and all such numbers with even length are divisible by 5)
* {X}P in base 34 (all such numbers with even length factored as difference of squares, and all such numbers with odd length are divisible by 5)
* 3{7} in base 36 (all such numbers factored as difference of squares)
* 3{Z} in base 36 (all such numbers factored as difference of squares)
* 8{Z} in base 36 (all such numbers factored as difference of squares)
* O{Z} in base 36 (all such numbers factored as difference of squares)
* {Z}B in base 36 (all such numbers factored as difference of squares)
* C{b} in base 38 (all such numbers are divisible by 3, 5, or 17)
* G{0}1 in base 38 (all such numbers are divisible by 3, 5, or 17)
* 3{c} in base 39 (all such numbers with odd length factored as difference of squares, and all such numbers with even length are divisible by 5)
* 1{9} in base 41 (all such numbers with odd length factored as difference of squares, and all such numbers with even length are divisible by 2)
* 1{0}8 in base 47 (all such numbers are divisible by 3, 5, or 13)
* 1{0}G in base 47 (all such numbers are divisible by 3, 5, or 17)
* 8{0}1 in base 47 (all such numbers are divisible by 3, 5, or 13)
* D{k} in base 47 (all such numbers are divisible by 3, 5, or 13)
* G{0}1 in base 47 (all such numbers are divisible by 3, 5, or 17)
* H{n} in base 50 (all such numbers with even length factored as difference of squares, and all such numbers with odd length are divisible by 17)
* 3{r} in base 54 (all such numbers with odd length factored as difference of squares, and all such numbers with even length are divisible by 5)
* 5{r} in base 54 (all such numbers with even length factored as difference of squares, and all such numbers with odd length are divisible by 5)
* 8{r} in base 54 (all such numbers with odd length factored as difference of squares, and all such numbers with even length are divisible by 5)
* jP{0}1 in base 55 (all such numbers with length == 2 mod 4 factored as x^4+4*y^4, all such numbers with odd length are divisible by 7, and all such numbers with length == 0 mod 4 are divisible by 17)
* 1{7} in base 57 (all such numbers with odd length factored as difference of squares, and all such numbers with even length are divisible by 2)
* 3{7} in base 57 (all such numbers with odd length factored as difference of squares, and all such numbers with even length are divisible by 2)
* F{7} in base 57 (all such numbers with odd length factored as difference of squares, and all such numbers with even length are divisible by 2)
* (12):{(64)}:(65) in base 81 (all such numbers factored as x^4+4*y^4)
* {(64)}:(65) in base 81 (all such numbers factored as x^4+4*y^4)
* (73):{(80)} in base 81 (all such numbers are divisible by 7, 13, or 73)
* (8):{(0)}:(1) in base 128 (such numbers with length n are divisible by 2^(2^valuation(7*n-4, 2))+1, and 7*n-4 cannot be power of 2, since all powers of 2 are == 1, 2, 4 mod 7)
* (32):{(0)}:(1) in base 128 (such numbers with length n are divisible by 2^(2^valuation(7*n-2, 2))+1, and 7*n-2 cannot be power of 2, since all powers of 2 are == 1, 2, 4 mod 7)
* (64):{(0)}:(1) in base 128 (such numbers with length n are divisible by 2^(2^valuation(7*n-1, 2))+1, and 7*n-1 cannot be power of 2, since all powers of 2 are == 1, 2, 4 mod 7)
* (16):{(0)}:(1) in base 200 (all such numbers with length == 3 mod 4 factored as x^4+4*y^4, all such numbers with even length are divisible by 7, and all such numbers with length == 1 mod 4 are divisible by 17)
* (73):{(337)} in base 338 (all such numbers are divisible by 3, 5, or 73)
* (21):{(130)} in base 391 (all such numbers with odd length factored as difference of squares, all such numbers with length == 1 mod 3 factored as difference of cubes, all such numbers with length == 2 mod 3 are divisible by 19, and all such numbers with length == 0 mod 6 are divisible by 109)
* (73):{(391)} in base 392 (all such numbers are divisible by 3, 5, or 73)
* (73):{(445)} in base 446 (all such numbers are divisible by 3, 7, 13, or 73)
* (1):(399):{(0)}:(1) in base 625 (all such numbers factored as x^4+4*y^4)
* (4):{(0)}:(1) in base 625 (all such numbers factored as x^4+4*y^4)
* (63):{(935)} in base 936 (all such numbers with odd length factored as difference of squares, all such numbers with length == 1 mod 3 factored as difference of cubes, all such numbers with length == 2 mod 6 are divisible by 37, and all such numbers with length == 0 mod 6 are divisible by 109)
* (63):{(956)} in base 957 (all such numbers with length == 1 mod 3 factored as difference of cubes, all such numbers with length == 2 mod 3 are divisible by 73, and all such numbers with length == 0 mod 3 are divisible by 19)

[sweety439]

Update pdf file for the proofs (not complete, continue updating ....)

[sweety439]

Now, I try to prove base 12 (may find some minimal primes not in my current list)

In base 12, the possible (first digit,last digit) for an element with >=3 digits in the minimal set of the strings for primes with at least two digits are

(1,1), (1,5), (1,7), (1,B), (2,1), (2,5), (2,7), (2,B), (3,1), (3,5), (3,7), (3,B), (4,1), (4,5), (4,7), (4,B), (5,1), (5,5), (5,7), (5,B), (6,1), (6,5), (6,7), (6,B), (7,1), (7,5), (7,7), (7,B), (8,1), (8,5), (8,7), (8,B), (9,1), (9,5), (9,7), (9,B), (A,1), (A,5), (A,7), (A,B), (B,1), (B,5), (B,7), (B,B)

* Case (1,1):

** 11 is prime, and thus the only minimal prime in this family.

* Case (1,5):

** 15 is prime, and thus the only minimal prime in this family.

* Case (1,7):

** 17 is prime, and thus the only minimal prime in this family.

* Case (1,B):

** 1B is prime, and thus the only minimal prime in this family.

* Case (2,1):

** Since 25, 27, 11, 31, 51, 61, 81, 91, 221, 241, 2A1, 2B1 are primes, we only need to consider the family 2{0}1 (since any digits 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B between them will produce smaller primes)

*** The smallest prime of the form 2{0}1 is 2001

* Case (2,5):

** 25 is prime, and thus the only minimal prime in this family.

* Case (2,7):

** 27 is prime, and thus the only minimal prime in this family.

* Case (2,B):

** Since 25, 27, 1B, 3B, 4B, 5B, 6B, 8B, AB, 2BB are primes, we only need to consider the family 2{0,2,9}B (since any digits 1, 3, 4, 5, 6, 7, 8, A, B between them will produce smaller primes)

*** Since 90B, 200B, 202B, 222B, 229B, 292B, 299B are primes, we only need to consider the numbers 20B, 22B, 29B, 209B, 220B (since any digits combo 00, 02, 22, 29, 90, 92, 99 between them will produce smaller primes)

**** None of 20B, 22B, 29B, 209B, 220B are primes.

* Case (3,1):

** 31 is prime, and thus the only minimal prime in this family.

* Case (3,5):

** 35 is prime, and thus the only minimal prime in this family.

* Case (3,7):

** 37 is prime, and thus the only minimal prime in this family.

* Case (3,B):

** 3B is prime, and thus the only minimal prime in this family.

[sweety439]

* Case (4,1):

** Since 45, 4B, 11, 31, 51, 61, 81, 91, 401, 421, 471 are primes, we only need to consider the family 4{4,A}1 (since any digit 0, 1, 2, 3, 5, 6, 7, 8, 9, B between them will produce smaller primes)

*** Since A41 and 4441 are primes, we only need to consider the families 4{A}1 and 44{A}1 (since any digit combo 44, A4 between them will produce smaller primes)

**** All numbers of the form 4{A}1 are divisible by 5, thus cannot be prime.

**** The smallest prime of the form 44{A}1 is 44AAA1

* Case (4,5):

** 45 is prime, and thus the only minimal prime in this family.

* Case (4,7):

** Since 45, 4B, 17, 27, 37, 57, 67, 87, A7, B7, 447, 497 are primes, we only need to consider the family 4{0,7}7 (since any digit 1, 2, 3, 4, 5, 6, 8, 9, A, B between them will produce smaller primes)

*** Since 4707 and 4777 are primes, we only need to consider the families 4{0}7 and 4{0}77 (since any digit combo 70, 77 between them will produce smaller primes)

**** All numbers of the form 4{0}7 are divisible by B, thus cannot be prime.

**** The smallest prime of the form 4{0}77 is 400000000000000000000000000000000000000077

* Case (4,B):

** 4B is prime, and thus the only minimal prime in this family.

[sweety439]

* Case (5,1):

** 51 is prime, and thus the only minimal prime in this family.

* Case (5,5):

** Since 51, 57, 5B, 15, 25, 35, 45, 75, 85, 95, B5, 565 are primes, we only need to consider the family 5{0,5,A}5 (since any digits 1, 2, 3, 4, 6, 7, 8, 9, B between them will produce smaller primes)

*** All numbers of the form 5{0,5,A}5 are divisible by 5, thus cannot be prime.

* Case (5,7):

** 57 is prime, and thus the only minimal prime in this family.

* Case (5,B):

** 5B is prime, and thus the only minimal prime in this family.

* Case (6,1):

** 61 is prime, and thus the only minimal prime in this family.

* Case (6,5):

** Since 61, 67, 6B, 15, 25, 35, 45, 75, 85, 95, B5, 655, 665 are primes, we only need to consider the family 6{0,A}5 (since any digits 1, 2, 3, 4, 5, 6, 7, 8, 9, B between them will produce smaller primes)

*** Since 6A05 and 6AA5 are primes, we only need to consider the families 6{0}5 and 6{0}A5 (since any digit combo A0, AA between them will produce smaller primes)

**** All numbers of the form 6{0}5 are divisible by B, thus cannot be prime.

**** The smallest prime of the form 6{0}A5 is 600A5

* Case (6,7):

** 67 is prime, and thus the only minimal prime in this family.

* Case (6,B):

** 6B is prime, and thus the only minimal prime in this family.

[sweety439]

* Case (7,1):

** Since 75, 11, 31, 51, 61, 81, 91, 701, 721, 771, 7A1 are primes, we only need to consider the family 7{4,B}1 (since any digits 0, 1, 2, 3, 5, 6, 7, 8, 9, A between them will produce smaller primes)

*** Since 7BB, 7441 and 7B41 are primes, we only need to consider the numbers 741, 7B1, 74B1

**** None of 741, 7B1, 74B1 are primes.

* Case (7,5):

** 75 is prime, and thus the only minimal prime in this family.

* Case (7,7):

** Since 75, 17, 27, 37, 57, 67, 87, A7, B7, 747, 797 are primes, we only need to consider the family 7{0,7}7 (since any digits 1, 2, 3, 4, 5, 6, 8, 9, A, B between them will produce smaller primes)

*** All numbers of the form 7{0,7}7 are divisible by 7, thus cannot be prime.

* Case (7,B):

** Since 75, 1B, 3B, 4B, 5B, 6B, 8B, AB, 70B, 77B, 7BB are primes, we only need to consider the family 7{2,9}B (since any digits 0, 1, 3, 4, 5, 6, 7, 8, A, B between them will produce smaller primes)

*** Since 222B, 729B is prime, we only need to consider the families 7{9}B, 7{9}2B, 7{9}22B (since any digits combo 222, 29 between them will produce smaller primes)

**** The smallest prime of the form 7{9}B is 7999B

**** The smallest prime of the form 7{9}2B is 79992B (not minimal prime, since 992B and 7999B are primes)

**** The smallest prime of the form 7{9}22B is 79922B (not minimal prime, since 992B is prime)

* Case (8,1):

** 81 is prime, and thus the only minimal prime in this family.

* Case (8,5):

** 85 is prime, and thus the only minimal prime in this family.

* Case (8,7):

** 87 is prime, and thus the only minimal prime in this family.

* Case (8,B):

** 8B is prime, and thus the only minimal prime in this family.

[sweety439]

* Case (9,1):

** 91 is prime, and thus the only minimal prime in this family.

* Case (9,5):

** 95 is prime, and thus the only minimal prime in this family.

* Case (9,7):

** Since 91, 95, 17, 27, 37, 57, 67, 87, A7, B7, 907 are primes, we only need to consider the family 9{4,7,9}7 (since any digit 0, 1, 2, 3, 5, 6, 8, A, B between them will produce smaller primes)

*** Since 447, 497, 747, 797, 9777, 9947, 9997 are primes, we only need to consider the numbers 947, 977, 997, 9477, 9977 (since any digits combo 44, 49, 74, 77, 79, 94, 99 between them will produce smaller primes)

**** None of 947, 977, 997, 9477, 9977 are primes.

* Case (9,B):

** Since 91, 95, 1B, 3B, 4B, 5B, 6B, 8B, AB, 90B, 9BB are primes, we only need to consider the family 9{2,7,9}B (since any digit 0, 1, 3, 4, 5, 6, 8, A, B between them will produce smaller primes)

*** Since 27, 77B, 929B, 992B, 997B are primes, we only need to consider the families 9{2,7}2{2}B, 97{2,9}B, 9{7,9}9{9}B (since any digits combo 27, 29, 77, 92, 97 between them will produce smaller primes)

**** For the 9{2,7}2{2}B family, since 27 and 77B are primes, we only need to consider the families 9{2}2{2}B and 97{2}2{2}B (since any digits combo 27, 77 between (9,2{2}B) will produce smaller primes)

***** The smallest prime of the form 9{2}2{2}B is 9222B (not minimal prime, since 222B is prime)

***** The smallest prime of the form 97{2}2{2}B is 9722222222222B (not minimal prime, since 222B is prime)

**** For the 97{2,9}B family, since 729B and 929B are primes, we only need to consider the family 97{9}{2}B (since any digits combo 29 between (97,B) will produce smaller primes)

***** Since 222B is prime, we only need to consider the families 97{9}B, 97{9}2B, 97{9}22B (since any digit combo 222 between (97,B) will produce smaller primes)

****** All numbers of the form 97{9}B are divisible by 11, thus cannot be prime.

****** The smallest prime of the form 97{9}2B is 979999992B (not minimal prime, since 9999B is prime)

****** All numbers of the form 97{9}22B are divisible by 11, thus cannot be prime.

**** For the 9{7,9}9{9}B family, since 77B and 9999B are primes, we only need to consider the numbers 99B, 999B, 979B, 9799B, 9979B

***** None of 99B, 999B, 979B, 9799B, 9979B are primes.

* Case (A,1):

** Since A7, AB, 11, 31, 51, 61, 81, 91, A41 are primes, we only need to consider the family A{0,2,A}1 (since any digits 1, 3, 4, 5, 6, 7, 8, 9, B between them will produce smaller primes)

*** Since 221, 2A1, A0A1, A201 are primes, we only need to consider the families A{A}{0}1 and A{A}{0}21 (since any digits combo 0A, 20, 22, 2A between them will produce smaller primes)

**** For the A{A}{0}1 family:

***** All numbers of the form A{0}1 are divisible by B, thus cannot be prime.

***** The smallest prime of the form AA{0}1 is AA000001

***** The smallest prime of the form AAA{0}1 is AAA0001

***** The smallest prime of the form AAAA{0}1 is AAAA1

****** Since this prime has no 0's, we do not need to consider the families {A}1, {A}01, {A}001, etc.

**** All numbers of the form A{A}{0}21 are divisible by 5, thus cannot be prime.

[sweety439]

* Case (A,5):

** Since A7, AB, 15, 25, 35, 45, 75, 85, 95, B5 are primes, we only need to consider the family A{0,5,6,A}5 (since any digits 1, 2, 3, 4, 7, 8, 9, B between them will produce smaller primes)

*** Since 565, 655, 665, A605, A6A5, AA65 are primes, we only need to consider the families A{0,5,A}5 and A{0}65 (since any digits combo 56, 60, 65, 66, 6A, A6 between them will produce smaller primes)

**** All numbers of the form A{0,5,A}5 are divisible by 5, thus cannot be prime.

**** The smallest prime of the form A{0}65 is A00065

* Case (A,7):

** A7 is prime, and thus the only minimal prime in this family.

* Case (A,B):

** AB is prime, and thus the only minimal prime in this family.

* Case (B,1):

** Since B5, B7, 11, 31, 51, 61, 81, 91, B21 are primes, we only need to consider the family B{0,4,A,B}1 (since any digits 1, 2, 3, 5, 6, 7, 8, 9 between them will produce smaller primes)

*** Since 4B, AB, 401, A41, B001, B0B1, BB01, BB41 are primes, we only need to consider the families B{A}0{4,A}1, B{0,4}4{4,A}1, B{0,4,A,B}A{0,A}1, B{B}B{A,B}1 (since any digits combo 00, 0B, 40, 4B, A4, AB, B0, B4 between them will produce smaller primes)

**** For the B{A}0{4,A}1 family, since A41 is prime, we only need consider the families B0{4}{A}1 and B{A}0{A}1

***** For the B0{4}{A}1 family, since B04A1 is prime, we only need to consider the families B0{4}1 and B0{A}1

****** The smallest prime of the form B0{4}1 is B04441 (not minimal prime, since 4441 is prime)

****** The smallest prime of the form B0{A}1 is B0AAAAA1 (not minimal prime, since AAAA1 is prime)

***** For the B{A}0{A}1 family, since A0A1 is prime, we only need to consider the families B{A}01 and B0{A}1

****** The smallest prime of the form B{A}01 is BAA01

****** The smallest prime of the form B0{A}1 is B0AAAAA1 (not minimal prime, since AAAA1 is prime)

**** For the B{0,4}4{4,A}1 family, since 4441 is prime, we only need to consider the families B{0}4{4,A}1 and B{0,4}4{A}1

***** For the B{0}4{4,A}1 family, since B001 is prime, we only need to consider the families B4{4,A}1 and B04{4,A}1

****** For the B4{4,A}1 family, since A41 is prime, we only need to consider the family B4{4}{A}1

******* Since 4441 and BAAA1 are primes, we only need to consider the numbers B41, B441, B4A1, B44A1, B4AA1, B44AA1

******** None of B41, B441, B4A1, B44A1, B4AA1, B44AA1 are primes.

****** For the B04{4,A}1 family, since B04A1 is prime, we only need to consider the family B04{4}1

******* The smallest prime of the form B04{4}1 is B04441 (not minimal prime, since 4441 is prime)

***** For the B{0,4}4{A}1 family, since 401, 4441, B001 are primes, we only need to consider the families B4{A}1, B04{A}1, B44{A}1, B044{A}1 (since any digits combo 00, 40, 44 between (B,4{A}1) will produce smaller primes)

****** The smallest prime of the form B4{A}1 is B4AAA1 (not minimal prime, since BAAA1 is prime)

****** The smallest prime of the form B04{A}1 is B04A1

****** The smallest prime of the form B44{A}1 is B44AAAAAAA1 (not minimal prime, since BAAA1 is prime)

****** The smallest prime of the form B044{A}1 is B044A1 (not minimal prime, since B04A1 is prime)

**** For the B{0,4,A,B}A{0,A}1 family, since all numbers in this family with 0 between (B,1) are in the B{A}0{4,A}1 family, and all numbers in this family with 4 between (B,1) are in the B{0,4}4{4,A}1 family, we only need to consider the family B{A,B}A{A}1

***** Since BAAA1 is prime, we only need to consider the families B{A,B}A1 and B{A,B}AA1

****** For the B{A,B}A1 family, since AB and BAAA1 are primes, we only need to consider the families B{B}A1 and B{B}AA1

******* All numbers of the form B{B}A1 are divisible by B, thus cannot be prime.

******* The smallest prime of the form B{B}AA1 is BBBAA1

****** For the B{A,B}AA1 family, since BAAA1 is prime, we only need to consider the families B{B}AA1

******* The smallest prime of the form B{B}AA1 is BBBAA1

**** For the B{B}B{A,B}1 family, since AB and BAAA1 are primes, we only need to consider the families B{B}B1, B{B}BA1, B{B}BAA1 (since any digits combo AB or AAA between (B{B}B,1) will produce smaller primes)

***** The smallest prime of the form B{B}B1 is BBBB1

***** All numbers of the form B{B}BA1 are divisible by B, thus cannot be prime.

***** The smallest prime of the form B{B}BAA1 is BBBAA1

* Case (B,5):

** B5 is prime, and thus the only minimal prime in this family.

* Case (B,7):

** B7 is prime, and thus the only minimal prime in this family.

* Case (B,B):

** Since B5, B7, 1B, 3B, 4B, 5B, 6B, 8B, AB, B2B are primes, we only need to consider the family B{0,9,B}B (since any digits 1, 2, 3, 4, 5, 6, 7, 8, A between them will produce smaller primes)

*** Since 90B and 9BB are primes, we only need to consider the families B{0,B}{9}B

**** Since 9999B is prime, we only need to consider the families B{0,B}B, B{0,B}9B, B{0,B}99B, B{0,B}999B

***** All numbers of the form B{0,B}B are divisible by B, thus cannot be prime.

***** For the B{0,B}9B family:

****** Since B0B9B and BB09B are primes, we only need to consider the families B{0}9B and B{B}9B (since any digits combo 0B, B0 between (B,9B) will produce smaller primes)

******* The smallest prime of the form B{0}9B is B0000000000000000000000000009B

******* All numbers of the from B{B}9B is either divisible by 11 (if totally number of B's is even) or factored as 10^(2*n)-21 = (10^n-5) * (10^n+5) (if totally number of B's is odd number 2*n-1), thus cannot be prime.

***** For the B{0,B}99B family:

****** Since B0B9B and BB09B are primes, we only need to consider the families B{0}99B and B{B}99B (since any digits combo 0B, B0 between (B,99B) will produce smaller primes)

******* The smallest prime of the form B{0}99B is B00099B

******* The smallest prime of the form B{B}99B is BBBBBB99B

***** For the B{0,B}999B family:

****** Since B0B9B and BB09B are primes, we only need to consider the families B{0}999B and B{B}999B (since any digits combo 0B, B0 between (B,999B) will produce smaller primes)

******* The smallest prime of the form B{0}999B is B0000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000999B (not minimal prime, since B00099B and B0000000000000000000000000009B are primes)

******* The smallest prime of the form B{B}999B is BBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBB999B (not minimal prime, since BBBBBB99B is prime)