mersenneforum.org - generalized minimal (probable) primes

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- sweety439 (https://www.mersenneforum.org/forumdisplay.php?f=137)

- - generalized minimal (probable) primes (https://www.mersenneforum.org/showthread.php?t=21819)

Unsolved families which are CRUS Sierpinski/Riesel problems but with k's > CK:

Base 32:

G{0}1: 16*32^n+1
UG{0}1: 976*32^n+1

Base 33:

FFF{W}: 16846*33^n-1

Base 41:

FZ{0}1: 650*41^n+1
R0R8{0}1: 1861982*41^n+1
S{0}1: 28*41^n+1
XL4{0}1: 56338*41^n+1
Z098{0}1: 2412612*41^n+1
Z0R{0}1: 58862*41^n+1
EF{e}: 590*41^n-1
PI{e}: 1044*41^n-1
UFM{e}: 51068*41^n-1
UX{e}: 1264*41^n-1
XOC{e}: 56470*41^n-1
XQO{e}: 56564*41^n-1
XQXXXX{e}: 3899055672*41^n-1
XQ{e}: 1380*41^n-1

Base 43:

Y6{0}1: 1468*43^n+1
6XF{0}1: 12528*43^n+1
8Q6{0}1: 15916*43^n+1
XZZ{g}: 62558*43^n-1
YFa{g}: 63548*43^n-1
dcU{g}: 73776*43^n-1
4ZZZ{g}: 384284*43^n-1
8OR{g}: 15852*43^n-1
9QQ{g}: 17786*43^n-1
FFFFFFFQ{g}: 4174357242012*43^n-1
FFFQ{g}: 1221012*43^n-1

All smallest generalized repunit prime base b are minimal prime base b, since they are of the form {1} in base b, for the smallest generalized repunit (probable) prime base b for b<=1024, see [URL="https://raw.githubusercontent.com/xayahrainie4793/Sierpinski-Riesel-for-fixed-k-and-variable-base/master/Riesel%20k1.txt"]https://raw.githubusercontent.com/xayahrainie4793/Sierpinski-Riesel-for-fixed-k-and-variable-base/master/Riesel%20k1.txt[/URL]

All smallest generalized Fermat prime base b (for even b) and all smallest generalized half Fermat prime base b (for odd b) are minimal prime base b, unless (b-1)/2 is prime for odd b, since they are of the form 1{0}1 in base b (for even b) or {(b-1)/2}(b+1)/2 in base b (for odd b), for the smallest generalized (half) Fermat (probable) prime base b for b<=1024, see [URL="https://raw.githubusercontent.com/xayahrainie4793/Sierpinski-Riesel-for-fixed-k-and-variable-base/master/Sierp%20k1.txt"]https://raw.githubusercontent.com/xayahrainie4793/Sierpinski-Riesel-for-fixed-k-and-variable-base/master/Sierp%20k1.txt[/URL]

There are no known generalized repunit (probable) primes in these bases <= 1024: (search limit: 100000)

{185, 269, 281, 380, 384, 385, 394, 452, 465, 511, 574, 601, 631, 632, 636, 711, 713, 759, 771, 795, 861, 866, 881, 938, 948, 951, 956, 963, 1005, 1015}

There are no known generalized (half) Fermat (probable) primes in these bases <= 1024: (search limit: 2^22 for GFN for even bases, 2^18 for half GFN for odd bases)

{31, 38, 50, 55, 62, 63, 67, 68, 77, 83, 86, 89, 91, 92, 97, 98, 99, 104, 107, 109, 122, 123, 127, 135, 137, 143, 144, 147, 149, 151, 155, 161, 168, 179, 182, 183, 186, 189, 197, 200, 202, 207, 211, 212, 214, 215, 218, 223, 227, 233, 235, 241, 244, 246, 247, 249, 252, 255, 257, 258, 263, 265, 269, 281, 283, 285, 286, 287, 291, 293, 294, 298, 302, 303, 304, 307, 308, 311, 319, 322, 324, 327, 338, 344, 347, 351, 354, 355, 356, 359, 362, 367, 368, 369, 377, 380, 383, 387, 389, 390, 394, 398, 401, 402, 404, 407, 410, 411, 413, 416, 417, 422, 423, 424, 437, 439, 443, 446, 447, 450, 454, 458, 467, 468, 469, 473, 475, 480, 482, 483, 484, 489, 493, 495, 497, 500, 509, 511, 514, 515, 518, 524, 528, 530, 533, 534, 538, 547, 549, 552, 555, 558, 563, 564, 572, 574, 578, 580, 590, 591, 593, 597, 601, 602, 603, 604, 608, 611, 615, 619, 620, 622, 626, 627, 629, 632, 635, 637, 638, 645, 647, 648, 650, 651, 653, 655, 659, 662, 663, 666, 667, 668, 670, 671, 675, 678, 679, 683, 684, 687, 691, 692, 694, 698, 706, 707, 709, 712, 720, 722, 724, 731, 734, 735, 737, 741, 743, 744, 746, 749, 752, 753, 754, 755, 759, 762, 766, 767, 770, 771, 773, 775, 783, 785, 787, 792, 794, 797, 802, 806, 807, 809, 812, 813, 814, 818, 823, 825, 836, 840, 842, 844, 848, 849, 851, 853, 854, 867, 868, 870, 872, 873, 878, 887, 888, 889, 893, 896, 899, 902, 903, 904, 907, 908, 911, 915, 922, 923, 924, 926, 927, 932, 937, 938, 939, 941, 942, 943, 944, 945, 947, 948, 953, 954, 958, 961, 964, 967, 968, 974, 975, 977, 978, 980, 983, 987, 988, 993, 994, 998, 999, 1002, 1003, 1006, 1009, 1014, 1016}

[QUOTE=sweety439;561623]There are no known generalized repunit (probable) primes in these bases <= 1024: (search limit: 100000)

{185, 269, 281, 380, 384, 385, 394, 452, 465, 511, 574, 601, 631, 632, 636, 711, 713, 759, 771, 795, 861, 866, 881, 938, 948, 951, 956, 963, 1005, 1015}

There are no known generalized (half) Fermat (probable) primes in these bases <= 1024: (search limit: 2^22 for GFN for even bases, 2^18 for half GFN for odd bases)

{31, 38, 50, 55, 62, 63, 67, 68, 77, 83, 86, 89, 91, 92, 97, 98, 99, 104, 107, 109, 122, 123, 127, 135, 137, 143, 144, 147, 149, 151, 155, 161, 168, 179, 182, 183, 186, 189, 197, 200, 202, 207, 211, 212, 214, 215, 218, 223, 227, 233, 235, 241, 244, 246, 247, 249, 252, 255, 257, 258, 263, 265, 269, 281, 283, 285, 286, 287, 291, 293, 294, 298, 302, 303, 304, 307, 308, 311, 319, 322, 324, 327, 338, 344, 347, 351, 354, 355, 356, 359, 362, 367, 368, 369, 377, 380, 383, 387, 389, 390, 394, 398, 401, 402, 404, 407, 410, 411, 413, 416, 417, 422, 423, 424, 437, 439, 443, 446, 447, 450, 454, 458, 467, 468, 469, 473, 475, 480, 482, 483, 484, 489, 493, 495, 497, 500, 509, 511, 514, 515, 518, 524, 528, 530, 533, 534, 538, 547, 549, 552, 555, 558, 563, 564, 572, 574, 578, 580, 590, 591, 593, 597, 601, 602, 603, 604, 608, 611, 615, 619, 620, 622, 626, 627, 629, 632, 635, 637, 638, 645, 647, 648, 650, 651, 653, 655, 659, 662, 663, 666, 667, 668, 670, 671, 675, 678, 679, 683, 684, 687, 691, 692, 694, 698, 706, 707, 709, 712, 720, 722, 724, 731, 734, 735, 737, 741, 743, 744, 746, 749, 752, 753, 754, 755, 759, 762, 766, 767, 770, 771, 773, 775, 783, 785, 787, 792, 794, 797, 802, 806, 807, 809, 812, 813, 814, 818, 823, 825, 836, 840, 842, 844, 848, 849, 851, 853, 854, 867, 868, 870, 872, 873, 878, 887, 888, 889, 893, 896, 899, 902, 903, 904, 907, 908, 911, 915, 922, 923, 924, 926, 927, 932, 937, 938, 939, 941, 942, 943, 944, 945, 947, 948, 953, 954, 958, 961, 964, 967, 968, 974, 975, 977, 978, 980, 983, 987, 988, 993, 994, 998, 999, 1002, 1003, 1006, 1009, 1014, 1016}[/QUOTE]

The GFN for these bases (always minimal primes):

{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}

The half GFN for these bases are also minimal primes:

{31, 55, 67, 77, 89, 91, 97, 99, 109, 127, 137, 149, 151, 155, 161, 183, 189, 197, 211, 223, 233, 235, 241, 247, 249, 257, 265, 269, 281, 283, 285, 287, 291, 293, 307, 311, 319, 351, 355, 367, 369, 377, 389, 401, 407, 411, 413, 417, 437, 439, 443, 469, 473, 475, 489, 493, 495, 497, 509, 511, 533, 547, 549, 591, 593, 597, 601, 603, 611, 619, 629, 637, 645, 647, 651, 653, 655, 659, 667, 671, 679, 683, 687, 691, 709, 731, 737, 741, 743, 749, 753, 755, 771, 773, 775, 783, 785, 787, 797, 807, 809, 813, 823, 825, 849, 851, 853, 873, 889, 893, 903, 907, 911, 937, 939, 941, 943, 945, 947, 953, 961, 967, 977, 987, 993, 1003, 1009}

However, the half GFN for these bases are not minimal primes, since (b-1)/2 is prime:

{63, 83, 107, 123, 135, 143, 147, 179, 207, 215, 227, 255, 263, 303, 327, 347, 359, 383, 387, 423, 447, 467, 483, 515, 555, 563, 615, 627, 635, 663, 675, 707, 735, 759, 767, 867, 887, 899, 915, 923, 927, 975, 983, 999}

The remain k < b (also including k > CK, if k < b, e.g. 28*41^n+1 and 27*34^n-1) in CRUS corresponding to minimal primes to base b if and only if:

* In Sierpinski case, k is not prime (since it is k{0}1 in base b)
* In Riesel case, neither k-1 nor b-1 is prime (since it is (k-1){(b-1)} in base b)

e.g. the smallest prime of the form 4*53^n+1 (already searched to 1.65M) will be minimal prime base 53, if it exists (CRUS conjectured that they all exist)

[QUOTE=sweety439;561624]The GFN for these bases (always minimal primes):

{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}

The half GFN for these bases are also minimal primes:

{31, 55, 67, 77, 89, 91, 97, 99, 109, 127, 137, 149, 151, 155, 161, 183, 189, 197, 211, 223, 233, 235, 241, 247, 249, 257, 265, 269, 281, 283, 285, 287, 291, 293, 307, 311, 319, 351, 355, 367, 369, 377, 389, 401, 407, 411, 413, 417, 437, 439, 443, 469, 473, 475, 489, 493, 495, 497, 509, 511, 533, 547, 549, 591, 593, 597, 601, 603, 611, 619, 629, 637, 645, 647, 651, 653, 655, 659, 667, 671, 679, 683, 687, 691, 709, 731, 737, 741, 743, 749, 753, 755, 771, 773, 775, 783, 785, 787, 797, 807, 809, 813, 823, 825, 849, 851, 853, 873, 889, 893, 903, 907, 911, 937, 939, 941, 943, 945, 947, 953, 961, 967, 977, 987, 993, 1003, 1009}

However, the half GFN for these bases are not minimal primes, since (b-1)/2 is prime:

{63, 83, 107, 123, 135, 143, 147, 179, 207, 215, 227, 255, 263, 303, 327, 347, 359, 383, 387, 423, 447, 467, 483, 515, 555, 563, 615, 627, 635, 663, 675, 707, 735, 759, 767, 867, 887, 899, 915, 923, 927, 975, 983, 999}[/QUOTE]

These bases are the bases <= 1024 which is not perfect odd power (of the form m^r with odd r>1) whose "minimal prime program" have GFN or half GFN remain, for the bases <= 1024 which is perfect odd power (of the form m^r with odd r>1):

* Cubes:

** Base 8: GFN in base 2 are either 2{0}1 or 4{0}1 in base 8, however, 2 and 401 are primes, thus, base 8 does not have GFN or half GFN remain.

** Base 27: half GFN in base 3 are either 1{D}E or 4{D}E in base 27, however, D is prime, thus, base 27 does not have GFN or half GFN remain.

** Base 64: GFN in base 2 are either 4{0}1 or G{0}1 in base 64, however, 41 and G01 are primes, thus, base 64 does not have GFN or half GFN remain.

** Base 125: half GFN in base 5 are either 2:{62}:63 or 12:{62}:63 in base 125, however, 2 is prime, but the family 12:{62}:63 does not have any known (probable) prime (the only known half GFN (probable) primes in base 5 are 3, 13, 2:63), thus, [B][I]base 125 has half GFN remain.[/I][/B]

** Base 216: GFN in base 6 are either 6:{0}:1 or 36:{0}:1 in base 216, however, 6:1 is prime, but the family 36:{0}:1 does not have any known prime (the only known GFN primes in base 6 are 7, 37, 6:1), thus, [B][I]base 216 has GFN remain.[/I][/B]

** Base 343: half GFN in base 7 are either 3:{171}:172 or 24:{171}:172 in base 343, however, 3 is prime, but the family 24:{171}:172 does not have any known (probable) prime (the only known half GFN (probable) prime in base 7 is 3:172), thus, [B][I]base 343 has half GFN remain.[/I][/B]

** Base 512: GFN in base 2 are 2:{0}:1, 4:{0}:1, 16:{0}:1, 32:{0}:1, 128:{0}:1, or 256:{0}:1 in base 512, however, 2 and 128:1 are primes, but the families 4:{0}:1, 16:{0}:1, 32:{0}:1, 256:{0}:1 do not have any known prime (the only known GFN primes in base 2 are 3, 5, 17, 257, 128:1), thus, [B][I]base 512 has GFN remain.[/I][/B]

** Base 729: half GFN in base 3 are either 4:{364}:365 or 40:{364}:365 in base 729, however, 40:364:365 and 4:364:364:364:364:365 are primes, thus, base 729 does not have GFN or half GFN remain.

** Base 1000: GFN in base 10 are either 10:{0}:1 or 100:{0}:1 in base 1000, and both families do not have any known prime (the only known GFN primes in base 10 are 11 and 101), thus, [B][I]base 1000 has GFN remain.[/I][/B]

* 5th powers:

** Base 32: GFN in base 2 are 2{0}1, 4{0}1, 8{0}1, or G{0}1 in base 32, however, 2 and 81 are primes, but the families 4{0}1 and G{0}1 do not have any known prime (the only known GFN primes in base 2 are 3, 5, H, 81, 2001), thus, [B][I]base 32 has GFN remain.[/I][/B]

** Base 243: half GFN in base 3 are 1:{121}:122, 4:{121}:122, 13:{121}:122, or 40:{121}:122 in base 243, however, 1:121:121:122, 4:121, 13, 40:121:121:121:121:121:121:121:121:121:121:121:122 are primes, thus, base 243 does not have GFN or half GFN remain.

** Base 1024: GFN in base 2 are 4:{0}:1, 16:{0}:1, 64:{0}:1, or 256:{0}:1 in base 1024, however, 64:1 is prime, but the families 4:{0}:1, 16:{0}:1, 256:{0}:1 do not have any known prime (the only known GFN primes in base 2 are 3, 5, 17, 257, 64:1), thus, [B][I]base 1024 has GFN remain.[/I][/B]

* 7th powers:

** Base 128: GFN in base 2 are 2:{0}:1, 4:{0}:1, or 16:{0}:1 in base 128, however, 2 and 4:0:1 are primes, but the family 16:{0}:1 do not have any known prime (the only known GFN primes in base 2 are 3, 5, 17, 2:1, 4:0:1), thus, [B][I]base 128 has GFN remain.[/I][/B]

There are about exp(gamma*k) minimal primes in base n, where k = number of 2-digit numbers [I]xy[/I] in base n such that none of [I]x[/I], [I]y[/I], [I]xy[/I] are primes, [I]x[/I] != 0, gcd([I]y[/I],n) = 1

where exp(x) = e^x (e is [URL="https://en.wikipedia.org/wiki/E_(mathematical_constant)"]the base of the natural logarithm[/URL] (2.718281828...), gamma is [URL="https://en.wikipedia.org/wiki/Euler%E2%80%93Mascheroni_constant"]Euler–Mascheroni constant[/URL] (0.5772156649...))

Also, there are about exp(gamma*k) minimal strings of primes with >=2 digits in base n (see thread [URL="https://mersenneforum.org/showthread.php?t=24972"]https://mersenneforum.org/showthread.php?t=24972[/URL]), where k = number of 2-digit numbers [I]xy[/I] in base n such that [I]xy[/I] is not prime, [I]x[/I] != 0, gcd([I]y[/I],n) = 1

[QUOTE=sweety439;562748]There are about exp(gamma*k) minimal primes in base n, where k = number of 2-digit numbers [I]xy[/I] in base n such that none of [I]x[/I], [I]y[/I], [I]xy[/I] are primes, [I]x[/I] != 0, gcd([I]y[/I],n) = 1

where exp(x) = e^x (e is [URL="https://en.wikipedia.org/wiki/E_(mathematical_constant)"]the base of the natural logarithm[/URL] (2.718281828...), gamma is [URL="https://en.wikipedia.org/wiki/Euler%E2%80%93Mascheroni_constant"]Euler–Mascheroni constant[/URL] (0.5772156649...))

Also, there are about exp(gamma*k) minimal strings of primes with >=2 digits in base n (see thread [URL="https://mersenneforum.org/showthread.php?t=24972"]https://mersenneforum.org/showthread.php?t=24972[/URL]), where k = number of 2-digit numbers [I]xy[/I] in base n such that [I]xy[/I] is not prime, [I]x[/I] != 0, gcd([I]y[/I],n) = 1[/QUOTE]

The reason is if and only if a 2-digit number [I]xy[/I] satisfies all these condition (none of [I]x[/I], [I]y[/I], [I]xy[/I] are primes, [I]x[/I] != 0, gcd([I]y[/I],n) = 1), then [I]xy[/I] can be the first and last digit of a "base n minimal prime" with >=3 digits (if we require the primes have >=2 digits, then the conditions [I]x[/I] is not prime, [I]y[/I] is not prime, are both not needed, only need [I]xy[/I] is not prime, [I]x[/I] != 0, gcd([I]y[/I],n) = 1)

The k for the original case (i.e. including the single-digit primes)

[CODE]
base,k
2,0
3,1
4,0
5,4
6,1
7,5
8,3
9,8
10,5
11,27
12,2
13,38
14,10
15,23
16,17
17,84
18,4
19,108
20,17
21,59
22,30
23,164
24,9
25,151
26,57
27,136
28,55
29,307
30,8
31,350
32,87
33,190
34,111
35,282
36,42
37,539
38,144
39,289
40,107
41,678
42,31
43,736
44,169
45,295
46,227
47,892
48,59
49,804
50,160
51,543
52,286
53,1194
54,85
55,842
56,284
57,731
58,416
59,1545
60,47
61,1627
62,464
63,738
64,508
65,1248
66,144
67,2031
68,537
69,1101
70,265
71,2296
72,190
73,2404
74,676
75,936
76,696
77,1943
78,203
79,2867
80,503
81,1623
82,912
83,3179
84,150
85,2275
86,999
87,1865
88,911
89,3750
90,110
91,2865
92,1121
93,2182
94,1285
95,3009
96,456
97,4603
98,1012
99,2249
100,901
101,4994
102,420
103,5158
104,1347
105,1500
106,1635
107,5562
108,539
109,5725
110,812
111,3123
112,1300
113,6178
114,502
115,4391
116,1852
117,3231
118,2048
119,5209
120,273
121,6478
122,2286
123,4081
124,2313
125,5810
126,536
127,8241
128,2568
[/CODE]

The k for the case for prime with >=2 digits:

[CODE]
base,k
2,0
3,2
4,2
5,10
6,2
7,25
8,14
9,30
10,15
11,75
12,15
13,111
14,40
15,70
16,72
17,202
18,43
19,260
20,82
21,163
22,126
23,394
24,88
25,375
26,187
27,348
28,196
29,648
30,88
31,749
32,335
33,470
34,348
35,627
36,221
37,1089
38,450
39,684
40,385
41,1350
42,231
43,1495
44,579
45,764
46,685
47,1802
48,425
49,1674
50,628
51,1237
52,846
53,2311
54,549
55,1742
56,891
57,1575
58,1138
59,2894
60,458
61,3099
62,1316
63,1701
64,1470
65,2512
66,724
67,3766
68,1539
69,2370
70,1021
71,4245
72,1034
73,4500
74,1927
75,2242
76,1964
77,3802
78,1076
79,5295
80,1716
81,3495
82,2395
83,5861
84,1109
85,4476
86,2654
87,3879
88,2521
89,6768
90,1142
91,5466
92,2970
93,4467
94,3202
95,5671
96,1922
97,8078
98,2914
99,4697
100,2756
101,8774
102,1984
103,9137
104,3656
105,3683
106,4130
107,9883
108,2480
109,10270
110,2942
111,6478
112,3859
113,11051
114,2551
115,8490
116,4876
117,6765
118,5170
119,9691
120,2152
121,11515
122,5547
123,8024
124,5614
125,10609
126,2682
127,14030
128,6259
[/CODE]

[QUOTE=sweety439;562832]The reason is if and only if a 2-digit number [I]xy[/I] satisfies all these condition (none of [I]x[/I], [I]y[/I], [I]xy[/I] are primes, [I]x[/I] != 0, gcd([I]y[/I],n) = 1), then [I]xy[/I] can be the first and last digit of a "base n minimal prime" with >=3 digits (if we require the primes have >=2 digits, then the conditions [I]x[/I] is not prime, [I]y[/I] is not prime, are both not needed, only need [I]xy[/I] is not prime, [I]x[/I] != 0, gcd([I]y[/I],n) = 1)

The k for the original case (i.e. including the single-digit primes)

[CODE]
base,k
2,0
3,1
4,0
5,4
6,1
7,5
8,3
9,8
10,5
11,27
12,2
13,38
14,10
15,23
16,17
17,84
18,4
19,108
20,17
21,59
22,30
23,164
24,9
25,151
26,57
27,136
28,55
29,307
30,8
31,350
32,87
33,190
34,111
35,282
36,42
37,539
38,144
39,289
40,107
41,678
42,31
43,736
44,169
45,295
46,227
47,892
48,59
49,804
50,160
51,543
52,286
53,1194
54,85
55,842
56,284
57,731
58,416
59,1545
60,47
61,1627
62,464
63,738
64,508
65,1248
66,144
67,2031
68,537
69,1101
70,265
71,2296
72,190
73,2404
74,676
75,936
76,696
77,1943
78,203
79,2867
80,503
81,1623
82,912
83,3179
84,150
85,2275
86,999
87,1865
88,911
89,3750
90,110
91,2865
92,1121
93,2182
94,1285
95,3009
96,456
97,4603
98,1012
99,2249
100,901
101,4994
102,420
103,5158
104,1347
105,1500
106,1635
107,5562
108,539
109,5725
110,812
111,3123
112,1300
113,6178
114,502
115,4391
116,1852
117,3231
118,2048
119,5209
120,273
121,6478
122,2286
123,4081
124,2313
125,5810
126,536
127,8241
128,2568
[/CODE]

The k for the case for prime with >=2 digits:

[CODE]
base,k
2,0
3,2
4,2
5,10
6,2
7,25
8,14
9,30
10,15
11,75
12,15
13,111
14,40
15,70
16,72
17,202
18,43
19,260
20,82
21,163
22,126
23,394
24,88
25,375
26,187
27,348
28,196
29,648
30,88
31,749
32,335
33,470
34,348
35,627
36,221
37,1089
38,450
39,684
40,385
41,1350
42,231
43,1495
44,579
45,764
46,685
47,1802
48,425
49,1674
50,628
51,1237
52,846
53,2311
54,549
55,1742
56,891
57,1575
58,1138
59,2894
60,458
61,3099
62,1316
63,1701
64,1470
65,2512
66,724
67,3766
68,1539
69,2370
70,1021
71,4245
72,1034
73,4500
74,1927
75,2242
76,1964
77,3802
78,1076
79,5295
80,1716
81,3495
82,2395
83,5861
84,1109
85,4476
86,2654
87,3879
88,2521
89,6768
90,1142
91,5466
92,2970
93,4467
94,3202
95,5671
96,1922
97,8078
98,2914
99,4697
100,2756
101,8774
102,1984
103,9137
104,3656
105,3683
106,4130
107,9883
108,2480
109,10270
110,2942
111,6478
112,3859
113,11051
114,2551
115,8490
116,4876
117,6765
118,5170
119,9691
120,2152
121,11515
122,5547
123,8024
124,5614
125,10609
126,2682
127,14030
128,6259
[/CODE][/QUOTE]

This is why base 34 is harder than base 17, base 38 is harder than base 19, but base 42 is easier than base 21

[CODE]
base number of unsolved families when searched to 10000 digits
17 2
34 33
19 5
38 77
21 3
42 0 (the largest prime has only 487 digits)
[/CODE]

[URL="https://cs.uwaterloo.ca/~cbright/talks/minimal-slides.pdf"]https://cs.uwaterloo.ca/~cbright/talks/minimal-slides.pdf[/URL]