20220406, 14:06  #320  
Feb 2017
Nowhere
5772_{10} Posts 
Quote:
In the present instance, 14...91 or 14... (24862044 digits) ...91 would be much clearer. The most significant digits of M_{p} are determined by the fractional part of p*log(2)/log(10). EDIT: Although there is no upper bound on the number of leading 1's in a Mersenne number 2^{n}  1, or AFAIK for 2^{p}  1 with p prime, there is an upper bound for the number of terminal 1's. Three. The proof is easy. (The Mersenne number 2^{888689}  1 has 4 leading 1's and 3 terminal 1's. M_{888689} has a small factor q = 24*p + 1, and the cofactor is proven composite.) Last fiddled with by Dr Sardonicus on 20220406 at 19:47 Reason: As indicated 

20220407, 03:12  #321  
"99(4^34019)99 palind"
Nov 2016
(P^81993)SZ base 36
2^{2}×5×13^{2} Posts 
Quote:
In base b=2, T={2}, since all numbers > 1 not divisible by 2 (i.e. the odd numbers > 1) have first digit 1 and last digit 1 in base b=2, thus the only minimal prime (start with b+1) 11 (decimal 3) must be a subsequence. In base b=3, T={2,3}; first, 10 (decimal 3) is needed, since this prime is needed to remove the numbers 1{0} (i.e. 100, 1000, 10000, 100000, ...); second, for the numbers not divisible by 2 or 3, such number cannot end with 0 (of course also cannot begin with 0) and must have an odd number of 1, thus either 111 (i.e. 3 1's) or 12 or 21 must be a subsequence, unless the number is 1, which is not allowed in this research. In base b=4, T={2,3}; since all numbers not divisible by 2 (i.e. end with 1 or 3) not contain any of {11, 13, 23, 31, 221} as subsequence are either of the form {0}2{0}1 or contain only 0 and 3, thus must be divisible by 3 In base b=5, T={2,3,5,7,11,13,17,23,31,41,47,251,691,887,5081,7219,82609,105367}, see factorization of 1{0}13 in base 5 Code:
prime use to remove 2 (2) 1{0,4}1, 1{0}3, 11{4}4, 2{0,2,4}2, 2{0,2,4}4, 30{0}01, 33{0}31, 4{4}11, 4{0,2,4}2, 4{0,2,4}4 3 (3) 11{0}3, 3{0,3}11, 3{0,3}3, 1(0^n)13 for all even n (including n=0) and n < 93 10 (5) all numbers ending with 0 12 (7) 144, 331, 1(0^n)13 for all n == 1 mod 6 and n < 93 21 (11) 3301, 441, 1(0^n)13 for all n divisible by 5 and n < 93 23 (13) 1(0^n)13 for all n == 3 mod 4 and n < 93 32 (17) 3031 43 (23) 1(0^81)13 111 (31) 30031 131 (41) 1(0^17)13, 1(0^57)13, 1(0^77)13 142 (47) 1(0^29)13 2001 (251) 1(0^89)13 10231 (691) 1(0^9)13 12022 (887) 1(0^21)13 130311 (5081) 1(0^41)13 212334 (7219) 1(0^69)13 10120414 (82609) 1(0^33)13 11332432 (105367) 1(0^53)13 

20220407, 03:25  #322 
"99(4^34019)99 palind"
Nov 2016
(P^81993)SZ base 36
3380_{10} Posts 
Since finding the set T requiring finding the set M(Lb) first, finding T is much harder than finding M(Lb)
For b = 8 and b = 9 and b = 12, T contains infinitely many numbers: * For b = 8, T contains the smallest prime factor of 8^(2^n)+1 for all n * For b = 9, T contains the smallest prime factor of (9^n1)/8 for all prime n, also contain the smallest prime factor of (25*9^n1)/8, 4*9^n49, 4*9^n1, 9^n4 for all n * For b = 12, T contains the smallest prime factor of 12^n25 for all even n (for all odd n, the smallest prime factor is 13) Theorem: For all base b, T must contain all primes <= b Proof: For primes p < b, consider the combination of the digits divisible by p (including 0), and for prime p = b, consider the family 1{0} We find the T for b = 10: * T must contain {2,3,5,7} * The smallest prime of the form 2{0}21 is 20021 > 221 (=13*17) and 2021 (=43*47) must be removed, thus either 13 or 17 is needed, and either 43 or 47 is needed * The smallest prime of the form 22{0}1 is 22000001 > 221, 2201, 22001, 220001, 2200001 must be removed, thus the primes {13,31,19,73} are needed (7 is already in the set) * The smallest prime of the form 5{5}1 is 555555555551 > All smaller numbers in this family must be removed * 581 must be removed, but 581 is divisible by 7 and 7 is already in the set * The smallest prime of the form 5{0}27 is 5000000000000000000000000000027 > All smaller numbers in this family must be removed * The smallest prime of the form 52{0}7 is 5200007 > 527, 5207, 52007, 520007 must be removed, thus the primes {17,41,131,23} are needed * 
20220408, 11:44  #323 
"99(4^34019)99 palind"
Nov 2016
(P^81993)SZ base 36
D34_{16} Posts 
Factorization of x{y}z families other than Cunningham numbers:
https://stdkmd.net/nrr/#factortables http://mklasson.com/factors/index.php https://cs.stanford.edu/people/rprop...ctors/3n2.txt https://oeis.org/A162491 https://oeis.org/A093810 https://oeis.org/A093817 https://oeis.org/A080443 https://oeis.org/A081715 https://oeis.org/A080798 https://oeis.org/A080892 
20220411, 02:13  #324  
"99(4^34019)99 palind"
Nov 2016
(P^81993)SZ base 36
2^{2}·5·13^{2} Posts 
Quote:
* A = numbers n such that 2*b^n1 is prime > family 1{z} * B = numbers n such that (2*b^n+1)/gcd(b1,3) is prime > for b not == 1 mod 3, family 2{0}1 (for b == 1 mod 3, family 2{0}1 can be ruled out as only contain composite numbers, as all numbers in this family are divisible by 3) * C = numbers n such that (b^n2)/gcd(b,2) is prime > for odd b, family {z}y (for even b, family {z}y can be ruled out as only contain composite numbers, as all numbers in this family are divisible by 2) * D = numbers n such that (b^n+2)/gcd(b,2)/gcd(b1,3) is prime > for b == 3 or 5 mod 6, family 1{0}2 (for b == 1 mod 3, family 1{0}2 can be ruled out as only contain composite numbers, as all numbers in this family are divisible by 3, also, for even b, family 1{0}2 can be ruled out as only contain composite numbers, as all numbers in this family are divisible by 2) for k = 3: * A = numbers n such that (3*b^n1)/gcd(b1,2) is prime > for even b, family 2{z} (for odd b, family 2{z} can be ruled out as only contain composite numbers, as all numbers in this family are divisible by 2) * B = numbers n such that (3*b^n+1)/gcd(b1,4) is prime > for even b, family 3{0}1 (for odd b, family 3{0}1 can be ruled out as only contain composite numbers, as all numbers in this family are divisible by 2) * C = numbers n such that (b^n3)/gcd(b,3)/gcd(b1,2) is prime > for b == 2 or 4 mod 6, family {z}x (for odd b, family {z}x can be ruled out as only contain composite numbers, as all numbers in this family are divisible by 2, also, for b divisible by 3, family {z}x can be ruled out as only contain composite numbers, as all numbers in this family are divisible by 3) * D = numbers n such that (b^n+3)/gcd(b,3)/gcd(b1,4) is prime > for b == 2 or 4 mod 6, family 1{0}3 (for odd b, family 1{0}3 can be ruled out as only contain composite numbers, as all numbers in this family are divisible by 2, also, for b divisible by 3, family 1{0}3 can be ruled out as only contain composite numbers, as all numbers in this family are divisible by 3) 

20220411, 02:39  #325 
"99(4^34019)99 palind"
Nov 2016
(P^81993)SZ base 36
2^{2}·5·13^{2} Posts 
We can expect the length of the largest minimal prime (start with b+1) in base b, and expect the number of unsolved families in base b when searched to certain limit (e.g. 10000 digits or 50000 digits), like expectations of the length of the largest left truncatable primes in base b and expectations of the length of the largest right truncatable primes in base b
There are examples of expectations related to this: https://mersenneforum.org/showpost.p...8&postcount=22 https://mersenneforum.org/showpost.p...0&postcount=20 https://mersenneforum.org/showpost.p...65&postcount=7 https://mersenneforum.org/showthread.php?t=26228 https://oeis.org/A055557/a055557.txt https://listserv.nodak.edu/cgibin/w...;417ab0d6.0906 http://www.fermatquotient.com/PrimSerien/GenRepu.txt http://www.fermatquotient.com/PrimSerien/GenRepuP.txt Last fiddled with by sweety439 on 20220411 at 02:42 
20220421, 18:25  #326 
"99(4^34019)99 palind"
Nov 2016
(P^81993)SZ base 36
2^{2}×5×13^{2} Posts 
Upload newest pdf file.
Last fiddled with by sweety439 on 20220430 at 06:23 
20220507, 12:39  #327 
"99(4^34019)99 palind"
Nov 2016
(P^81993)SZ base 36
6464_{8} Posts 
upload the base 9 file.

20220508, 14:14  #328 
"99(4^34019)99 palind"
Nov 2016
(P^81993)SZ base 36
2^{2}·5·13^{2} Posts 
bases 9, 10, 12

20220512, 12:59  #329 
"99(4^34019)99 palind"
Nov 2016
(P^81993)SZ base 36
2^{2}·5·13^{2} Posts 
The possible length of a prime in the families (in any base b) are: (the length is possible if and only if the corresponding polynomial of the base b satisfies the condition in Bunyakovsky conjecture, and if Bunyakovsky conjecture is true, then there are infinitely many bases b such that the number with these lengths in the families are primes)
1{0}1: length is of the form 2^r+1 (b^n+1 is divisible by b^k+1 if k divides n and n/k is odd > 1, and b^n+1 satisfies both conditions in Bunyakovsky conjecture if n is of the form 2^r) 1{0}2: any length >= 2 (b^n+2 always satisfies both conditions in Bunyakovsky conjecture) 1{0}3: any length >= 2 (b^n+3 always satisfies both conditions in Bunyakovsky conjecture) 1{0}4: length not == 1 mod 4 (b^n+4 = (b^(n/2)2*b^(n/4)+2) * (b^(n/2)+2*b^(n/4)+2) if n == 0 mod 4, and b^n+4 satisfies both conditions in Bunyakovsky conjecture if n not == 0 mod 4) 1{0}z: length not == 0 mod 6 (b^n+(b1) has factor b^2b+1 (z1 in base b) if n == 5 mod 6, and b^n+(b1) satisfies both conditions in Bunyakovsky conjecture if n not == 5 mod 6) {1}: length is prime ((b^n1)/(b1) is divisible by (b^k1)/(b1) if k divides n and k > 1 and n/k > 1 (for k = 1, (b^k1)/(b1) = 1), and (b^n1)/(b1) satisfies both conditions in Bunyakovsky conjecture if n is prime) 1{z}: any length >= 2 (2*b^n1 always satisfies both conditions in Bunyakovsky conjecture) 2{0}1: any length >= 2 (2*b^n+1 always satisfies both conditions in Bunyakovsky conjecture) 2{z}: any length >= 2 (3*b^n1 always satisfies both conditions in Bunyakovsky conjecture) 3{0}1: any length >= 2 (3*b^n+1 always satisfies both conditions in Bunyakovsky conjecture) 3{z}: length == 0 mod 2 (4*b^n1 = (2*b^(n/2)1) * (2*b^(n/2)+1) if n == 0 mod 2, and 4*b^n1 satisfies both conditions in Bunyakovsky conjecture if n == 1 mod 2) 4{0}1: length not == 1 mod 4 (4*b^n+1 = (2*b^(n/2)2*b^(n/4)+1) * (2*b^(n/2)+2*b^(n/4)+1) if n == 0 mod 4, and 4*b^n+1 satisfies both conditions in Bunyakovsky conjecture if n not == 0 mod 4) {y}z: any length >= 2 (((b2)*b^n+1)/(b1) always satisfies both conditions in Bunyakovsky conjecture) y{z}: length not == 5 mod 6 ((b1)*b^n1 has factor b^2b+1 (z1 in base b) if n == 4 mod 6, and (b1)*b^n1 satisfies both conditions in Bunyakovsky conjecture if n not == 4 mod 6) z{0}1: length 2 and length not == 2 mod 6 ((b1)*b^n+1 has factor b^2b+1 (z1 in base b) if n == 1 mod 6 and n > 1 (for n = 1, (b1)*b^n+1 is exactly b^2b+1 (z1 in base b) itself), and (b1)*b^n+1 satisfies both conditions in Bunyakovsky conjecture if n not == 1 mod 6) {z}1: length 2 and length not == 2 mod 6 (b^n(b1) has factor b^2b+1 (z1 in base b) if n == 2 mod 6 and n > 2 (for n = 2, b^n(b1) is exactly b^2b+1 (z1 in base b) itself), and b^n(b1) satisfies both conditions in Bunyakovsky conjecture if n not == 2 mod 6) {z}w: length == 1 mod 2 (b^n4 = (b^(n/2)2) * (b^(n/2)+2) if n == 0 mod 2, and b^n4 satisfies both conditions in Bunyakovsky conjecture if n == 1 mod 2) {z}x: any length >= 2 (b^n3 always satisfies both conditions in Bunyakovsky conjecture) {z}y: any length >= 2 (b^n2 always satisfies both conditions in Bunyakovsky conjecture) These are the cyclotomic polynomials Phi(n,b) written in base b for n = 1 to 10000, since all cyclotomic polynomials satisfy both conditions in Bunyakovsky conjecture, it is conjectured that there are infinitely many bases b such that the number is prime, the smallest such base b are listed in A085398 (note: 1 cannot be the base of a numeral system) Interesting things: except GFN and GRU (see this post) (the smallest GFN in base b and the smallest GRU in base b is always both minimal prime (start with b+1) in base b and unique prime in base b (with period length "power of 2" and "prime", respectively), for some bases b, there are other numbers which are both minimal prime (start with b+1) in base b and unique prime in base b (the period length is neither prime nor power of 2) * z1 (period = 6): see next post (if z1 is prime, then it is always such number, it is interesting not only because unique prime, but also because the Williams families z{0}1 and {z}1, whose 2digit numbers are both z1) * z0z1 (period = 10): 12, 20, 37, 38, 47, 71, 126, 131, 157, 158, 180, 223, 251, 257, 268, 276, 342, 361, 363, 375, 377, 385, 388, 438, 452, 462, 487, 507, 551, 588, 603, 628, 641, 652, 658, 676, 693, 707, 716, 738, 758, 760, 768, 782, 791, 792, 812, 850, 935, 973, 978, ... (z0z1 is prime, but z1, z01, zz1 are all composite) * zz01 (period = 12): 12, 17, 27, 30, 38, 56, 61, 65, 94, 105, 109, 114, 126, 131, 166, 172, 183, 185, 196, 198, 225, 229, 231, 257, 261, 263, 269, 270, 296, 302, 308, 313, 324, 328, 339, 346, 350, 363, 374, 407, 412, 413, 424, 426, 429, 434, 437, 438, 443, 447, 463, 468, 480, 485, 502, 503, 507, 511, 515, 533, 545, 549, 558, 559, 571, 582, 593, 595, 599, 612, 614, 632, 638, 640, 641, 653, 659, 676, 699, 701, 707, 716, 718, 724, 725, 738, 742, 749, 767, 768, 777, 794, 797, 798, 801, 805, 811, 832, 842, 862, 868, 871, 872, 893, 905, 909, 923, 949, 953, 958, 963, 967, 974, 978, 1009, 1015, 1018 (zz01 is prime, but z1, z01, zz1 are all composite) (see also this post for references of unique primes) Code:
b primes which are both minimal prime (start with b+1) in base b and unique prime in base b 2 11 (3, period 2, both GFN and GRU) 3 12 (5, period 4, GFN), 21 (7, period 6), 111 (13, period 3, GRU) 4 11 (5, period 2, both GFN and GRU), 13 (7, period 3), 31 (13, period 6), 221 (41, period 10, this prime is completely a coincidence, since 41 = Phi(10,4)/gcd(Phi(10,4),10) and gcd(Phi(10,4),10) > 1) 5 12 (7, period 6), 23 (13, period 4, GFN), 111 (31, period 3, GRU) 6 11 (7, period 2, both GFN and GRU), 51 (31, period 6) 7 25 (19, period 3), 61 (43, period 6), 3334 (1201, period 8, GFN), 11111 (2801, period 5, GRU) 8 23 (19, period 6), 111 (73, period 3, GRU) 9 45 (41, period 4, GFN), 81 (73, period 6) 10 11 (11, period 2, both GFN and GRU), 37 (period 3) 11 34 (37, period 6), 61 (period 4, GFN) 12 11 (13, period 2, both GFN and GRU), B0B1 (19141, period 10), BB01 (20593, period 12) Last fiddled with by sweety439 on 20220513 at 10:40 
20220512, 12:59  #330 
"99(4^34019)99 palind"
Nov 2016
(P^81993)SZ base 36
2^{2}×5×13^{2} Posts 
The minimal prime in family 1{z} in base b (always minimal prime (start with b+1) base b) has length 2 if and only if the minimal prime in family 1{0}z in base b (always minimal prime (start with b+1) base b) has length 2, since the corresponding primes are both 1z in base b
Such bases b are 2, 3, 4, 6, 7, 9, 10, 12, 15, 16, 19, 21, 22, 24, 27, 30, 31, 34, 36, 37, 40, 42, 45, 49, 51, 52, 54, 55, 57, 64, 66, 69, 70, 75, 76, 79, 82, 84, 87, 90, 91, 96, 97, 99, 100, 106, 112, 114, 115, 117, 120, 121, 126, 129, 132, 135, 136, 139, 141, 142, 147, 154, 156, 157, 159, 166, 169, 174, 175, 177, 180, 184, 187, 190, 192, 195, 199, 201, 205, 210, 211, 216, 217, 220, 222, 225, 229, 231, 232, 234, 240, 244, 246, 250, 252, 255, 261, 262, 271, 274, 279, 282, 285, 286, 289, 294, 297, 300, 301, 304, 307, 309, 310, 316, 321, 322, 324, 327, 330, 331, 337, 339, 342, 346, 351, 355, 360, 364, 367, 370, 372, 376, 379, 381, 385, 387, 394, 399, 405, 406, 411, 412, 414, 415, 420, 427, 429, 430, 432, 439, 441, 442, 444, 454, 456, 460, 465, 469, 471, 474, 477, 484, 486, 489, 492, 496, 499, 505, 507, 510, 511, 516, 517, 520, 525, 526, 531, 532, 535, 544, 546, 547, 549, 552, 555, 559, 562, 565, 576, 577, 582, 586, 591, 594, 597, 601, 607, 609, 612, 615, 616, 619, 625, 630, 639, 640, 642, 645, 646, 649, 651, 652, 654, 660, 661, 664, 681, 684, 687, 691, 700, 705, 712, 714, 715, 717, 720, 724, 726, 727, 730, 736, 741, 742, 744, 745, 747, 750, 756, 762, 766, 772, 775, 777, 780, 784, 786, 790, 792, 799, 801, 804, 805, 807, 810, 811, 814, 819, 829, 832, 834, 835, 847, 849, 850, 855, 861, 862, 867, 871, 874, 877, 880, 889, 892, 894, 895, 901, 906, 912, 916, 924, 931, 934, 936, 937, 939, 940, 945, 951, 954, 957, 966, 967, 975, 976, 987, 990, 994, 997, 999, 1000, 1002, 1006, 1009, 1014, 1015, 1020, ..., they are listed in https://oeis.org/A006254, in these bases b, 1z is minimal prime (start with b+1) The minimal prime in family {z}1 in base b (always minimal prime (start with b+1) base b) has length 2 if and only if the minimal prime in family z{0}1 in base b (always minimal prime (start with b+1) base b) has length 2, since the corresponding primes are both z1 in base b Such bases b are 2, 3, 4, 6, 7, 9, 13, 15, 16, 18, 21, 22, 25, 28, 34, 39, 42, 51, 55, 58, 60, 63, 67, 70, 72, 76, 78, 79, 81, 90, 91, 100, 102, 106, 111, 112, 118, 120, 132, 139, 142, 144, 148, 151, 154, 156, 162, 163, 165, 168, 169, 174, 177, 189, 190, 193, 195, 204, 207, 210, 216, 219, 232, 237, 246, 247, 267, 273, 279, 280, 288, 289, 291, 294, 310, 315, 330, 333, 337, 343, 345, 349, 352, 358, 370, 379, 382, 384, 393, 396, 399, 403, 405, 406, 415, 417, 427, 435, 436, 448, 454, 456, 457, 477, 490, 496, 501, 513, 519, 526, 531, 532, 534, 538, 541, 552, 555, 561, 567, 568, 573, 580, 583, 585, 604, 606, 610, 613, 622, 625, 627, 636, 643, 645, 669, 672, 678, 687, 697, 702, 721, 727, 729, 736, 744, 748, 756, 762, 763, 769, 774, 783, 786, 793, 799, 802, 813, 819, 820, 826, 828, 837, 840, 847, 856, 858, 861, 865, 876, 879, 891, 895, 898, 900, 912, 916, 919, 921, 928, 951, 960, 961, 970, 975, 982, 988, 991, 993, 994, 1002, 1003, 1008, 1012, 1017, 1021, 1023, ..., they are listed in https://oeis.org/A055494, in these bases b, z1 is minimal prime (start with b+1) The minimal prime in family y{z} in base b (always minimal prime (start with b+1) base b) has length 2 if and only if the minimal prime in family {y}z in base b (always minimal prime (start with b+1) base b) has length 2, since the corresponding primes are both yz in base b (note: yz does not exist in base 2, since it has leading zeros) Such bases b are 3, 4, 5, 6, 7, 9, 10, 11, 12, 14, 16, 17, 20, 21, 22, 25, 27, 29, 31, 32, 36, 39, 40, 42, 45, 46, 47, 49, 51, 54, 55, 56, 57, 60, 61, 65, 66, 67, 69, 71, 77, 84, 86, 87, 90, 94, 95, 97, 101, 102, 104, 115, 116, 121, 126, 127, 131, 132, 135, 139, 141, 142, 145, 146, 149, 150, 154, 155, 156, 159, 160, 161, 164, 165, 170, 172, 175, 177, 181, 182, 185, 187, 189, 192, 194, 196, 197, 200, 204, 207, 210, 216, 219, 220, 221, 226, 231, 232, 234, 237, 241, 242, 245, 247, 249, 259, 260, 264, 265, 266, 269, 282, 286, 289, 291, 295, 297, 299, 302, 304, 306, 307, 310, 315, 320, 324, 331, 332, 336, 340, 344, 350, 351, 352, 355, 359, 361, 362, 365, 374, 375, 379, 381, 386, 387, 392, 394, 396, 397, 402, 406, 409, 419, 420, 421, 424, 429, 432, 434, 446, 449, 450, 451, 454, 456, 457, 462, 464, 469, 472, 474, 475, 476, 479, 482, 487, 491, 495, 496, 497, 500, 501, 502, 505, 507, 511, 512, 516, 519, 520, 529, 531, 542, 545, 549, 550, 551, 552, 561, 562, 570, 572, 574, 579, 582, 589, 590, 592, 595, 596, 597, 599, 605, 607, 611, 617, 619, 625, 626, 630, 634, 640, 641, 645, 647, 649, 652, 655, 656, 660, 669, 671, 674, 677, 684, 687, 692, 694, 696, 700, 706, 707, 709, 710, 716, 717, 721, 725, 729, 740, 744, 747, 751, 754, 755, 762, 764, 766, 770, 777, 780, 781, 782, 787, 795, 799, 804, 805, 810, 815, 816, 817, 826, 834, 839, 846, 849, 857, 864, 874, 882, 885, 886, 890, 891, 902, 907, 916, 922, 924, 926, 931, 940, 942, 945, 947, 951, 956, 957, 964, 966, 967, 969, 975, 977, 982, 985, 990, 995, 997, 999, 1001, 1002, 1006, 1007, 1014, 1015, 1017, 1024, ..., they are listed in https://oeis.org/A002328, in these bases b, yz is minimal prime (start with b+1) The minimal prime in family {z}y in base b (always minimal prime (start with b+1) base b) has length 2 if and only if the minimal prime in family z{y} in base b (always minimal prime (start with b+1) base b) has length 2, since the corresponding primes are both zy in base b (note: zy does not exist in base 2, since it has trailing zeros) Such bases b are 3, 5, 7, 9, 13, 15, 19, 21, 27, 29, 33, 35, 37, 43, 47, 49, 55, 61, 63, 69, 71, 75, 77, 89, 93, 103, 107, 117, 119, 121, 127, 131, 135, 139, 145, 155, 161, 169, 173, 177, 183, 191, 205, 211, 217, 223, 231, 233, 237, 239, 247, 253, 257, 259, 265, 267, 273, 279, 285, 293, 299, 301, 303, 309, 313, 315, 323, 335, 337, 341, 355, 357, 359, 371, 387, 391, 399, 405, 415, 421, 425, 429, 435, 441, 443, 447, 449, 467, 469, 481, 489, 491, 495, 497, 505, 513, 523, 525, 531, 541, 545, 561, 565, 569, 573, 575, 583, 587, 595, 607, 609, 615, 621, 637, 645, 653, 663, 667, 671, 677, 713, 715, 719, 727, 733, 743, 747, 751, 757, 761, 763, 775, 779, 785, 789, 797, 807, 811, 817, 825, 841, 863, 875, 881, 897, 901, 931, 943, 945, 951, 959, 973, 979, 987, 993, 995, 1003, 1013, 1021, 1023, ..., they are listed in https://oeis.org/A028870, in these bases b, zy is minimal prime (start with b+1) The minimal prime in family {1} in base b (always minimal prime (start with b+1) base b) has length 2 if and only if the minimal prime in family 1{0}1 in base b (always minimal prime (start with b+1) base b) has length 2, since the corresponding primes are both 11 in base b Such bases b are 2, 4, 6, 10, 12, 16, 18, 22, 28, 30, 36, 40, 42, 46, 52, 58, 60, 66, 70, 72, 78, 82, 88, 96, 100, 102, 106, 108, 112, 126, 130, 136, 138, 148, 150, 156, 162, 166, 172, 178, 180, 190, 192, 196, 198, 210, 222, 226, 228, 232, 238, 240, 250, 256, 262, 268, 270, 276, 280, 282, 292, 306, 310, 312, 316, 330, 336, 346, 348, 352, 358, 366, 372, 378, 382, 388, 396, 400, 408, 418, 420, 430, 432, 438, 442, 448, 456, 460, 462, 466, 478, 486, 490, 498, 502, 508, 520, 522, 540, 546, 556, 562, 568, 570, 576, 586, 592, 598, 600, 606, 612, 616, 618, 630, 640, 642, 646, 652, 658, 660, 672, 676, 682, 690, 700, 708, 718, 726, 732, 738, 742, 750, 756, 760, 768, 772, 786, 796, 808, 810, 820, 822, 826, 828, 838, 852, 856, 858, 862, 876, 880, 882, 886, 906, 910, 918, 928, 936, 940, 946, 952, 966, 970, 976, 982, 990, 996, 1008, 1012, 1018, 1020, ..., they are listed in https://oeis.org/A006093 (except 1, since 1 cannot be the base of a numeral system), in these bases b, 11 is minimal prime (start with b+1) The minimal prime in family 1{0}11 in base b (always minimal prime (start with b+1) base b) has length 3 if and only if the minimal prime in family 11{0}1 in base b (always minimal prime (start with b+1) base b) has length 3, since the corresponding primes are both 111 in base b, and both of them do not have the length 2 number, 111 is minimal prime (start with b+1) in bases b = 3, 5, 8, 14, 15, 17, 20, 21, 24, 27, 33, 38, 41, 50, 54, 57, 59, 62, 69, 71, 75, 77, 80, 89, 90, 99, 101, 105, 110, 111, 117, 119, 131, 141, 143, 147, 153, 155, 161, 164, 167, 168, 173, 176, 188, 189, 194, 203, 206, 209, 215, 218, 231, 236, 245, 246, 266, 272, 278, 279, 287, 288, 290, 293, 309, 314, 329, 332, 342, 344, 351, 357, 369, 381, 383, 392, 395, 398, 402, 404, 405, 414, 416, 426, 434, 435, 447, 453, 455, 476, 489, 495, 500, 512, 518, 525, 530, 531, 533, 537, 551, 554, 560, 566, 567, 572, 579, 582, 584, 603, 605, 609, 621, 624, 626, 635, 644, 668, 671, 677, 686, 696, 701, 720, 728, 735, 743, 747, 755, 761, 762, 773, 782, 785, 792, 798, 801, 812, 818, 819, 825, 827, 836, 839, 846, 855, 857, 860, 864, 875, 878, 890, 894, 897, 899, 911, 915, 920, 927, 950, 959, 960, 969, 974, 981, 987, 992, 993, 1001, 1002, 1007, 1011, 1016, 1022, ..., they are the set difference https://oeis.org/A002384 \ https://oeis.org/A006093, and they are also the bases b such that the minimal prime in family {1} in base b (always minimal prime (start with b+1) base b) has length 3 The number "101" (in base b) is minimal prime (start with b+1) in these bases b: 14, 20, 24, 26, 54, 56, 74, 84, 90, 94, 110, 116, 120, 124, 134, 146, 160, 170, 176, 184, 204, 206, 224, 230, 236, 260, 264, 284, 300, 314, 326, 340, 350, 384, 386, 406, 436, 440, 444, 464, 470, 474, 496, 536, 544, 584, 594, 634, 636, 644, 654, 674, 680, 686, 696, 704, 714, 716, 740, 764, 780, 784, 816, 860, 864, 890, 920, 930, 950, 960, 986, 1004, 1010 They are the bases b such that ... * The minimal prime in family 10{1} has length 3 (and "11" is not prime) * The minimal prime in family 1{0}1 has length 3 * The minimal prime in family {1}01 has length 3 (and "11" is not prime) The number "102" (in base b) is minimal prime (start with b+1) in these bases b: 33, 117, 123, 219, 243, 273, 297, 303, 321, 363, 369, 375, 423, 453, 513, 549, 573, 603, 609, 711, 753, 777, 867, 897, 903, 933, 957, ... They are the bases b such that ... * The minimal prime in family 10{2} has length 3 (and "12" is not prime) * The minimal prime in family 1{0}2 has length 3 * The minimal prime in family {1}02 has length 3 (and "12" is not prime) The number "201" (in base b) is minimal prime (start with b+1) in these bases b: 24, 27, 42, 45, 66, 72, 87, 93, 102, 123, 132, 162, 177, 201, 237, 240, 255, 264, 297, 324, 327, 351, 357, 360, 387, 399, 417, 423, 450, 456, 462, 474, 486, 489, 492, 537, 555, 570, 588, 597, 621, 633, 636, 657, 666, 693, 705, 750, 756, 759, 762, 768, 786, 792, 819, 837, 864, 885, 897, 918, 951, 960, 963, 978, 990, 1011, 1017, 1020, ... They are the bases b such that ... * The minimal prime in family 20{1} has length 3 (and "21" is not prime) * The minimal prime in family 2{0}1 has length 3 * The minimal prime in family {2}01 has length 3 (and "21" is not prime) The number "10z" (in base b) is minimal prime (start with b+1) in these bases b: 5, 8, 11, 13, 20, 26, 28, 35, 38, 39, 41, 44, 46, 48, 50, 53, 56, 59, 60, 65, 68, 83, 85, 86, 89, 93, 94, 101, 103, 125, 130, 131, 134, 138, 140, 144, 145, 148, 149, 153, 155, 158, 160, 163, 164, 171, 176, 181, 186, 188, 191, 193, 196, 203, 206, 209, 215, 218, 219, 230, 233, 236, 241, 248, 258, 259, 263, 264, 265, 268, 281, 288, 290, 296, 298, 303, 305, 306, 314, 319, 323, 335, 343, 349, 350, 354, 358, 361, 373, 374, 378, 380, 386, 391, 393, 395, 396, 401, 408, 418, 419, 423, 428, 431, 433, 445, 448, 449, 450, 453, 455, 461, 463, 468, 473, 475, 478, 481, 490, 494, 495, 500, 501, 504, 506, 515, 518, 519, 528, 530, 541, 548, 550, 551, 560, 561, 569, 571, 573, 578, 581, 588, 589, 595, 596, 598, 604, 606, 610, 618, 624, 629, 633, 644, 648, 655, 659, 668, 670, 673, 676, 683, 686, 693, 695, 699, 706, 708, 709, 716, 728, 739, 743, 746, 753, 754, 761, 763, 765, 769, 776, 779, 781, 794, 798, 803, 809, 815, 816, 825, 833, 838, 845, 848, 856, 863, 873, 881, 884, 885, 890, 915, 921, 923, 925, 930, 941, 944, 946, 950, 955, 956, 963, 965, 968, 974, 981, 984, 989, 996, 998, 1001, 1005, 1013, 1016, 1023, 1024, ... They are the bases b such that ... * The minimal prime in family 10{z} has length 3 (and "1z" is not prime) * The minimal prime in family 1{0}z has length 3 * The minimal prime in family {1}0z has length 3 (and "1z" is not prime) The number "z01" (in base b) is minimal prime (start with b+1) in these bases b: 5, 8, 14, 24, 26, 29, 33, 35, 36, 40, 43, 45, 48, 49, 50, 80, 83, 89, 93, 96, 99, 101, 104, 110, 115, 121, 124, 133, 135, 138, 140, 149, 161, 170, 173, 181, 191, 194, 201, 203, 205, 206, 209, 211, 215, 224, 226, 230, 253, 254, 258, 259, 260, 266, 274, 278, 281, 285, 286, 300, 301, 309, 311, 318, 320, 321, 323, 325, 334, 336, 344, 348, 353, 355, 365, 366, 373, 380, 386, 390, 395, 400, 404, 411, 414, 423, 425, 428, 433, 441, 444, 446, 451, 458, 465, 474, 483, 484, 489, 491, 495, 500, 505, 506, 514, 518, 523, 525, 528, 539, 560, 565, 566, 579, 584, 586, 594, 608, 616, 619, 623, 629, 654, 663, 666, 670, 679, 681, 695, 710, 714, 723, 734, 750, 754, 755, 764, 765, 773, 785, 789, 803, 806, 808, 815, 818, 824, 848, 853, 854, 855, 859, 873, 880, 881, 883, 884, 899, 908, 910, 925, 946, 954, 965, 969, 980, 983, 1011, 1014, ... They are the bases b such that ... * The minimal prime in family z0{1} has length 3 (and "z1" is not prime) * The minimal prime in family z{0}1 has length 3 * The minimal prime in family {z}01 has length 3 (and "z1" is not prime) The number "1zz" (in base b) is minimal prime (start with b+1) in these bases b: 8, 11, 13, 17, 18, 25, 28, 38, 39, 41, 43, 46, 50, 56, 59, 62, 63, 73, 80, 81, 85, 92, 95, 98, 102, 108, 109, 113, 118, 125, 127, 134, 137, 140, 143, 153, 155, 158, 160, 164, 165, 171, 172, 178, 179, 181, 183, 185, 186, 188, 196, 197, 200, 204, 206, 214, 228, 237, 238, 242, 245, 248, 249, 251, 256, 259, 263, 265, 266, 270, 273, 277, 280, 288, 291, 293, 295, 305, 311, 312, 323, 329, 332, 335, 353, 363, 365, 374, 375, 378, 384, 386, 391, 402, 409, 410, 416, 419, 426, 433, 435, 437, 440, 447, 448, 452, 455, 458, 459, 466, 468, 470, 472, 475, 482, 487, 498, 501, 508, 512, 514, 519, 528, 533, 539, 542, 556, 563, 567, 570, 571, 573, 580, 595, 602, 606, 608, 610, 617, 620, 622, 623, 634, 636, 650, 659, 671, 675, 679, 689, 690, 693, 696, 701, 703, 706, 708, 710, 713, 718, 721, 729, 731, 735, 739, 743, 749, 757, 759, 764, 774, 795, 797, 815, 816, 818, 820, 827, 839, 840, 848, 851, 854, 857, 869, 876, 882, 885, 893, 896, 899, 910, 918, 923, 930, 941, 944, 948, 958, 959, 962, 970, 973, 979, 981, 984, 993, 1001, 1004, 1016, 1018, ... They are the bases b such that ... * The minimal prime in family 1{z} has length 3 * The minimal prime in family {1}zz has length 3 (and "1z" is not prime) The number "zz1" (in base b) is minimal prime (start with b+1) in these bases b: 10, 11, 14, 19, 26, 31, 35, 41, 45, 50, 52, 54, 57, 62, 75, 77, 84, 85, 87, 92, 95, 99, 101, 116, 125, 129, 130, 134, 136, 140, 141, 147, 155, 167, 176, 202, 221, 230, 239, 242, 244, 245, 252, 256, 259, 262, 264, 265, 271, 272, 274, 277, 286, 290, 292, 295, 297, 299, 305, 307, 316, 321, 322, 327, 329, 340, 347, 351, 354, 356, 362, 364, 372, 376, 391, 392, 414, 419, 441, 442, 446, 449, 455, 465, 466, 470, 472, 479, 482, 486, 491, 494, 497, 504, 510, 516, 521, 525, 540, 547, 554, 557, 570, 574, 584, 587, 591, 594, 605, 617, 619, 626, 629, 631, 642, 647, 650, 655, 656, 670, 682, 684, 685, 686, 689, 696, 712, 722, 732, 739, 741, 745, 746, 750, 752, 757, 759, 761, 766, 771, 776, 780, 787, 789, 790, 809, 815, 817, 824, 827, 831, 834, 836, 839, 851, 857, 860, 867, 869, 882, 901, 904, 915, 922, 924, 932, 934, 946, 950, 957, 959, 971, 972, 976, 977, 987, 1000, 1005, 1019, ... They are the bases b such that ... * The minimal prime in family {z}1 has length 3 * The minimal prime in family zz{1} has length 3 (and "z1" is not prime) The number "yzz" (in base b) is minimal prime (start with b+1) in these bases b: (note: this family does not exist in base b=2) 13, 18, 30, 33, 48, 58, 75, 79, 85, 96, 99, 100, 106, 111, 114, 117, 118, 120, 129, 133, 138, 144, 153, 162, 171, 174, 186, 195, 199, 202, 222, 223, 243, 246, 252, 273, 276, 279, 298, 303, 328, 334, 342, 345, 348, 366, 372, 376, 378, 393, 400, 403, 418, 426, 435, 436, 447, 459, 460, 463, 465, 468, 471, 480, 493, 498, 504, 508, 510, 525, 526, 535, 541, 556, 558, 565, 567, 594, 612, 613, 636, 651, 667, 682, 688, 690, 702, 703, 705, 711, 732, 733, 736, 738, 745, 759, 763, 789, 790, 792, 796, 798, 831, 832, 835, 844, 855, 859, 862, 865, 871, 877, 898, 903, 915, 925, 933, 946, 948, 961, 970, 976, 981, 987, 993, 1000, 1009, ... They are the bases b such that ... * The minimal prime in family y{z} has length 3 * The minimal prime in family {y}zz has length 3 (and "yz" is not prime) The number "zzy" (in base b) is minimal prime (start with b+1) in these bases b: (note: this family does not exist in base b=2) 31, 67, 91, 99, 109, 129, 151, 165, 187, 189, 201, 207, 241, 277, 289, 319, 367, 369, 411, 417, 427, 439, 445, 457, 459, 477, 511, 549, 555, 619, 651, 657, 669, 691, 697, 741, 745, 771, 787, 819, 859, 861, 879, 927, 939, 949, 967, 1015, ... They are the bases b such that ... * The minimal prime in family {z}y has length 3 * The minimal prime in family zz{y} has length 3 (and "zy" is not prime) Last fiddled with by sweety439 on 20220512 at 20:41 
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