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

- - **Minimal set of the strings for primes with at least two digits**
(*https://www.mersenneforum.org/showthread.php?t=24972*)

[QUOTE=LaurV;531660]I didn't look yet how good is your code, but my former one is lousy, so there are chances that yours is better. I mean, not the code, but my method itself was lousy, to look at all primes one by one. The authors of that paper you linked describe a method which is much better and somehow similar to what I am doing now.
Right now, I split the problem in two steps, first I let the zero apart, and solve the problem with "digits" from 1 to b-1, by starting from the end with all possible cases in a set. Starting from the end or from the beginning makes no difference, but in the case the base is even, I only have n/2 elements in the initial set (because numbers ending in 2, 4, 6, etc, can never be prime), so the search dimension is reduced in half. Then, for all elements in set, I check what digit I can add in front of them and still avoiding conflicts. If any of the resulting numbers is prime, I add it to the set. Here is where the algorithm "strikes", because I can do this in about linear time, by creating a matrix with the possible candidates, and then eliminating them from the matrix, by different criteria (like, it produces conflict, it is a prime and I add it to the list, or it is always composite regardless of how you extend it, etc), and sometimes full rows and columns can be eliminated. This gives me the complete set, excluding the numbers that contain zero, in just minutes. The second part comes from the realization that the numbers that contain zero and have to be in the set, if we delete zeroes from them, the new created are (1) still not in the set, and (2) can not be covered with numbers in the set, and (3) are the same magnitude as the numbers in the set except maybe the first digit, that can repeat indefinitely till the first prime is found. The (3) is very important (and it can be proved) so the second part of the algorithm is to create a list with all such numbers (like 5-6 digit numbers in our case) and see which one becomes a prime when it is "stuffed" with zeroes, which is piece of cake. Mind that the zeros have to be "between" the digits, as "leading zeros do not count"[sup](TM)[/sup] and numbers ending in zero in any base are not prime.[/QUOTE] Unfortunately, my program also look at all primes one by one Is there a better program to write all minimal prime <= 1000 digits in <= 5 minute? Like that we can use program to write all repunit prime <= 1000 digits in <= 5 minute Can we take all forms that may have primes? Like [URL="https://github.com/curtisbright/mepn-data/blob/master/data/unsolved.25.txt"]https://github.com/curtisbright/mepn-data/blob/master/data/unsolved.25.txt[/URL] (base 25) and [URL="https://github.com/RaymondDevillers/primes/blob/master/left31"]https://github.com/RaymondDevillers/primes/blob/master/left31[/URL] (base 31), etc. |

The cited GitHub repositories don't provide the programs to calculate the sets of minimal base-n representations, but the lists themselves are given.
See e.g. for n=8:[URL="https://github.com/curtisbright/mepn-data/blob/master/data/minimal.8.txt"]minimal.8.txt[/URL] From the discussion here I have learned that the length of the lists with the shortest entries are not in the OEIS. That's why I added the corresponding entries A[OEIS]330048[/OEIS] and A[OEIS]330049[/OEIS] With the filling of such defects I am merciless and fast. I could also insert a link to this discussion, or you could provide a PARI program to compute the initial terms of the sequence. No need to be efficient, but demonstrating the principle. |

[QUOTE=yae9911;531687]The cited GitHub repositories don't provide the programs to calculate the sets of minimal base-n representations, but the lists themselves are given.
See e.g. for n=8:[URL="https://github.com/curtisbright/mepn-data/blob/master/data/minimal.8.txt"]minimal.8.txt[/URL] From the discussion here I have learned that the length of the lists with the shortest entries are not in the OEIS. That's why I added the corresponding entries A[OEIS]330048[/OEIS] and A[OEIS]330049[/OEIS] With the filling of such defects I am merciless and fast. I could also insert a link to this discussion, or you could provide a PARI program to compute the initial terms of the sequence. No need to be efficient, but demonstrating the principle.[/QUOTE] My problem is not for the set of minimal base-n representations of the primes, it is for the set of minimal base-n representations of the [B][I]primes >= n[/I][/B], i.e. single-digit primes are not counted. Thus, e.g. for base 5: original set is {2, 3, 10, 111, 401, 414, 14444, 44441} new set is {10, 12, 21, 23, 32, 34, 43, 111, 131, 133, 313, 401, 414, 14444, 30301, 33001, 33331, 44441, 300031} For base 6: original set is {2, 3, 5, 11, 4401, 4441, 40041} new set is {11, 15, 21, 25, 31, 35, 45, 51, 4401, 4441, 40041} |

[QUOTE=yae9911;531687]The cited GitHub repositories don't provide the programs to calculate the sets of minimal base-n representations, but the lists themselves are given.
See e.g. for n=8:[URL="https://github.com/curtisbright/mepn-data/blob/master/data/minimal.8.txt"]minimal.8.txt[/URL] From the discussion here I have learned that the length of the lists with the shortest entries are not in the OEIS. That's why I added the corresponding entries A[OEIS]330048[/OEIS] and A[OEIS]330049[/OEIS] With the filling of such defects I am merciless and fast. I could also insert a link to this discussion, or you could provide a PARI program to compute the initial terms of the sequence. No need to be efficient, but demonstrating the principle.[/QUOTE] However, [URL="https://oeis.org/A326609"]https://oeis.org/A326609[/URL] is in OEIS, A330049(n) is the length of A326609(n) in base n. A330048(17) is either 1279 or 1280, A330048(19) is either 3462 or 3463, A330048(21) is either 2599 or 2600, 17597<=A330048(25)<=17609, 5662<=A330048(26)<=5664, also, A330048(30)=220, 6295<=A330048(36)<=6297, 37773<=A330048(40)<=37774 (I found the prime Qa{U[SUB]12380[/SUB]}X in base 40), A330048(42)=4551, 29103<=A330048(48)<=29109, see [URL="https://github.com/RaymondDevillers/primes/"]https://github.com/RaymondDevillers/primes/[/URL] A330049(30)=1024, A330049(42)=487. Besides, I saw A327282, this is A327282(n) for 28<=n<=48: [CODE] n,A327282(n) 28,131 29,123 30,207 31,147 32,160 33,163 34,201 35,169 36,216 37,173 38,185 39,195 40,242 41,205 42,331 43,229 44,242 45,252 46,277 47,261 48,411 [/CODE] (I only searched up to 4 digits, I assume that there are no minimal composites with >=5 digits in these bases) Also, all A330048, A330049 and A327282 should have the keyword "base". |

[QUOTE=sweety439;531721]However, [URL="https://oeis.org/A326609"]https://oeis.org/A326609[/URL] is in OEIS, A330049(n) is the length of A326609(n) in base n.
A330048(17) is either 1279 or 1280, A330048(19) is either 3462 or 3463, A330048(21) is either 2599 or 2600, 17597<=A330048(25)<=17609, 5662<=A330048(26)<=5664, also, A330048(30)=220, 6295<=A330048(36)<=6297, 37773<=A330048(40)<=37774 (I found the prime Qa{U[SUB]12380[/SUB]}X in base 40), A330048(42)=4551, 29103<=A330048(48)<=29109, see [URL="https://github.com/RaymondDevillers/primes/"]https://github.com/RaymondDevillers/primes/[/URL] A330049(30)=1024, A330049(42)=487. Besides, I saw A327282, this is A327282(n) for 28<=n<=48: [CODE] n,A327282(n) 28,131 29,123 30,207 31,147 32,160 33,163 34,201 35,169 36,216 37,173 38,185 39,195 40,242 41,205 42,331 43,229 44,242 45,252 46,277 47,261 48,411 [/CODE] (I only searched up to 4 digits, I assume that there are no minimal composites with >=5 digits in these bases) Also, all A330048, A330049 and A327282 should have the keyword "base".[/QUOTE] A327282(n) for 49<=n<=75: [CODE] 49,294 50,292 51,290 52,322 53,299 54,438 55,331 56,304 57,331 58,356 59,339 60,659 61,375 62,379 63,404 64,461 65,412 66,613 67,416 68,419 69,449 70,647 71,464 72,696 73,505 74,499 75,538 [/CODE] This is enough to fill the "data" section of A327282 |

[QUOTE=sweety439;531436]....
Now, let's consider: if our set is [B]the set of prime numbers >= b[/B] written in radix b (i.e. the prime numbers with at least two digits in radix b), then we get the sets: [CODE] b, we get the set 2: {10, 11} 3: {10, 12, 21, 111} 4: {11, 13, 23, 31, 221} 5: {10, 12, 21, 23, 32, 34, 43, 111, 131, 133, 313, 401, 414, 14444, 30301, 33001, 33331, 44441, 300031} 6: {11, 15, 21, 25, 31, 35, 45, 51, 4401, 4441, 40041} 7: {10, 14, 16, 23, 25, 32, 41, 43, 52, 56, 61, 65, 113, 115, 131, 133, 155, 212, 221, 304, 313, 335, 344, 346, 364, 445, 515, 533, 535, 544, 551, 553, 1112, 1211, 1222, 2111, 3031, 3055, 3334, 3503, 3505, 3545, 4504, 4555, 5011, 5455, 5545, 5554, 6034, 6634, 11111, 30011, 31111, 33001, 33311, 35555, 40054, 300053, 33333301} 8: {13, 15, 21, 23, 27, 35, 37, 45, 51, 53, 57, 65, 73, 75, 107, 111, 117, 141, 147, 161, 177, 225, 255, 301, 343, 361, 401, 407, 417, 431, 433, 463, 467, 471, 631, 643, 661, 667, 701, 711, 717, 747, 767, 3331, 3411, 4043, 4443, 4611, 5205, 6007, 6101, 6441, 6477, 6707, 6777, 7461, 7641, 47777, 60171, 60411, 60741, 444641, 500025, 505525, 3344441, 4444477, 5500525, 5550525, 55555025, 444444441, 744444441} [/CODE] However, I do not think that my base 7 and 8 sets are complete (I use PARI program to find these primes (all written in base b), but I only searched the primes with <= 8 digits, so there may be missing primes), I proved that my base 2, 3, 4, 5 and 6 sets are complete. Can someone complete my base 7 and 8 set? Also find the sets of bases 9 to 36.[/QUOTE] For bases 9 to 12: [CODE] b, we get the set 9: {12, 14, 18, 21, 25, 32, 34, 41, 45, 47, 52, 58, 65, 67, 74, 78, 81, 87, 117, 131, 135, 151, 155, 175, 177, 238, 272, 308, 315, 331, 337, 355, 371, 375, 377, 438, 504, 515, 517, 531, 537, 557, 564, 601, 638, 661, 702, 711, 722, 735, 737, 751, 755, 757, 771, 805, 838, 1011, 1015, 1101, 1701, 2027, 2207, 3017, 3057, 3101, 3501, 3561, 3611, 3688, 3868, 5035, 5051, 5071, 5101, 5501, 5554, 5705, 5707, 7017, 7075, 7105, 7301, 8535, 8544, 8555, 8854, 20777, 22227, 22777, 30161, 33388, 50161, 50611, 53335, 55111, 55535, 55551, 57061, 57775, 70631, 71007, 77207, 100037, 100071, 100761, 105007, 270707, 301111, 305111, 333035, 333385, 333835, 338885, 350007, 500075, 530005, 555611, 631111, 720707, 2770007, 3030335, 7776662, 30300005, 30333335, 38333335, 51116111, 70000361, 300030005, 300033305, 351111111, 1300000007} 10: {11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 227, 251, 257, 277, 281, 349, 409, 449, 499, 521, 557, 577, 587, 727, 757, 787, 821, 827, 857, 877, 881, 887, 991, 2087, 2221, 5051, 5081, 5501, 5581, 5801, 5851, 6469, 6949, 8501, 9001, 9049, 9221, 9551, 9649, 9851, 9949, 20021, 20201, 50207, 60649, 80051, 666649, 946669, 5200007, 22000001, 60000049, 66000049, 66600049, 80555551} 11: {10, 12, 16, 18, 21, 27, 29, 34, 38, 3A, 43, 49, 54, 56, 61, 65, 67, 72, 76, 81, 89, 92, 94, 98, 9A, A3, 115, 117, 133, 139, 153, 155, 171, 193, 197, 199, 1AA, 225, 232, 236, 25A, 263, 315, 319, 331, 335, 351, 353, 362, 373, 379, 391, 395, 407, 414, 452, 458, 478, 47A, 485, 4A5, 4A7, 502, 508, 511, 513, 533, 535, 539, 551, 571, 579, 588, 595, 623, 632, 70A, 711, 715, 731, 733, 737, 755, 759, 775, 791, 797, 7AA, 803, 847, 858, 85A, 874, 885, 887, 913, 919, 931, 937, 957, 959, 975, 995, A07, A1A, A25, A45, A74, A7A, A85, AA1, AA7, 11A9, 1451, 1457, 15A7, 175A, 17A5, 17A9, 2023, 2045, 2052, 2083, 20A5, 2333, 2A05, 2A52, 3013, 3026, 3059, 3097, 3206, 3222, 3233, 3307, 3332, 3505, 4025, 4151, 4157, 4175, 4405, 4445, 4487, 450A, 4575, 5017, 5031, 5059, 5075, 5097, 5099, 515A, 517A, 520A, 5301, 5583, 5705, 577A, 5853, 5873, 5909, 5A17, 5A57, 5A77, 5A8A, 6683, 66A9, 7019, 7073, 7079, 7088, 7093, 7095, 7309, 7451, 7501, 7507, 7578, 757A, 75A7, 7787, 7804, 7844, 7848, 7853, 7877, 78A4, 7A04, 7A57, 7A79, 7A95, 8078, 8245, 8333, 8355, 8366, 8375, 8425, 8553, 8663, 8708, 8777, 878A, 8A05, 9053, 9305, 9505, 9703, A052, A119, A151, A175, A515, A517, A575, A577, A5A8, A719, A779, A911, AAA9, 11131, 11144, 11191, 1141A, 114A1, 13757, 1411A, 14477, 144A4, 14A11, 1774A, 17777, 177A4, 17A47, 1A114, 1A411, 20005, 20555, 22203, 25228, 25282, 25552, 25822, 28522, 30037, 30701, 30707, 31113, 33777, 35009, 35757, 39997, 40045, 4041A, 40441, 4045A, 404A1, 4111A, 411A1, 42005, 44401, 44474, 444A1, 44555, 44577, 445AA, 44744, 44A01, 47471, 47477, 47701, 5057A, 50903, 5228A, 52A22, 52A55, 52A82, 55007, 550A9, 55205, 55522, 55557, 55593, 55805, 57007, 57573, 57773, 57807, 5822A, 58307, 58505, 58A22, 59773, 59917, 59973, 59977, 59999, 5A015, 5A2A2, 5AA99, 60836, 60863, 68636, 6A609, 6A669, 6A696, 6A906, 6A966, 70048, 70471, 70583, 70714, 71474, 717A4, 74084, 74444, 74448, 74477, 744A8, 74747, 74774, 7488A, 74A48, 75773, 77144, 77401, 77447, 77799, 77A09, 78008, 78783, 7884A, 78888, 788A8, 79939, 79993, 79999, 7A051, 7A444, 7A471, 80005, 80252, 80405, 80522, 80757, 80AA5, 83002, 84045, 85307, 86883, 88863, 8A788, 90073, 90707, 90901, 95003, 97779, 97939, 99111, 99177, 99973, A0111, A0669, A0966, A0999, A0A09, A4177, A4401, A4717, A5228, A52AA, A5558, A580A, A5822, A58AA, A5A59, A5AA2, A6096, A6966, A6999, A7051, A7778, A7808, A9055, A9091, A9699, A9969, AA52A, AA58A, 222223, 222823, 300202, 300323, 303203, 307577, 332003, 370777, 400555, 401A11, 404001, 404111, 405AAA, 440A41, 451777, 455555, 470051, 470444, 474404, 4A0401, 4A4041, 500015, 500053, 500077, 500507, 505577, 522A2A, 525223, 528A2A, 550777, 553707, 5555A9, 555A99, 557707, 55A559, 5807A7, 580A0A, 580A55, 58A0AA, 590007, 599907, 5A2228, 5A2822, 5A2AAA, 5A552A, 5AA22A, 5AAA22, 60A069, 683006, 6A0096, 6A0A96, 6A9099, 6A9909, 700778, 701777, 704408, 704417, 704457, 704484, 707041, 707441, 707708, 707744, 707784, 740008, 74484A, 770441, 770744, 770748, 770771, 777017, 777071, 777448, 777484, 777701, 7778A8, 777A19, 777A48, 778883, 78A808, 790003, 7A4408, 7A7708, 80A555, 828283, 828883, 840555, 850505, 868306, 873005, 883202, 900701, 909739, 909979, 909991, 970771, 977701, 979909, 990739, 990777, 990793, 997099, 999709, 999901, A00009, A00599, A05509, A0A058, A0A955, A555A2, A55999, A59991, A5A222, A5A22A, A60609, A66069, A66906, A69006, A79005, A87888, A90099, A90996, A96006, A96666, A97177, A97771, AA0A58, AA5A22, AAA522, 111114A, 1111A14, 1111A41, 1144441, 14A4444, 1A44444, 2555505, 2845055, 3030023, 3333397, 4000111, 4011111, 41A1111, 4411111, 444441A, 4444771, 4470004, 4505005, 4744417, 4774441, 4777404, 4777417, 4777747, 4A11111, 4A40001, 5000093, 50005A7, 5005777, 5050553, 5055503, 5070777, 5222222, 5222AAA, 52AAAA2, 52AAAAA, 5505053, 5552AAA, 5555599, 5555A58, 5558A0A, 5558A55, 5558AAA, 55A0009, 55AAA52, 580000A, 5822222, 58AAAAA, 5A2222A, 5AA2222, 6000A69, 6000A96, 6A00069, 7000417, 7000741, 7000835, 7000857, 7007177, 7008305, 7014447, 7017444, 7074177, 7077477, 7077741, 7077747, 717444A, 7400404, 7700717, 7707778, 7774004, 777741A, 7777441, 777774A, 7777A47, 7779003, 777A008, 777A778, 777A808, 77A4777, 7900399, 8305007, 8500707, 8555707, 8883022, 8AA5222, 9000035, 9007999, 9009717, 9009777, 9009997, 9090997, 9099907, 9355555, 9790099, 9900991, 9900997, 9907909, 9909079, 9979009, 9990079, 9990091, 9990907, 9999771, 9999799, 9999979, A000696, A000991, A006906, A040041, A0AAA58, A141111, A5222A2, A600A69, A906606, A909009, A990009, A997701, AA55A52, AAA5552, AAAAA52, 22888823, 28888223, 30555777, 33000023} 12: {11, 15, 17, 1B, 25, 27, 31, 35, 37, 3B, 45, 4B, 51, 57, 5B, 61, 67, 6B, 75, 81, 85, 87, 8B, 91, 95, A7, AB, B5, B7, 221, 241, 2A1, 2B1, 2BB, 401, 421, 447, 471, 497, 565, 655, 665, 701, 70B, 721, 747, 771, 77B, 797, 7A1, 7BB, 907, 90B, 9BB, A41, B21, B2B, 2001, 200B, 202B, 222B, 229B, 292B, 299B, 4441, 4707, 4777, 6A05, 6AA5, 729B, 7441, 7B41, 929B, 9777, 992B, 9947, 997B, 9997, A0A1, A201, A605, A6A5, AA65, B001, B0B1, BB01, BB41, 600A5, 7999B, 9999B, AAAA1, B04A1, B0B9B, BAA01, BAAA1, BB09B, BBBB1, 44AAA1, A00065, BBBAA1, AAA0001, B00099B, AA000001} [/CODE] Can someone complete them? |

For base 11, I found these numbers: (for the primes with at least two digits)
10, 12, 16, 18, 21, 27, 29, 34, 38, 3A, 43, 49, 54, 56, 61, 65, 67, 72, 76, 81, 89, 92, 94, 98, 9A, A3, 115, 117, 133, 139, 153, 155, 171, 193, 197, 199, 1AA, 225, 232, 236, 25A, 263, 315, 319, 331, 335, 351, 353, 362, 373, 379, 391, 395, 407, 414, 452, 458, 478, 47A, 485, 4A5, 4A7, 502, 508, 511, 513, 533, 535, 539, 551, 571, 579, 588, 595, 623, 632, 70A, 711, 715, 731, 733, 737, 755, 759, 775, 791, 797, 7AA, 803, 847, 858, 85A, 874, 885, 887, 913, 919, 931, 937, 957, 959, 975, 995, A07, A1A, A25, A45, A74, A7A, A85, AA1, AA7, 11A9, 1451, 1457, 15A7, 175A, 17A5, 17A9, 2023, 2045, 2052, 2083, 20A5, 2333, 2A05, 2A52, 3013, 3026, 3059, 3097, 3206, 3222, 3233, 3307, 3332, 3505, 4025, 4151, 4157, 4175, 4405, 4445, 4487, 450A, 4575, 5017, 5031, 5059, 5075, 5097, 5099, 515A, 517A, 520A, 5301, 5583, 5705, 577A, 5853, 5873, 5909, 5A17, 5A57, 5A77, 5A8A, 6683, 66A9, 7019, 7073, 7079, 7088, 7093, 7095, 7309, 7451, 7501, 7507, 7578, 757A, 75A7, 7787, 7804, 7844, 7848, 7853, 7877, 78A4, 7A04, 7A57, 7A79, 7A95, 8078, 8245, 8333, 8355, 8366, 8375, 8425, 8553, 8663, 8708, 8777, 878A, 8A05, 9053, 9305, 9505, 9703, A052, A119, A151, A175, A515, A517, A575, A577, A5A8, A719, A779, A911, AAA9, 11131, 11144, 11191, 1141A, 114A1, 13757, 1411A, 14477, 144A4, 14A11, 1774A, 17777, 177A4, 17A47, 1A114, 1A411, 20005, 20555, 22203, 25228, 25282, 25552, 25822, 28522, 30037, 30701, 30707, 31113, 33777, 35009, 35757, 39997, 40045, 4041A, 40441, 4045A, 404A1, 4111A, 411A1, 42005, 44401, 44474, 444A1, 44555, 44577, 445AA, 44744, 44A01, 47471, 47477, 47701, 5057A, 50903, 5228A, 52A22, 52A55, 52A82, 55007, 550A9, 55205, 55522, 55557, 55593, 55805, 57007, 57573, 57773, 57807, 5822A, 58307, 58505, 58A22, 59773, 59917, 59973, 59977, 59999, 5A015, 5A2A2, 5AA99, 60836, 60863, 68636, 6A609, 6A669, 6A696, 6A906, 6A966, 70048, 70471, 70583, 70714, 71474, 717A4, 74084, 74444, 74448, 74477, 744A8, 74747, 74774, 7488A, 74A48, 75773, 77144, 77401, 77447, 77799, 77A09, 78008, 78783, 7884A, 78888, 788A8, 79939, 79993, 79999, 7A051, 7A444, 7A471, 80005, 80252, 80405, 80522, 80757, 80AA5, 83002, 84045, 85307, 86883, 88863, 8A788, 90073, 90707, 90901, 95003, 97779, 97939, 99111, 99177, 99973, A0111, A0669, A0966, A0999, A0A09, A4177, A4401, A4717, A5228, A52AA, A5558, A580A, A5822, A58AA, A5A59, A5AA2, A6096, A6966, A6999, A7051, A7778, A7808, A9055, A9091, A9699, A9969, AA52A, AA58A, 222223, 222823, 300202, 300323, 303203, 307577, 332003, 370777, 400555, 401A11, 404001, 404111, 405AAA, 440A41, 451777, 455555, 470051, 470444, 474404, 4A0401, 4A4041, 500015, 500053, 500077, 500507, 505577, 522A2A, 525223, 528A2A, 550777, 553707, 5555A9, 555A99, 557707, 55A559, 5807A7, 580A0A, 580A55, 58A0AA, 590007, 599907, 5A2228, 5A2822, 5A2AAA, 5A552A, 5AA22A, 5AAA22, 60A069, 683006, 6A0096, 6A0A96, 6A9099, 6A9909, 700778, 701777, 704408, 704417, 704457, 704484, 707041, 707441, 707708, 707744, 707784, 740008, 74484A, 770441, 770744, 770748, 770771, 777017, 777071, 777448, 777484, 777701, 7778A8, 777A19, 777A48, 778883, 78A808, 790003, 7A4408, 7A7708, 80A555, 828283, 828883, 840555, 850505, 868306, 873005, 883202, 900701, 909739, 909979, 909991, 970771, 977701, 979909, 990739, 990777, 990793, 997099, 999709, 999901, A00009, A00599, A05509, A0A058, A0A955, A555A2, A55999, A59991, A5A222, A5A22A, A60609, A66069, A66906, A69006, A79005, A87888, A90099, A90996, A96006, A96666, A97177, A97771, AA0A58, AA5A22, AAA522, 111114A, 1111A14, 1111A41, 1144441, 14A4444, 1A44444, 2555505, 2845055, 3030023, 3333397, 4000111, 4011111, 41A1111, 4411111, 444441A, 4444771, 4470004, 4505005, 4744417, 4774441, 4777404, 4777417, 4777747, 4A11111, 4A40001, 5000093, 50005A7, 5005777, 5050553, 5055503, 5070777, 5222222, 5222AAA, 52AAAA2, 52AAAAA, 5505053, 5552AAA, 5555599, 5555A58, 5558A0A, 5558A55, 5558AAA, 55A0009, 55AAA52, 580000A, 5822222, 58AAAAA, 5A2222A, 5AA2222, 6000A69, 6000A96, 6A00069, 7000417, 7000741, 7000835, 7000857, 7007177, 7008305, 7014447, 7017444, 7074177, 7077477, 7077741, 7077747, 717444A, 7400404, 7700717, 7707778, 7774004, 777741A, 7777441, 777774A, 7777A47, 7779003, 777A008, 777A778, 777A808, 77A4777, 7900399, 8305007, 8500707, 8555707, 8883022, 8AA5222, 9000035, 9007999, 9009717, 9009777, 9009997, 9090997, 9099907, 9355555, 9790099, 9900991, 9900997, 9907909, 9909079, 9979009, 9990079, 9990091, 9990907, 9999771, 9999799, 9999979, A000696, A000991, A006906, A040041, A0AAA58, A141111, A5222A2, A600A69, A906606, A909009, A990009, A997701, AA55A52, AAA5552, AAAAA52, 22888823, 28888223, 30555777, 33000023, 40A00041, 45000055, 47040004, 50377777, 50555553, 5282AAA2, 55505003, 555A5A52, 555AAA2A, 55A5A552, 5AAAAA2A, 60A99999, 70000057, 70070474, 70074704, 70700078, 70700474, 70704704, 70777177, 74470001, 77000177, 77070477, 77470004, 77700404, 77748808, 7774A888, 77770078, 77770474, 77774704, 77777008, 77777404, 77777778, 80555055, 88828823, 88888326, 88888823, 8A522222, 90097909, 90700999, 90977777, 97000001, 97000717, 97770007, 99000001, 99000771, 99077001, 99090097, 99777707, 99900097, 99970717, 99999097, 99999707, A0000058, A0004041, A00055A9, A000A559, A5555009, A5A55552, A9700001, A9909006, A9990006, A9990606, A9999917, A9999966, 300577777, 40000A401, 447771777, 44A444441, 474000004, 477700004, 477777774, 505000003, 55555AA2A, 5555A5A2A, 700000147, 700044004, 700077774, 704000044, 704040004, 707070774, 707077704, 707770704, 707777004, 770000078, 770004704, 770070747, 770070774, 770700008, 770700084, 770707074, 777000044, 777000774, 777770477, 777770747, 7777777A4, 77A700008, 888888302, 900000091, 900090799, 970009099, 990990007, 997000077, 999999997, A0000AA58, A00990001, A05555559, A44444111, A44444777, A44477777, A66666669, A90000606, A99999006, A99999099, |

In base 8, 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), (1,3), (1,5), (1,7), (2,1), (2,3), (2,5), (2,7), (3,1), (3,3), (3,5), (3,7), (4,1), (4,3), (4,5), (4,7), (5,1), (5,3), (5,5), (5,7), (6,1), (6,3), (6,5), (6,7), (7,1), (7,3), (7,5), (7,7) * Case (1,1): ** Since 13, 15, 21, 51, [B]111[/B], [B]141[/B], [B]161[/B] are primes, we only need to consider the family 1{0,7}1 (since any digits 1, 2, 3, 4, 5, 6 between them will produce smaller primes) *** Since 107, 177, 701 are primes, we only need to consider the number 171 and the family 1{0}1 (since any digits combo 07, 70, 77 between them will produce smaller primes) **** 171 is not prime. **** All numbers of the form 1{0}1 factored as 10^n+1 = (2^n+1) * (4^n-2^n+1), thus cannot be prime. * Case (1,3): ** [B]13[/B] is prime, and thus the only minimal prime in this family. * Case (1,5): ** [B]15[/B] is prime, and thus the only minimal prime in this family. * Case (1,7): ** Since 13, 15, 27, 37, 57, [B]107[/B], [B]117[/B], [B]147[/B], [B]177[/B] are primes, we only need to consider the family 1{6}7 (since any digits 0, 1, 2, 3, 4, 5, 7 between them will produce smaller primes) *** The smallest prime of the form 1{6}7 is 16667 (not minimal prime, since 667 is prime) * Case (2,1): ** [B]21[/B] is prime, and thus the only minimal prime in this family. * Case (2,3): ** [B]23[/B] is prime, and thus the only minimal prime in this family. * Case (2,5): ** Since 21, 23, 27, 15, 35, 45, 65, 75, [B]225[/B], [B]255[/B] are primes, we only need to consider the family 2{0}5 (since any digits 1, 2, 3, 4, 5, 6, 7 between them will produce smaller primes) *** All numbers of the form 2{0}5 are divisible by 7, thus cannot be prime. * Case (2,7): ** [B]27[/B] is prime, and thus the only minimal prime in this family. |

* Case (3,1):
** Since 35, 37, 21, 51, [B]301[/B], [B]361[/B] are primes, we only need to consider the family 3{1,3,4}1 (since any digits 0, 2, 5, 6, 7 between them will produce smaller primes) *** Since 13, 343, 111, 131, 141, 431, [B]3331[/B], [B]3411[/B] are primes, we only need to consider the families 3{3}11, 33{1,4}1, 3{3,4}4{4}1 (since any digits combo 11, 13, 14, 33, 41, 43 between them will produce smaller primes) **** All numbers of the form 3{3}11 are divisible by 3, thus cannot be prime. **** For the 33{1,4}1 family, since 111 and 141 are primes, we only need to consider the families 33{4}1 and 33{4}11 (since any digits combo 11, 14 between them will produce smaller primes) ***** The smallest prime of the form 33{4}1 is [B]3344441[/B] ***** All numbers of the form 33{4}11 are divisible by 301, thus cannot be prime. **** For the 3{3,4}4{4}1 family, since 3331 and 3344441 are primes, we only need to consider the families 3{4}1, 3{4}31, 3{4}341, 3{4}3441, 3{4}34441 (since any digits combo 33 or 34444 between (3,1) will produce smaller primes) ***** All numbers of the form 3{4}1 are divisible by 31, thus cannot be prime. ***** Since 4443 is prime, we only need to consider the numbers 3431, 34431, 34341, 344341, 343441, 3443441, 3434441, 34434441 (since any digit combo 444 between (3,3{4}1) will produce smaller primes) ****** None of 3431, 34431, 34341, 344341, 343441, 3443441, 3434441, 34434441 are primes. |

* Case (3,3):
** Since 35, 37, 13, 23, 53, 73, [B]343[/B] are primes, we only need to consider the family 3{0,3,6}3 (since any digits 1, 2, 4, 5, 7 between them will produce smaller primes) *** All numbers of the form 3{0,3,6}3 are divisible by 3, thus cannot be prime. * Case (3,5): ** [B]35[/B] is prime, and thus the only minimal prime in this family. * Case (3,7): ** [B]37[/B] is prime, and thus the only minimal prime in this family. |

* Case (4,1):
** Since 45, 21, 51, [B]401[/B], [B]431[/B], [B]471[/B] are primes, we only need to consider the family 4{1,4,6}1 (since any digits 0, 2, 3, 5, 7 between them will produce smaller primes) *** Since 111, 141, 161, 661, [B]4611[/B] are primes, we only need to consider the families 4{4}11, 4{4,6}4{1,4,6}1, 4{4}6{4}1 (since any digits combo 11, 14, 16, 61, 66 between them will produce smaller primes) **** The smallest prime of the form 4{4}11 is 44444444444444411 (not minimal prime, since 444444441 is prime) **** For the 4{4,6}4{1,4,6}1 family, we can separate this family to 4{4,6}41, 4{4,6}411, 4{4,6}461 ***** For the 4{4,6}41 family, since 661 and 6441 are primes, we only need to consider the families 4{4}41 and 4{4}641 (since any digits combo 64 or 66 between (4,41) will produce smaller primes) ****** The smallest prime of the form 4{4}41 is [B]444444441[/B] ****** The smallest prime of the form 4{4}641 is [B]444641[/B] ***** For the 4{4,6}411 family, since 661 and 6441 are primes, we only need to consider the families 4{4}411 and 4{4}6411 (since any digits combo 64 or 66 between (4,411) will produce smaller primes) ****** The smallest prime of the form 4{4}411 is [B]444444441[/B] ****** The smallest prime of the form 4{4}6411 is 4444444444444446411 (not minimal prime, since 444444441 and 444641 are primes) ***** For the 4{4,6}461 family, since 661 is prime, we only need to consider the family 4{4}461 ****** The smallest prime of the form 4{4}461 is 4444444461 (not minimal prime, since 444444441 is prime) **** For the 4{4}6{4}1 family, since 6441 is prime, we only need to consider the families 4{4}61 and 4{4}641 (since any digits combo 44 between (4{4}6,1) will produce smaller primes) ***** The smallest prime of the form 4{4}61 is 4444444461 (not minimal prime, since 444444441 is prime) ***** The smallest prime of the form 4{4}641 is [B]444641[/B] * Case (4,3): ** Since 45, 13, 23, 53, 73, [B]433[/B], [B]463[/B] are primes, we only need to consider the family 4{0,4}3 (since any digits 1, 2, 3, 5, 6, 7 between them will produce smaller primes) *** Since [B]4043[/B] and [B]4443[/B] are primes, we only need to consider the families 4{0}3 and 44{0}3 (since any digits combo 04, 44 between them will produce smaller primes) **** All numbers of the form 4{0}3 are divisible by 7, thus cannot be prime. **** All numbers of the form 44{0}3 are divisible by 3, thus cannot be prime. * Case (4,5): ** [B]45[/B] is prime, and thus the only minimal prime in this family. * Case (4,7): ** Since 45, 27, 37, 57, [B]407[/B], [B]417[/B], [B]467[/B] are primes, we only need to consider the family 4{4,7}7 (since any digits 0, 1, 2, 3, 5, 6 between them will produce smaller primes) *** Since 747 is prime, we only need to consider the families 4{4}7, 4{4}77, 4{7}7, 44{7}7 (since any digits combo 74 between (4,7) will produce smaller primes) **** The smallest prime of the form 4{4}7 is [B]44444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444447[/B], with 220 4's, which can be written as 4[SUB]220[/SUB]7 and equal the prime (2^665+17)/7 **** The smallest prime of the form 4{4}77 is [B]4444477[/B] **** The smallest prime of the form 4{7}7 is [B]47777[/B] **** The smallest prime of the form 44{7}7 is 4477777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777, with 851 7's, which can be written as 447[SUB]851[/SUB] and equal the prime 37*2^2553-1 (not minimal prime, since 47777 is prime) |

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