20191129, 09:08  #12  
Nov 2016
2^{2}×3×5×47 Posts 
Quote:
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 https://github.com/curtisbright/mepn...nsolved.25.txt (base 25) and https://github.com/RaymondDevillers/.../master/left31 (base 31), etc. Last fiddled with by sweety439 on 20191129 at 09:11 

20191129, 14:39  #13 
"Hugo"
Jul 2019
Germany
31 Posts 
The cited GitHub repositories don't provide the programs to calculate the sets of minimal basen representations, but the lists themselves are given.
See e.g. for n=8:minimal.8.txt 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 A330048 and A330049 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. 
20191130, 00:26  #14  
Nov 2016
2^{2}×3×5×47 Posts 
Quote:
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} Last fiddled with by sweety439 on 20191130 at 00:41 

20191130, 00:59  #15  
Nov 2016
2^{2}×3×5×47 Posts 
Quote:
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_{12380}}X in base 40), A330048(42)=4551, 29103<=A330048(48)<=29109, see https://github.com/RaymondDevillers/primes/ 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 Also, all A330048, A330049 and A327282 should have the keyword "base". Last fiddled with by sweety439 on 20191130 at 01:00 

20191130, 06:43  #16  
Nov 2016
2^{2}·3·5·47 Posts 
Quote:
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 Last fiddled with by sweety439 on 20191130 at 06:44 

20191130, 13:42  #17  
Nov 2016
101100000100_{2} Posts 
Quote:
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} Last fiddled with by sweety439 on 20191130 at 13:42 

20191203, 03:58  #18 
Nov 2016
2^{2}·3·5·47 Posts 
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, 
20201124, 03:50  #19 
Nov 2016
2^{2}×3×5×47 Posts 
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, 111, 141, 161 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^n2^n+1), thus cannot be prime. * Case (1,3): ** 13 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): ** Since 13, 15, 27, 37, 57, 107, 117, 147, 177 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): ** 21 is prime, and thus the only minimal prime in this family. * Case (2,3): ** 23 is prime, and thus the only minimal prime in this family. * Case (2,5): ** Since 21, 23, 27, 15, 35, 45, 65, 75, 225, 255 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): ** 27 is prime, and thus the only minimal prime in this family. Last fiddled with by sweety439 on 20201227 at 06:08 
20201125, 04:52  #20 
Nov 2016
2^{2}×3×5×47 Posts 
* Case (3,1):
** Since 35, 37, 21, 51, 301, 361 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, 3331, 3411 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 3344441 ***** 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. Last fiddled with by sweety439 on 20201227 at 06:09 
20201125, 04:57  #21 
Nov 2016
B04_{16} Posts 
* Case (3,3):
** Since 35, 37, 13, 23, 53, 73, 343 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): ** 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. Last fiddled with by sweety439 on 20201227 at 06:09 
20201212, 10:39  #22 
Nov 2016
B04_{16} Posts 
* Case (4,1):
** Since 45, 21, 51, 401, 431, 471 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, 4611 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 444444441 ****** The smallest prime of the form 4{4}641 is 444641 ***** 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 444444441 ****** 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 444641 * Case (4,3): ** Since 45, 13, 23, 53, 73, 433, 463 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 4043 and 4443 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): ** 45 is prime, and thus the only minimal prime in this family. * Case (4,7): ** Since 45, 27, 37, 57, 407, 417, 467 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 44444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444447, with 220 4's, which can be written as 4_{220}7 and equal the prime (2^665+17)/7 **** The smallest prime of the form 4{4}77 is 4444477 **** The smallest prime of the form 4{7}7 is 47777 **** The smallest prime of the form 44{7}7 is 4477777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777, with 851 7's, which can be written as 447_{851} and equal the prime 37*2^25531 (not minimal prime, since 47777 is prime) Last fiddled with by sweety439 on 20210113 at 20:38 
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