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Minimal set of the strings for primes with at least two digits
[COLOR="Red"]Edit:
The article for this project: [URL="https://docs.google.com/document/d/e/2PACX-1vQct6Hx-IkJd5-iIuDuOKkKdw2teGmmHW-P75MPaxqBXB37u0odFBml5rx0PoLa0odTyuW67N_vn96J/pub"]https://docs.google.com/document/d/e/2PACX-1vQct6Hx-IkJd5-iIuDuOKkKdw2teGmmHW-P75MPaxqBXB37u0odFBml5rx0PoLa0odTyuW67N_vn96J/pub[/URL] (new link from GoogleDrive: [URL="https://docs.google.com/document/d/17RtAuTOGMJOYjbyf24zcPJqEHQQQwC_1A6EN154rpFs/edit?usp=sharing"]https://docs.google.com/document/d/17RtAuTOGMJOYjbyf24zcPJqEHQQQwC_1A6EN154rpFs/edit?usp=sharing[/URL]) The GitHub page for this project: [URL="https://github.com/xayahrainie4793/non-single-digit-primes"]https://github.com/xayahrainie4793/non-single-digit-primes[/URL] The Excel file for the smallest (probable) primes in families which [I]always[/I] produce minimal primes (start with b+1) for 2<=b<=1024: [URL="https://docs.google.com/spreadsheets/d/e/2PACX-1vTKkSNKGVQkUINlp1B3cXe90FWPwiegdA07EE7-U7sqXntKAEQrynoI1sbFvvKriieda3LfkqRwmKME/pubhtml"]https://docs.google.com/spreadsheets/d/e/2PACX-1vTKkSNKGVQkUINlp1B3cXe90FWPwiegdA07EE7-U7sqXntKAEQrynoI1sbFvvKriieda3LfkqRwmKME/pubhtml[/URL] [/COLOR] ----------------------------------------------------------I am the dividing line---------------------------------------------------------- [URL="http://primes.utm.edu/glossary/xpage/MinimalPrime.html"]http://primes.utm.edu/glossary/xpage/MinimalPrime.html[/URL] In 1996, Jeffrey Shallit [Shallit96] suggested that we view prime numbers as strings of digits. He then used concepts from formal language theory to define an interesting set of primes called the minimal primes: A string a is a subsequence of another string b, if a can be obtained from b by deleting zero or more of the characters in b. For example, 514 is a substring of 251664. The empty string is a subsequence of every string. Two strings a and b are comparable if either a is a substring of b, or b is a substring of a. A surprising result from formal language theory is that every set of pairwise incomparable strings is finite [Lothaire83]. This means that from any set of strings we can find its minimal elements. A string a in a set of strings S is minimal if whenever b (an element of S) is a substring of a, we have b = a. This set must be finite! For example, if our set is the set of prime numbers (written in radix 10), then we get the set {2, 3, 5, 7, 11, 19, 41, 61, 89, 409, 449, 499, 881, 991, 6469, 6949, 9001, 9049, 9649, 9949, 60649, 666649, 946669, 60000049, 66000049, 66600049}, and if our set is the set of composite numbers (written in radix 10), then we get the set {4, 6, 8, 9, 10, 12, 15, 20, 21, 22, 25, 27, 30, 32, 33, 35, 50, 51, 52, 55, 57, 70, 72, 75, 77, 111, 117, 171, 371, 711, 713, 731} Besides, if our set is the set of prime numbers written in radix b, then we get these sets: [CODE] b, we get the set 2: {10, 11} 3: {2, 10, 111} 4: {2, 3, 11} 5: {2, 3, 10, 111, 401, 414, 14444, 44441} 6: {2, 3, 5, 11, 4401, 4441, 40041} 7: {2, 3, 5, 10, 14, 16, 41, 61, 11111} 8: {2, 3, 5, 7, 111, 141, 161, 401, 661, 4611, 6101, 6441, 60411, 444641, 444444441} 9: {2, 3, 5, 7, 14, 18, 41, 81, 601, 661, 1011, 1101} 10: {2, 3, 5, 7, 11, 19, 41, 61, 89, 409, 449, 499, 881, 991, 6469, 6949, 9001, 9049, 9649, 9949, 60649, 666649, 946669, 60000049, 66000049, 66600049} 11: {2, 3, 5, 7, 10, 16, 18, 49, 61, 81, 89, 94, 98, 9A, 199, 1AA, 414, 919, A1A, AA1, 11A9, 66A9, A119, A911, AAA9, 11144, 11191, 1141A, 114A1, 1411A, 144A4, 14A11, 1A114, 1A411, 4041A, 40441, 404A1, 4111A, 411A1, 44401, 444A1, 44A01, 6A609, 6A669, 6A696, 6A906, 6A966, 90901, 99111, A0111, A0669, A0966, A0999, A0A09, A4401, A6096, A6966, A6999, A9091, A9699, A9969, 401A11, 404001, 404111, 440A41, 4A0401, 4A4041, 60A069, 6A0096, 6A0A96, 6A9099, 6A9909, 909991, 999901, A00009, A60609, A66069, A66906, A69006, A90099, A90996, A96006, A96666, 111114A, 1111A14, 1111A41, 1144441, 14A4444, 1A44444, 4000111, 4011111, 41A1111, 4411111, 444441A, 4A11111, 4A40001, 6000A69, 6000A96, 6A00069, 9900991, 9990091, A000696, A000991, A006906, A040041, A141111, A600A69, A906606, A909009, A990009, 40A00041, 60A99999, 99000001, A0004041, A9909006, A9990006, A9990606, A9999966, 40000A401, 44A444441, 900000091, A00990001, A44444111, A66666669, A90000606, A99999006, A99999099, 600000A999, A000144444, A900000066, A0000000001, A0014444444, 40000000A0041, A000000014444, A044444444441, A144444444411, 40000000000401, A0000044444441, A00000000444441, 11111111111111111, 14444444444441111, 44444444444444111, A1444444444444444, A9999999999999996, 1444444444444444444, 4000000000000000A041, A999999999999999999999, A44444444444444444444444441, 40000000000000000000000000041, 440000000000000000000000000001, 999999999999999999999999999999991, 444444444444444444444444444444444444444444441} 12: {2, 3, 5, 7, B, 11, 61, 81, 91, 401, A41, 4441, A0A1, AAAA1, 44AAA1, AAA0001, AA000001} [/CODE] these are already researched in [URL="https://cs.uwaterloo.ca/~cbright/reports/mepn.pdf"]https://cs.uwaterloo.ca/~cbright/reports/mepn.pdf[/URL]. Now, let's consider: if our set is [B]the set of prime numbers > b[/B] written in radix b, then we get the sets: [CODE] b, we get the set 2: {11} 3: {12, 21, 111} 4: {11, 13, 23, 31, 221} 5: {12, 21, 23, 32, 34, 43, 104, 111, 131, 133, 313, 401, 414, 3101, 10103, 14444, 30301, 33001, 33331, 44441, 300031, 100000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000013} 6: {11, 15, 21, 25, 31, 35, 45, 51, 4401, 4441, 40041} 7: {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, 1022, 1051, 1112, 1202, 1211, 1222, 2111, 3031, 3055, 3334, 3503, 3505, 3545, 4504, 4555, 5011, 5455, 5545, 5554, 6034, 6634, 11111, 11201, 30011, 30101, 31001, 31111, 33001, 33311, 35555, 40054, 100121, 150001, 300053, 351101, 531101, 1100021, 33333301, 5100000001, ..., 33333333333333331, ...} 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, 77774444441, 7777777777771, 555555555555525, 44444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444447} 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, 5161111111, 8333333335, ..., 300000000035, ..., 311111111161, ..., 544444444444, ..., 2000000000007, ..., 5700000000001, ..., 100000000000507, ..., 5111111111111161, ..., 8888888888888888888335, ..., 30000000000000000000051, ..., 1000000000000000000000000057, ..., 56111111111111111111111111111111111111, ..., 7666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666662, ..., 27777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777707, ..., 300000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000011, ...} 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, 555555555551, 5000000000000000000000000000027} 11: {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, 10A, 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, 1101, 11A9, 1305, 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, 5105, 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, 9101, 9107, 9305, 9505, 9703, A052, A119, A151, A175, A515, A517, A575, A577, A5A8, A719, A779, A911, AAA9, 10011, 10075, 10091, 10109, 10411, 10444, 10705, 10709, 10774, 10901, 11104, 11131, 11144, 11191, 1141A, 114A1, 13757, 1411A, 14477, 144A4, 14A04, 14A11, 17045, 17704, 1774A, 17777, 177A4, 17A47, 1A091, 1A109, 1A114, 1A404, 1A411, 1A709, 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, 70103, 70471, 70583, 70714, 71474, 717A4, 71A09, 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, A1404, A4177, A4401, A4717, A5228, A52AA, A5558, A580A, A5822, A58AA, A5A59, A5AA2, A6096, A6966, A6999, A7051, A7778, A7808, A9055, A9091, A9699, A9969, AA52A, AA58A, 100019, 100079, 101113, 101119, 101911, 107003, 140004, 144011, 144404, 1A0019, 1A0141, 1A5001, 1A7005, 1A9001, 222223, 222823, 300107, 300202, 300323, 303203, 307577, 310007, 332003, 370777, 400555, 401A11, 404001, 404111, 405AAA, 41A011, 440A41, 441011, 451777, 455555, 470051, 470444, 474404, 4A0401, 4A4041, 500015, 500053, 500077, 500507, 505577, 522A2A, 525223, 528A2A, 531707, 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, 701074, 701777, 704408, 704417, 704457, 704484, 707041, 707441, 707708, 707744, 707784, 710777, 717044, 717077, 740008, 74484A, 770441, 770744, 770748, 770771, 777017, 777071, 777448, 777484, 777701, 7778A8, 777A19, 777A48, 778883, 78A808, 790003, 7A1009, 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, A01901, A05509, A0A058, A0A955, A10114, A555A2, A55999, A59991, A5A222, A5A22A, A60609, A66069, A66906, A69006, A79005, A87888, A90099, A90996, A96006, A96666, A97177, A97771, AA0A58, AA5A22, AAA522, 1000501, 1011141, 1030007, 1070047, 111114A, 1111A14, 1111A41, 1144441, 14A4444, 1700005, 1700474, 1A44444, 2555505, 2845055, 3030023, 3100003, 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, 7100447, 7174404, 717444A, 7400404, 7700717, 7701077, 7701707, 7707778, 7774004, 7777104, 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, A001091, A006906, A010044, A040041, A0AAA58, A141111, A5222A2, A600A69, A906606, A909009, A990009, A997701, AA55A52, AAA5552, AAAAA52, 10004747, 10005007, 17000744, 22888823, 28888223, 30010111, 30555777, 31011111, 33000023, 40A00041, 45000055, 47040004, 50377777, 50555553, 5282AAA2, 55505003, 555A5A52, 555AAA2A, 55A5A552, 5AAAAA2A, 60A99999, 70000057, 70070474, 70074704, 70174004, 70700078, 70700474, 70704704, 70710707, 70771007, 70777177, 71074004, 74470001, 77000177, 77070477, 77100077, 77470004, 77700404, 77710007, 77717707, 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, A1900001, A5555009, A5A55552, A9700001, A9909006, A9990006, A9990606, A9999917, A9999966, 100000507, 100035077, 100050777, 100057707, 101111114, 15A000001, 170000447, 300577777, 40000A401, 447771777, 44A444441, 474000004, 477700004, 477777774, 505000003, 55555AA2A, 5555A5A2A, 700000147, 700017004, 700044004, 700077774, 700170004, 701000047, 701700004, 704000044, 704040004, 707070774, 707077704, 707770704, 707777004, 717000004, 717700007, 770000078, 770004704, 770070747, 770070774, 770700008, 770700084, 770707074, 777000044, 777000774, 777717007, 777770477, 777770747, 7777777A4, 77A700008, 888888302, 900000091, 900090799, 970009099, 990990007, 997000077, 999999997, A0000AA58, A00990001, A05555559, A44444111, A44444777, A44477777, A66666669, A90000606, A99999006, A99999099, 1000007447, 1005000007, 1500000001, ..., 3700000001, ..., 4000000005, ..., 600000A999, ..., A000144444, ..., A900000066, ..., 33333333337, ..., 44444444447, ..., A0000000001, ..., A0014444444, ..., 100000000057, ..., 40000000A0041, ..., A000000014444, ..., A044444444441, ..., A144444444411, ..., 40000000000401, ..., A0000044444441, ..., A00000000444441, ..., 11111111111111111, ..., 14444444444441111, ..., 44444444444444111, ..., 70000000000000004, ..., A1444444444444444, ..., A9999999999999996, ..., 888888888888888883, ..., 1444444444444444444, ..., 7777777777777777771, ..., 4000000000000000A041, ..., 45AAAAAAAAAAAAAAAAAAAA, ..., 9777777777777777777707, ..., A999999999999999999999, ..., 10000000000000000000747, ..., 3577777777777777777777777, ..., 10000000000000000000000044, ..., 77700000000000000000000008, ..., A44444444444444444444444441, ..., 1500000000000000000000000007, ..., 40000000000000000000000000041, ..., 440000000000000000000000000001, ..., 999999999999999999999999999999991, ..., 1900000000000000000000000000000000001, ..., 7777777777777777777777777777777777704, ..., A477777777777777777777777777777777777777777, ..., 444444444444444444444444444444444444444444441, ..., 8055555555555555555555555555555555555555555555555555555555555, ..., 44777777777777777777777777777777777777777777777777777777777777777, ..., 99777777777777777777777777777777777777777777777777777777777777777, ..., 10000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000051, ..., 555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555552A, ..., 5077777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777, ..., 77777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777744, ..., 55777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777, ...} 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, BBBBBB99B, B0000000000000000000000000009B, 400000000000000000000000000000000000000077} [/CODE] However, I cannot prove that my base 7, 9, 11 sets are complete (I use PARI program to find these primes (all written in base b), but I only searched the primes up to certain limits (about 2^32) and found some unsolved families (e.g. 1{0}13 in base 5, {3}1 in base 7, {4}7 in base 8, {5}25 in base 8, {7}1 in base 8, 5{0}27 in base 10, {5}1 in base 10, 4{0}77 in base 12, B{0}9B in base 12) and found the smallest prime in these families, so there may be missing primes), I proved that my base 2, 3, 4, 5, 6, 8, 10, 12 sets are complete. Can someone complete my base 7, 9, 11 set? Also find the sets of bases 13 to 36. |
Your sets for 2 and 3 are complete, but the others are not.
For example, all Fermat numbers that may be prime are of the form "1000.....1" in base 4, and additionally, you will have and endless number of combinations of patterns of 0 and 1 that may be prime, and yet not appear in your list, so your base 4 may look like: [CODE]List(["2", "3", "11", "101", "10001", "100000001", "100001001001001", "100100100001001"])[/CODE] 5 seems to be "endless", too... [code] List(["2", "3", "10", "111", "401", "414", "1404", "4041", "14004", "14404", "14411", "14444", "40041", "40441", "41141", "44001 ", "44441", "144404", "404001", "404441", "1440044", "1444114", "4000001", "4000144", "4001411", "4114001", "11440001", "1400000 4", "14000114", "14000404", "14000411", "40400001", "40400014", "40400041", "44000441", "44400014", "44440001", "140000404", "14 0000411", "140000444", "140011411", "140014001", "141144004", "144000044", "144000114", "144000444", "400040014", "400114001", " 400140004", "400144004", "404000041", "404004001", "404040001", "404400001", "404400041", "440040001", "1400040004", "1400044114 ", "1411400011", "1440000404", "1440004114", "1444000114", "1444004404", "4000000041", "4000400041", "4000400441", "4004400041", "4040000041", "4040440001", "4044000041", "4044004001", "4400000001", "4400011441", "4404000441", "4444400041", "11400000001", "11400000041", "11400114441", "11440000144", "11444000014", "14000011444", "14000014441", "14000040011", "14000040044", "1400004 4004", "14000044114", "14000140001", "14001440001", "14114000011", "14114000444", "14400000004", "14400000011", "14400001441", " 14400004114", "14400040044", "14400400141", "14400400444", "14400404044", "14440000004", "14440004044", "14440040004", "14440040 444", "40000000041", "40000001411", "40000004441", "40000040014", "40000141144", "40000400441", "40001400004", "40004001141", "4 0004004441", "40011400001", "40014000044", "40014001144", "40014440044", "40040000001", "40040040001", "40044004001", "400440044 41", "40044400001", "40044411441", "40400004441", "40440400441", "40444000041", "40444004441", "41144400001", "44000400014", "44 004440001", "44040004441", "44044040001", "44114000441", "44114400001", "44400001444", "44400004001", "44400004441", "4444000000 1", "44440000144", "44444000041"]) [/code]etc... |
[QUOTE=LaurV;531466]Your sets for 2 and 3 are complete, but the others are not.
For example, all Fermat numbers that may be prime are of the form "1000.....1" in base 4, and additionally, you will have and endless number of combinations of patterns of 0 and 1 that may be prime, and yet not appear in your list, so your base 4 may look like: [CODE]List(["2", "3", "11", "101", "10001", "100000001", "100001001001001", "100100100001001"])[/CODE][/QUOTE] "11" is already on the list. That excludes all your additions (they contain two 1s). [QUOTE] 5 seems to be "endless", too... [code] List(["2", "3", "10", "111", "401", "414", "1404", "4041"]) [/code]etc...[/QUOTE]Invalid, as 401 is comparable to 4041 (40(4)1) and so on. |
Ok, scratch the former post, I am stupid.
It looked odd, and I could not believe those paper were right, then I went and read them (skimmed) and realized the sequence does not need to be consecutive. In fact, you stated as much as that, and your example with 251664 shows very clearly what you mean. Sorry for being stupid :redface: (in fact, my excuse is that you post so much rubbish here that I don't read your post more than fast skimming mode :razz:) Give me 5 minutes... (look how I am spending my lunch break!) (edit @uau: crosspost, I was talking to sweety) |
Ok, here is the code, and I searched much higher, there are no new numbers.
[CODE] \\finds text substrings in text strings, or small-sub-vecteors in large small-vectors :P \\(kind of efficient, but it still could be optimized...) find(subtxt,txt,startfrom=1)= { my(i,j,s=Vecsmall(subtxt), t=Vecsmall(txt)); i=startfrom-1; while(i<=#t-#s, j=1; while(j<=#s, if(s[j]!=t[i+j], break ); j++ ); if(j>#s, return(i+1) ); i++ ); return(0); } \\similar, but the strings do not need to be consecutive findweak(subtxt,txt,startfrom=1)= { my(i,j,s=Vecsmall(subtxt), t=Vecsmall(txt)); i=startfrom; j=1; while(i<=#t && j<=#s, if(s[j]==t[i], j++ ); i++ ); if(j>#s, return(1), return(0) ); } \\base from 2 to 36, no validation! bprint(n,base=10)= { if(base==10, return(Str(n)) ); my(v=digits(n,base)); for(i=1,#v, if(v[i]<10, v[i]+=48, v[i]+=55 ) ); return(Strchr(v)); } minprime(startfrom=2, base=10)= { my(n,k,x,cnt); lstminprimes=List([]); n=nextprime(startfrom); listput(lstminprimes,bprint(n,base)); print(lstminprimes); while(1, \\stop it with ctrl+c when fed-up with it, etc n=nextprime(n+1); x=bprint(n,base); k=1; while(k<=#lstminprimes, \\ if(find(lstminprimes[k],x), if(findweak(lstminprimes[k],x), break ); k++ ); if(k>#lstminprimes, listput(lstminprimes,x); print(lstminprimes) ); cnt++; if(cnt%10000==0, printf("...%d...%c",n,13) ) ); } [/CODE]Interesting that for 7 (when the restriction of n>=base is not required) it seems "easy" to prove, that the set is maximal, because combinations of only 0, 4, and 6 can not be prime (in any base, not only base 7). I still don't get it why they say it is hard to prove... Maybe there is a case I am missing... |
I let that script running in background for 8, during I was doing some real job-related work, and it still produced 77774444441.
Meantime I realized that it is terrible inefficient, and it can be done much cleverer. So. factory stopped for now, I may revisit it in the future (probably weekend). I also have an idea how 7 and 8 (starting from 2 digits) can be "proved" formally. I go home now, 18:20 here. |
I found an easy way to generate those sets, and to prove that they are complete.
For the "starting from two digits" version, neither one of the exposed sets for 7 and 8 are complete. Some larger primes are still lurking in the dark there. I have the complete sets for both 8, and 7 for the both cases when the base itself is included in the set or not*, but I don't want to spoil the puzzle, this is an interesting little problem... hehe... Hint: [CODE] gp > a=(7^17-5)/2 %1 = 116315256993601 gp > isprime(a) %2 = 1 gp > digits(a,7) %3 = [3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 1] gp > [/CODE]--------- *when the base is prime, like for 5 and 7, the sets are different; including the base results in automatic elimination of all possible extension numbers with "0 after 1" from the set, which is quite restrictive, so I also calculated the lists for the "base is not included" version, i.e. base-5 starting from 6, and base-7 starting from 8; in this case, for example, base-5 will include numbers like 104 and 10103 which are prime, and base-7 list will include 1022, 1051, 1202, .... 1100021 ... etc, they are "enriched" compared with the case when the first "10" is included. So I have the complete list for 8, and the complete two lists for 7, the normal one, and the "enriched" one. Base-5 is easy, in any case. |
[QUOTE=LaurV;531632]I found an easy way to generate those sets, and to prove that they are complete.
For the "starting from two digits" version, neither one of the exposed sets for 7 and 8 are complete. Some larger primes are still lurking in the dark there. I have the complete sets for both 8, and 7 for the both cases when the base itself is included in the set or not*, but I don't want to spoil the puzzle, this is an interesting little problem... hehe... Hint: [CODE] gp > a=(7^17-5)/2 %1 = 116315256993601 gp > isprime(a) %2 = 1 gp > digits(a,7) %3 = [3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 1] gp > [/CODE]--------- *when the base is prime, like for 5 and 7, the sets are different; including the base results in automatic elimination of all possible extension numbers with "0 after 1" from the set, which is quite restrictive, so I also calculated the lists for the "base is not included" version, i.e. base-5 starting from 6, and base-7 starting from 8; in this case, for example, base-5 will include numbers like 104 and 10103 which are prime, and base-7 list will include 1022, 1051, 1202, .... 1100021 ... etc, they are "enriched" compared with the case when the first "10" is included. So I have the complete list for 8, and the complete two lists for 7, the normal one, and the "enriched" one. Base-5 is easy, in any case.[/QUOTE] So, what is the complete list of 7? Is it just my list {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} plus the prime 33333333333333331? And what is the complete list of 8? You said that you have it. |
[QUOTE=LaurV;531470]Ok, here is the code, and I searched much higher, there are no new numbers.
[CODE] \\finds text substrings in text strings, or small-sub-vecteors in large small-vectors :P \\(kind of efficient, but it still could be optimized...) find(subtxt,txt,startfrom=1)= { my(i,j,s=Vecsmall(subtxt), t=Vecsmall(txt)); i=startfrom-1; while(i<=#t-#s, j=1; while(j<=#s, if(s[j]!=t[i+j], break ); j++ ); if(j>#s, return(i+1) ); i++ ); return(0); } \\similar, but the strings do not need to be consecutive findweak(subtxt,txt,startfrom=1)= { my(i,j,s=Vecsmall(subtxt), t=Vecsmall(txt)); i=startfrom; j=1; while(i<=#t && j<=#s, if(s[j]==t[i], j++ ); i++ ); if(j>#s, return(1), return(0) ); } \\base from 2 to 36, no validation! bprint(n,base=10)= { if(base==10, return(Str(n)) ); my(v=digits(n,base)); for(i=1,#v, if(v[i]<10, v[i]+=48, v[i]+=55 ) ); return(Strchr(v)); } minprime(startfrom=2, base=10)= { my(n,k,x,cnt); lstminprimes=List([]); n=nextprime(startfrom); listput(lstminprimes,bprint(n,base)); print(lstminprimes); while(1, \\stop it with ctrl+c when fed-up with it, etc n=nextprime(n+1); x=bprint(n,base); k=1; while(k<=#lstminprimes, \\ if(find(lstminprimes[k],x), if(findweak(lstminprimes[k],x), break ); k++ ); if(k>#lstminprimes, listput(lstminprimes,x); print(lstminprimes) ); cnt++; if(cnt%10000==0, printf("...%d...%c",n,13) ) ); } [/CODE]Interesting that for 7 (when the restriction of n>=base is not required) it seems "easy" to prove, that the set is maximal, because combinations of only 0, 4, and 6 can not be prime (in any base, not only base 7). I still don't get it why they say it is hard to prove... Maybe there is a case I am missing...[/QUOTE] Well, I already have the code: [CODE] a(n,k,b)=v=[];for(r=1,length(digits(n,b)),if(r+length(digits(k,2))-length(digits(n,b))>0 && digits(k,2)[r+length(digits(k,2))-length(digits(n,b))]==1,v=concat(v,digits(n,b)[r])));fromdigits(v,b) g(n)=if(n<10,n+48,if(n<36,n+55,if(n<62,n+61,if(n<77,n-29,if(n<84,n-19,if(n<90,n+7,if(n<94,n+33,n+67))))))) f(n,b)=for(k=1,length(digits(n,b)),print1(Strchr(g(digits(n,b)[k])))) is(n,b)=for(k=1,2^length(digits(n,b))-2,if(ispseudoprime(a(n,k,b)) && a(n,k,b)>=b+1,return(0)));1 c(b)=forprime(p=b+1,2^32,if(is(p,b),f(p,b);print1(", "))) [/CODE] and for the original problem in [URL="https://primes.utm.edu/glossary/page.php?sort=MinimalPrime"]https://primes.utm.edu/glossary/page.php?sort=MinimalPrime[/URL] (single-digit prime is also included), this is the code: [CODE] a(n,k,b)=v=[];for(r=1,length(digits(n,b)),if(r+length(digits(k,2))-length(digits(n,b))>0 && digits(k,2)[r+length(digits(k,2))-length(digits(n,b))]==1,v=concat(v,digits(n,b)[r])));fromdigits(v,b) g(n)=if(n<10,n+48,if(n<36,n+55,if(n<62,n+61,if(n<77,n-29,if(n<84,n-19,if(n<90,n+7,if(n<94,n+33,n+67))))))) f(n,b)=for(k=1,length(digits(n,b)),print1(Strchr(g(digits(n,b)[k])))) is(n,b)=for(k=1,2^length(digits(n,b))-2,if(ispseudoprime(a(n,k,b)),return(0)));1 c(b)=forprime(p=2,2^32,if(is(p,b),f(p,b);print1(", "))) [/CODE] Enter c(n) to print all minimal prime in base n (written in base n) |
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. |
This is the C code for this problem: (need run with [URL="https://gmplib.org/"]GMP[/URL])
This code computes minimal primes for bases between l and h, possibly along with a set of unsolved families. This code can be used for bases <= 64 (0-9 for digit values 0-9, A-Z for digit values 10-35, a-z for digit values 36-61, # for digit value 62, $ for digit value 63) Copy this C code and name "minimal.c" Usage: minimal l h To resume base b from iter i: minimal resume b i Note: This code will not remove the families with all or partial algebraic factorization of x^4+4*y^4 (as I do not know how to add it), thus this code will print base 16 unsolved families "C*D" and "C*DD" (both with full algebraic factorization of x^4+4*y^4) and base 55 unsolved family "jP0*1" (with partial algebraic factorization of x^4+4*y^4), although these three families can be ruled out as only contain composites (only count numbers > base). [CODE] #include <stdio.h> #include <stdlib.h> #include <gmp.h> #include <string.h> #define MAXSTRING 512 #ifdef PRINTALL #define PRINTDATA #define PRINTITER #define PRINTSUMMARY #define PRINTDIVISOR #define PRINTDIVISORTWO #define PRINTDIVISORTHREE #define PRINTDIVISORFOUR #define PRINTDIVISORFIVE #define PRINTDIVISORSQUARE #define PRINTDIVISORCUBE #define PRINTSTATS #define PRINTUNSOLVED #define PRINTSPLIT #define PRINTSPLITDOUBLE #define PRINTSPLITTRIPLE #define PRINTSPLITQUAD #define PRINTSPLITQUINT #define PRINTPRIMES #define PRINTSUBWORD #define PRINTEXPLORE #define PRINTTRIVIAL #define PRINTRESUME #define PRINTDIVISOREXT #define PRINTSPLITEXT #endif #if defined(PRINTDATA) || defined(PRINTITER) #include <sys/stat.h> #endif typedef struct { int len; int* numrepeats; char* digit; char** repeats; } family; typedef struct { int size; char** primes; } kernel; typedef struct { int size; family* fam; char* split; } list; void familystring(char* str, family p); void clearfamily(family* f); void copyfamily(family* newf, family f); void adddigit(family* f, char d, char* r, int n); void familyinit(family* p); void addtolist(list* l, family f, char split); void simplefamilystring(char* str, family p); int issimple(family f); int base; int depth; int iter; kernel K; int prsize; char* pr; void listinit(list* l) { l->size = 0; l->fam = NULL; l->split = NULL; } void copylist(list* out, list in) { out->size = in.size; out->fam = malloc(in.size*sizeof(family)); out->split = malloc(in.size*sizeof(char)); for(int i=0; i<in.size; i++) { familyinit(&(out->fam[i])); copyfamily(&(out->fam[i]), in.fam[i]); out->split[i] = in.split[i]; } } void clearlist(list* l) { for(int i=0; i<l->size; i++) clearfamily(&(l->fam[i])); free(l->fam); free(l->split); listinit(l); } void removedupes(list* unsolved) { if(unsolved->size==0) return; list newlist; listinit(&newlist); int n = 1; char** strlist = malloc(n*sizeof(char*)); int* listpos = malloc(n*sizeof(int)); char* str = malloc(MAXSTRING*sizeof(char)); if(issimple(unsolved->fam[0])) simplefamilystring(str, unsolved->fam[0]); else familystring(str, unsolved->fam[0]); strlist[0] = str; listpos[0] = 0; addtolist(&newlist, unsolved->fam[0], unsolved->split[0]); for(int i=1; i<unsolved->size; i++) { str = malloc(MAXSTRING*sizeof(char)); if(issimple(unsolved->fam[i])) simplefamilystring(str, unsolved->fam[i]); else familystring(str, unsolved->fam[i]); int addedtolist = 0; char* temp; char* last; int inttemp; int intlast; for(int j=0; j<n; j++) { if(addedtolist) { temp = strlist[j]; strlist[j] = last; last = temp; inttemp = listpos[j]; listpos[j] = intlast; intlast = inttemp; } else if(strcmp(str,strlist[j])>0) { addedtolist = 1; last = strlist[j]; strlist[j] = str; intlast = listpos[j]; listpos[j] = n; addtolist(&newlist, unsolved->fam[i], unsolved->split[i]); } else if(strcmp(str,strlist[j])==0) { if(issimple(unsolved->fam[i])) { char str1[MAXSTRING], str2[MAXSTRING]; familystring(str1, unsolved->fam[i]); familystring(str2, newlist.fam[listpos[j]]); if(strlen(str1)<strlen(str2)) { clearfamily(&(newlist.fam[listpos[j]])); copyfamily(&(newlist.fam[listpos[j]]), unsolved->fam[i]); } } break; } else if(j==n-1) { addedtolist = 1; last = str; intlast = n; addtolist(&newlist, unsolved->fam[i], unsolved->split[i]); } } if(addedtolist) { n++; strlist = realloc(strlist, n*sizeof(char*)); strlist[n-1] = last; listpos = realloc(listpos, n*sizeof(int)); listpos[n-1] = intlast; } else free(str); } clearlist(unsolved); copylist(unsolved, newlist); clearlist(&newlist); for(int i=0; i<n; i++) free(strlist[i]); free(strlist); free(listpos); } int issimple(family f) { int hasrepeat = 0; for(int i=0; i<f.len; i++) { if(f.numrepeats[i]>1) return 0; if(f.numrepeats[i]==1) { if(hasrepeat) return 0; hasrepeat = 1; } } return (hasrepeat==1); } int onlysimple(list l) { for(int i=0; i<l.size; i++) { if(!issimple(l.fam[i])) return 0; } return 1; } void printlist(list l) { for(int i=0; i<l.size; i++) { char str[MAXSTRING]; familystring(str, l.fam[i]); printf("%s\n", str); } } void simpleprintlist(list l) { for(int i=0; i<l.size; i++) { char str[MAXSTRING]; simplefamilystring(str, l.fam[i]); printf("%s\n", str); } } void addtolist(list* l, family f, char split) { int size = ++l->size; l->fam = realloc(l->fam, size*sizeof(family)); familyinit(&((l->fam)[size-1])); copyfamily(&((l->fam)[size-1]), f); l->split = realloc(l->split, size*sizeof(char)); l->split[size-1] = split; } void kernelinit() { K.size = 0; K.primes = NULL; } void addtokernel(char* p) { int size = ++K.size; K.primes = realloc(K.primes, size*sizeof(char*)); K.primes[size-1] = p; } void clearkernel() { for(int i=0; i<K.size; i++) free(K.primes[i]); free(K.primes); kernelinit(); } int nosubword(char* p) { for(int i=0; i<K.size; i++) { int k = 0; for(int j=0; j<strlen(p); j++) { if(K.primes[i][k]==p[j]) k++; if(k==strlen(K.primes[i])) return 0; } } return 1; } int nosubwordskip(char* p, int skip) { for(int i=0; i<K.size; i++) { if(i==skip) continue; int k = 0; for(int j=0; j<strlen(p); j++) { if(K.primes[i][k]==p[j]) k++; if(k==strlen(K.primes[i])) return 0; } } return 1; } int isprime(char* p) { mpz_t temp; mpz_init(temp); mpz_set_str(temp, p, base); if(mpz_probab_prime_p(temp, 25) > 0 && temp > base) { //gmp_printf("%Zd is prime\n", temp); mpz_clear(temp); return 1; } else { //gmp_printf("%Zd is not prime\n", temp); mpz_clear(temp); return 0; } } int newminimal(char* p) { if(!nosubword(p)) return 0; return isprime(p); } void familyinit(family* p) { p->len = 0; p->numrepeats = NULL; p->digit = NULL; p->repeats = NULL; } void adddigit(family* f, char d, char* r, int n) { int len = ++f->len; f->digit = realloc(f->digit, len*sizeof(char)); f->digit[len-1] = d; f->numrepeats = realloc(f->numrepeats, len*sizeof(int)); f->numrepeats[len-1] = n; f->repeats = realloc(f->repeats, len*sizeof(char*)); f->repeats[len-1] = r; } void clearfamily(family* f) { free(f->digit); for(int i=0; i<f->len; i++) free(f->repeats[i]); free(f->numrepeats); free(f->repeats); familyinit(f); } char digitchar(unsigned char digit) { if(digit==255) return 0; else if(digit==63) return '$'; else if(digit==62) return '#'; else if(digit>=36) return digit+'a'-36; else if(digit>=10) return digit+'A'-10; else return digit+'0'; } char invdigitchar(char input) { if(input>='0' && input<='9') return input-'0'; else if(input>='A' && input<='Z') return input-'A'+10; else if(input>='a' && input<='z') return input-'a'+36; else if(input=='#') return 62; else if(input=='$') return 63; } void familystring(char* str, family p) { sprintf(str, "%c", 0); for(int i=0; i<p.len; i++) { sprintf(str, "%s%c", str, digitchar(p.digit[i])); if(p.numrepeats[i]>0) { sprintf(str, "%s{", str); for(int j=0; j<p.numrepeats[i]; j++) sprintf(str, "%s%c", str, digitchar(p.repeats[i][j])); sprintf(str, "%s}*", str); } } } void simplefamilystring(char* str, family p) { sprintf(str, "%c", 0); char repeateddigit; int repeatedpos; for(int i=0; i<p.len; i++) if(p.numrepeats[i]==1) { repeateddigit = p.repeats[i][0]; repeatedpos = i; break; } int j=-1; for(int i=repeatedpos; i>=0; i--) if(p.digit[i]!=repeateddigit && (unsigned char)p.digit[i]!=255) { j = i; break; } int k=p.len; for(int i=repeatedpos+1; i<p.len; i++) if(p.digit[i]!=repeateddigit && (unsigned char)p.digit[i]!=255) { k = i; break; } for(int i=0; i<=j; i++) sprintf(str, "%s%c", str, digitchar(p.digit[i])); sprintf(str, "%s%c*", str, digitchar(repeateddigit)); for(int i=k; i<p.len; i++) sprintf(str, "%s%c", str, digitchar(p.digit[i])); } void startinstancestring(char* str, family p, int length) { sprintf(str, "%c", 0); for(int i=0; i<=length; i++) sprintf(str, "%s%c", str, digitchar(p.digit[i])); } void endinstancestring(char* str, family p, int start) { sprintf(str, "%c", 0); for(int i=start+1; i<p.len; i++) sprintf(str, "%s%c", str, digitchar(p.digit[i])); } void emptyinstancestring(char* str, family p) { sprintf(str, "%c", 0); for(int i=0; i<p.len; i++) sprintf(str, "%s%c", str, digitchar(p.digit[i])); } void instancestring(char* str, family p, int x, int y) { sprintf(str, "%c", 0); for(int i=0; i<p.len; i++) { sprintf(str, "%s%c", str, digitchar(p.digit[i])); if(i==x) sprintf(str, "%s%c", str, digitchar(p.repeats[x][y])); } } void doubleinstancestring(char* str, family p, int x1, int y1, int x2, int y2) { sprintf(str, "%c", 0); for(int i=0; i<p.len; i++) { sprintf(str, "%s%c", str, digitchar(p.digit[i])); if(i==x1) sprintf(str, "%s%c", str, digitchar(p.repeats[x1][y1])); if(i==x2) sprintf(str, "%s%c", str, digitchar(p.repeats[x2][y2])); } } void tripleinstancestring(char* str, family p, int x1, int y1, int x2, int y2, int x3, int y3) { sprintf(str, "%c", 0); for(int i=0; i<p.len; i++) { sprintf(str, "%s%c", str, digitchar(p.digit[i])); if(i==x1) sprintf(str, "%s%c", str, digitchar(p.repeats[x1][y1])); if(i==x2) sprintf(str, "%s%c", str, digitchar(p.repeats[x2][y2])); if(i==x3) sprintf(str, "%s%c", str, digitchar(p.repeats[x3][y3])); } } void quadinstancestring(char* str, family p, int x1, int y1, int x2, int y2, int x3, int y3, int x4, int y4) { sprintf(str, "%c", 0); for(int i=0; i<p.len; i++) { sprintf(str, "%s%c", str, digitchar(p.digit[i])); if(i==x1) sprintf(str, "%s%c", str, digitchar(p.repeats[x1][y1])); if(i==x2) sprintf(str, "%s%c", str, digitchar(p.repeats[x2][y2])); if(i==x3) sprintf(str, "%s%c", str, digitchar(p.repeats[x3][y3])); if(i==x4) sprintf(str, "%s%c", str, digitchar(p.repeats[x4][y4])); } } void quintinstancestring(char* str, family p, int x1, int y1) { sprintf(str, "%c", 0); for(int i=0; i<p.len; i++) { sprintf(str, "%s%c", str, digitchar(p.digit[i])); if(i==x1) { sprintf(str, "%s%c%c%c%c%c%c%c%c%c%c", str, digitchar(p.repeats[x1][y1]), digitchar(p.repeats[x1][y1]), digitchar(p.repeats[x1][y1]), digitchar(p.repeats[x1][y1]), digitchar(p.repeats[x1][y1]), digitchar(p.repeats[x1][y1]), digitchar(p.repeats[x1][y1]), digitchar(p.repeats[x1][y1]), digitchar(p.repeats[x1][y1]), digitchar(p.repeats[x1][y1])); } } } void copyfamily(family* newf, family f) { for(int i=0; i<f.len; i++) { char* repeatscopy = malloc(f.numrepeats[i]*sizeof(char)); memcpy(repeatscopy, f.repeats[i], f.numrepeats[i]*sizeof(char)); adddigit(newf, f.digit[i], repeatscopy, f.numrepeats[i]); } } int hasdivisor(family p) { mpz_t gcd, temp, gcd1, gcd2, x, y, z, temp2, temp3, temp4, temp5, temp6, temp7, temp8, temp9, temp10, empty; mpz_inits(gcd, temp, gcd1, gcd2, x, y, z, temp2, temp3, temp4, temp5, temp6, temp7, temp8, temp9, temp10, empty, NULL); char str[MAXSTRING]; int numrepeats = 0; emptyinstancestring(str, p); mpz_set_str(empty, str, base); mpz_set_str(gcd, str, base); int allsingledigit = 1; for(int i=0; i<p.len; i++) { for(int j=0; j<p.numrepeats[i]; j++) { instancestring(str, p, i, j); mpz_set_str(temp, str, base); mpz_gcd(gcd, gcd, temp); if(temp > base) allsingledigit = 0; } if(p.numrepeats[i]>0) numrepeats++; } if(allsingledigit) return 1; if(numrepeats==0) { #ifdef PRINTDIVISOR familystring(str, p); printf("%s is trivial\n", str); #endif mpz_clears(gcd, temp, gcd1, gcd2, x, y, z, temp2, temp3, temp4, temp5, temp6, temp7, temp8, temp9, temp10, empty, NULL); return 0; } if(mpz_cmp_ui(gcd, 1)>0 && mpz_cmp(empty, gcd)>0) { #ifdef PRINTDIVISOR familystring(str, p); gmp_printf("%s has a divisor %Zd\n", str, gcd); #endif mpz_clears(gcd, temp, gcd1, gcd2, x, y, z, temp2, temp3, temp4, temp5, temp6, temp7, temp8, temp9, temp10, empty, NULL); return 1; } for(int m=0; m<p.len; m++) { if(p.numrepeats[m]==0) continue; emptyinstancestring(str, p); mpz_set_str(gcd1, str, base); for(int i=0; i<p.len; i++) { if(i==m) continue; for(int j=0; j<p.numrepeats[i]; j++) { instancestring(str, p, i, j); mpz_set_str(temp, str, base); mpz_gcd(gcd1, gcd1, temp); } } for(int i=0; i<p.numrepeats[m]; i++) for(int j=0; j<p.numrepeats[m]; j++) { doubleinstancestring(str, p, m, i, m, j); mpz_set_str(temp, str, base); mpz_gcd(gcd1, gcd1, temp); } int gcdbeenset = 0; for(int n=0; n<p.numrepeats[m]; n++) { instancestring(str, p, m, n); mpz_set_str(temp, str, base); if(gcdbeenset) mpz_gcd(gcd2, gcd2, temp); else { mpz_set(gcd2, temp); gcdbeenset = 1; } for(int i=0; i<p.len; i++) { if(i==m) continue; for(int j=0; j<p.numrepeats[i]; j++) { doubleinstancestring(str, p, i, j, m, n); mpz_set_str(temp, str, base); mpz_gcd(gcd2, gcd2, temp); } } for(int i=0; i<p.numrepeats[m]; i++) for(int j=0; j<p.numrepeats[m]; j++) for(int k=0; k<p.numrepeats[m]; k++) { tripleinstancestring(str, p, m, i, m, j, m, k); mpz_set_str(temp, str, base); mpz_gcd(gcd2, gcd2, temp); } } if(mpz_cmp_ui(gcd1, 1)>0 && mpz_cmp_ui(gcd2, 1)>0 && mpz_cmp(empty, gcd1)>0 && mpz_cmp(empty, gcd2)>0) { #ifdef PRINTDIVISORTWO familystring(str, p); gmp_printf("%s has two divisors %Zd and %Zd\n", str, gcd1, gcd2); #endif mpz_clears(gcd, temp, gcd1, gcd2, x, y, z, temp2, temp3, temp4, temp5, temp6, temp7, temp8, temp9, temp10, empty, NULL); return 1; } mpz_set_ui(gcd1, 0); mpz_set_ui(gcd2, 0); } int i; int gcdbeenset = 0; for(i=0; i<p.len; i++) { for(int j=0; j<p.numrepeats[i]; j++) { instancestring(str, p, i, j); mpz_set_str(temp, str, base); if(gcdbeenset) mpz_gcd(gcd2, gcd2, temp); else { gcdbeenset = 1; mpz_set(gcd2, temp); } for(int k=0; k<p.len; k++) for(int l=0; l<p.numrepeats[k]; l++) for(int n=0; n<p.numrepeats[k]; n++) { tripleinstancestring(str, p, i, j, k, l, k, n); mpz_set_str(temp, str, base); mpz_gcd(gcd2, gcd2, temp); } } if(p.numrepeats[i]>0) break; } int firstrepeat = i; if(numrepeats==2) { emptyinstancestring(str, p); mpz_set_str(gcd1, str, base); for(int i=0; i<p.len; i++) for(int j=0; j<p.numrepeats[i]; j++) for(int l=0; l<p.numrepeats[i]; l++) { doubleinstancestring(str, p, i, j, i, l); mpz_set_str(temp, str, base); mpz_gcd(gcd1, gcd1, temp); } gcdbeenset = 0; for(i=firstrepeat+1; i<p.len; i++) { for(int j=0; j<p.numrepeats[i]; j++) { instancestring(str, p, i, j); mpz_set_str(temp, str, base); if(gcdbeenset) mpz_gcd(temp2, temp2, temp); else { gcdbeenset = 1; mpz_set(temp2, temp); } for(int k=0; k<p.len; k++) for(int l=0; l<p.numrepeats[k]; l++) for(int n=0; n<p.numrepeats[k]; n++) { tripleinstancestring(str, p, i, j, k, l, k, n); mpz_set_str(temp, str, base); mpz_gcd(temp2, temp2, temp); } } if(p.numrepeats[i]>0) break; } int secondrepeat = i; gcdbeenset = 0; for(int j=0; j<p.numrepeats[firstrepeat]; j++) for(int l=0; l<p.numrepeats[secondrepeat]; l++) { doubleinstancestring(str, p, firstrepeat, j, secondrepeat, l); mpz_set_str(temp, str, base); if(gcdbeenset) mpz_gcd(temp3, temp3, temp); else { gcdbeenset = 1; mpz_set(temp3, temp); } } for(i=0; i<p.numrepeats[firstrepeat]; i++) for(int j=0; j<p.numrepeats[firstrepeat]; j++) for(int k=0; k<p.numrepeats[firstrepeat]; k++) for(int l=0; l<p.numrepeats[secondrepeat]; l++) { quadinstancestring(str, p, firstrepeat, i, firstrepeat, j, firstrepeat, k, secondrepeat, l); mpz_set_str(temp, str, base); mpz_gcd(temp3, temp3, temp); } for(i=0; i<p.numrepeats[firstrepeat]; i++) for(int j=0; j<p.numrepeats[secondrepeat]; j++) for(int k=0; k<p.numrepeats[secondrepeat]; k++) for(int l=0; l<p.numrepeats[secondrepeat]; l++) { quadinstancestring(str, p, firstrepeat, i, secondrepeat, j, secondrepeat, k, secondrepeat, l); mpz_set_str(temp, str, base); mpz_gcd(temp3, temp3, temp); } if(mpz_cmp_ui(gcd1, 1)>0 && mpz_cmp_ui(gcd2, 1)>0 && mpz_cmp(empty, gcd1)>0 && mpz_cmp(empty, gcd2)>0 && mpz_cmp_ui(temp2, 1)>0 && mpz_cmp_ui(temp3, 1)>0 && mpz_cmp(empty, temp2)>0 && mpz_cmp(empty, temp3)>0) { #ifdef PRINTDIVISORTWONEW familystring(str, p); gmp_printf("%s has four divisors %Zd, %Zd, %Zd, and %Zd\n", str, gcd1, gcd2, temp2, temp3); #endif mpz_clears(gcd, temp, gcd1, gcd2, x, y, z, temp2, temp3, temp4, temp5, temp6, temp7, temp8, temp9, temp10, empty, NULL); return 1; } } char end[MAXSTRING], start[MAXSTRING], middle[2]; if(numrepeats==1) { for(int i=0; i<p.len; i++) { if(p.numrepeats[i]==1) { endinstancestring(str, p, i); int zlen = strlen(str); mpz_set_str(z, str, base); mpz_set_ui(y, p.repeats[i][0]); startinstancestring(str, p, i); mpz_set_str(x, str, base); endinstancestring(end, p, i); sprintf(middle, "%c", digitchar(p.repeats[i][0])); startinstancestring(start, p, i); sprintf(str, "%s%s\n", start, end); mpz_set_str(temp, str, base); sprintf(str, "%s%s%s\n", start, middle, end); mpz_set_str(temp2, str, base); sprintf(str, "%s%s%s%s\n", start, middle, middle, end); mpz_set_str(temp3, str, base); sprintf(str, "%s%s%s%s%s\n", start, middle, middle, middle, end); mpz_set_str(temp4, str, base); sprintf(str, "%s%s%s%s%s%s\n", start, middle, middle, middle, middle, end); mpz_set_str(temp5, str, base); sprintf(str, "%s%s%s%s%s%s%s\n", start, middle, middle, middle, middle, middle, end); mpz_set_str(temp6, str, base); mpz_gcd(temp, temp, temp4); mpz_gcd(temp2, temp2, temp5); mpz_gcd(temp3, temp3, temp6); if(mpz_cmp_ui(temp, 1)>0 && mpz_cmp_ui(temp2, 1)>0 && mpz_cmp_ui(temp3, 1)>0 && mpz_cmp(empty, temp)>0 && mpz_cmp(empty, temp2)>0 && mpz_cmp(empty, temp3)>0) { #ifdef PRINTDIVISORTHREE familystring(str, p); gmp_printf("%s has three divisors %Zd, %Zd, and %Zd\n", str, temp, temp2, temp3); #endif mpz_clears(gcd, temp, gcd1, gcd2, x, y, z, temp2, temp3, temp4, temp5, temp6, temp7, temp8, temp9, temp10, empty, NULL); return 1; } sprintf(str, "%s%s\n", start, end); mpz_set_str(temp, str, base); sprintf(str, "%s%s%s\n", start, middle, end); mpz_set_str(temp2, str, base); sprintf(str, "%s%s%s%s\n", start, middle, middle, end); mpz_set_str(temp3, str, base); sprintf(str, "%s%s%s%s%s\n", start, middle, middle, middle, end); mpz_set_str(temp4, str, base); sprintf(str, "%s%s%s%s%s%s\n", start, middle, middle, middle, middle, end); mpz_set_str(temp5, str, base); sprintf(str, "%s%s%s%s%s%s%s\n", start, middle, middle, middle, middle, middle, end); mpz_set_str(temp6, str, base); sprintf(str, "%s%s%s%s%s%s%s%s\n", start, middle, middle, middle, middle, middle, middle, end); mpz_set_str(temp7, str, base); sprintf(str, "%s%s%s%s%s%s%s%s%s\n", start, middle, middle, middle, middle, middle, middle, middle, end); mpz_set_str(temp8, str, base); mpz_gcd(temp, temp, temp5); mpz_gcd(temp2, temp2, temp6); mpz_gcd(temp3, temp3, temp7); mpz_gcd(temp4, temp4, temp8); if(mpz_cmp_ui(temp, 1)>0 && mpz_cmp_ui(temp2, 1)>0 && mpz_cmp_ui(temp3, 1)>0 && mpz_cmp_ui(temp4, 1)>0 && mpz_cmp(empty, temp)>0 && mpz_cmp(empty, temp2)>0 && mpz_cmp(empty, temp3)>0 && mpz_cmp(empty, temp4)>0) { #ifdef PRINTDIVISORFOUR familystring(str, p); gmp_printf("%s has four divisors %Zd, %Zd, %Zd, and %Zd\n", str, temp, temp2, temp3, temp4); #endif mpz_clears(gcd, temp, gcd1, gcd2, x, y, z, temp2, temp3, temp4, temp5, temp6, temp7, temp8, temp9, temp10, empty, NULL); return 1; } sprintf(str, "%s%s\n", start, end); mpz_set_str(temp, str, base); sprintf(str, "%s%s%s\n", start, middle, end); mpz_set_str(temp2, str, base); sprintf(str, "%s%s%s%s\n", start, middle, middle, end); mpz_set_str(temp3, str, base); sprintf(str, "%s%s%s%s%s\n", start, middle, middle, middle, end); mpz_set_str(temp4, str, base); sprintf(str, "%s%s%s%s%s%s\n", start, middle, middle, middle, middle, end); mpz_set_str(temp5, str, base); sprintf(str, "%s%s%s%s%s%s%s\n", start, middle, middle, middle, middle, middle, end); mpz_set_str(temp6, str, base); sprintf(str, "%s%s%s%s%s%s%s%s\n", start, middle, middle, middle, middle, middle, middle, end); mpz_set_str(temp7, str, base); sprintf(str, "%s%s%s%s%s%s%s%s%s\n", start, middle, middle, middle, middle, middle, middle, middle, end); mpz_set_str(temp8, str, base); sprintf(str, "%s%s%s%s%s%s%s%s%s%s\n", start, middle, middle, middle, middle, middle, middle, middle, middle, end); mpz_set_str(temp9, str, base); sprintf(str, "%s%s%s%s%s%s%s%s%s%s%s\n", start, middle, middle, middle, middle, middle, middle, middle, middle, middle, end); mpz_set_str(temp10, str, base); mpz_gcd(temp, temp, temp6); mpz_gcd(temp2, temp2, temp7); mpz_gcd(temp3, temp3, temp8); mpz_gcd(temp4, temp4, temp9); mpz_gcd(temp5, temp5, temp10); if(mpz_cmp_ui(temp, 1)>0 && mpz_cmp_ui(temp2, 1)>0 && mpz_cmp_ui(temp3, 1)>0 && mpz_cmp_ui(temp4, 1)>0 && mpz_cmp_ui(temp5, 1)>0 && mpz_cmp(empty, temp)>0 && mpz_cmp(empty, temp2)>0 && mpz_cmp(empty, temp3)>0 && mpz_cmp(empty, temp4)>0 && mpz_cmp(empty, temp5)>0) { #ifdef PRINTDIVISORFIVE familystring(str, p); gmp_printf("%s has five divisors %Zd, %Zd, %Zd, %Zd, and %Zd\n", str, temp, temp2, temp3, temp4, temp5); #endif mpz_clears(gcd, temp, gcd1, gcd2, x, y, z, temp2, temp3, temp4, temp5, temp6, temp7, temp8, temp9, temp10, empty, NULL); return 1; } mpz_gcd_ui(temp10, y, base-1); int g = mpz_get_ui(temp10); mpz_divexact_ui(temp, y, g); mpz_set(temp2, temp); mpz_addmul_ui(temp, x, (base-1)/g); mpz_ui_pow_ui(temp3, base, zlen); mpz_mul(temp, temp, temp3); mpz_mul(temp2, temp2, temp3); mpz_submul_ui(temp2, z, (base-1)/g); if(mpz_root(temp3, temp, 2)!=0 && mpz_sgn(temp2)>=0 && mpz_root(temp4, temp2, 2)!=0) { mpz_sub(temp5, temp3, temp4); mpz_set_ui(temp6, base); if(mpz_cmp_ui(temp5, (base-1)/g)>0 && mpz_root(temp6, temp6, 2)!=0) { #ifdef PRINTDIVISORSQUARE familystring(str, p); gmp_printf("%s factors as a difference of squares\n", str); gmp_printf("%s(%s)^n%s = %Zd + %d^%d*%Zd*(%d^n-1)/%d + %d^(n+%d)*%Zd = (%Zd*%d^n-%Zd)/%d = (%Zd*%Zd^n-%Zd)*(%Zd*%Zd^n+%Zd)/%d\n", start, middle, end, z, base, zlen, y, base, base-1, base, zlen, x, temp, base, temp2, (base-1)/g, temp3, temp6, temp4, temp3, temp6, temp4, (base-1)/g); #endif mpz_clears(gcd, temp, gcd1, gcd2, x, y, z, temp2, temp3, temp4, temp5, temp6, temp7, temp8, temp9, temp10, empty, NULL); return 1; } else if(mpz_cmp_ui(temp5, (base-1)/g)>0 && mpz_cmp_ui(gcd2, 1)>0 && mpz_cmp(empty, gcd2)>0) { #ifdef PRINTDIVISORSQUARE familystring(str, p); gmp_printf("%s factors as a difference of squares for even n, and has a factor %Zd for odd n\n", str, gcd2); gmp_printf("%s(%s)^n%s = %Zd + %d^%d*%Zd*(%d^n-1)/%d + %d^(n+%d)*%Zd = (%Zd*%d^n-%Zd)/%d = (%Zd*%d^(n/2)-%Zd)*(%Zd*%d^(n/2)+%Zd)/%d\n", start, middle, end, z, base, zlen, y, base, base-1, base, zlen, x, temp, base, temp2, (base-1)/g, temp3, base, temp4, temp3, base, temp4, (base-1)/g); #endif mpz_clears(gcd, temp, gcd1, gcd2, x, y, z, temp2, temp3, temp4, temp5, temp6, temp7, temp8, temp9, temp10, empty, NULL); return 1; } } if(mpz_root(temp3, temp, 3)!=0 && mpz_root(temp4, temp2, 3)!=0) { mpz_sub(temp5, temp3, temp4); mpz_set_ui(temp6, base); if(mpz_cmp_ui(temp5, (base-1)/g)>0 && mpz_root(temp6, temp6, 3)!=0) { #ifdef PRINTDIVISORCUBE familystring(str, p); gmp_printf("%s factors as a difference of cubes\n", str); if(mpz_sgn(temp2)>=0) gmp_printf("%s(%s)^n%s = (%Zd*%d^n-%Zd)/%d = (%Zd*%Zd^n-%Zd)*((%Zd*%Zd^n)^2+%Zd*%Zd^n*%Zd+%Zd^2)/%d\n", start, middle, end, temp, base, temp2, (base-1)/g, temp3, temp6, temp4, temp3, temp6, temp3, temp6, temp4, temp4, (base-1)/g); else gmp_printf("%s(%s)^n%s = (%Zd*%d^n-(%Zd))/%d = (%Zd*%Zd^n-(%Zd))*((%Zd*%Zd^n)^2+%Zd*%Zd^n*(%Zd)+(%Zd)^2)/%d\n", start, middle, end, temp, base, temp2, (base-1)/g, temp3, temp6, temp4, temp3, temp6, temp3, temp6, temp4, temp4, (base-1)/g); #endif mpz_clears(gcd, temp, gcd1, gcd2, x, y, z, temp2, temp3, temp4, temp5, temp6, temp7, temp8, temp9, temp10, empty, NULL); return 1; } } if(mpz_root(temp3, temp, 5)!=0 && mpz_root(temp4, temp2, 5)!=0) { mpz_sub(temp5, temp3, temp4); mpz_set_ui(temp6, base); if(mpz_cmp_ui(temp5, (base-1)/g)>0 && mpz_root(temp6, temp6, 3)!=0) { #ifdef PRINTDIVISORFIFTHPOWER familystring(str, p); gmp_printf("%s factors as a difference of fifth powers\n", str); if(mpz_sgn(temp2)>=0) gmp_printf("%s(%s)^n%s = (%Zd*%d^n-%Zd)/%d = (%Zd*%Zd^n-%Zd)*((%Zd*%Zd^n)^2+%Zd*%Zd^n*%Zd+%Zd^2)/%d\n", start, middle, end, temp, base, temp2, (base-1)/g, temp3, temp6, temp4, temp3, temp6, temp3, temp6, temp4, temp4, (base-1)/g); else gmp_printf("%s(%s)^n%s = (%Zd*%d^n-(%Zd))/%d = (%Zd*%Zd^n-(%Zd))*((%Zd*%Zd^n)^4+(%Zd*%Zd^n)^3*(%Zd)+(%Zd*%Zd^n)^2*(%Zd)^2+%Zd*%Zd^n*(%Zd)^3+(%Zd)^4)/%d\n", start, middle, end, temp, base, temp2, (base-1)/g, temp3, temp6, temp4, temp3, temp6, temp3, temp6, temp4, temp4, (base-1)/g); #endif mpz_clears(gcd, temp, gcd1, gcd2, x, y, z, temp2, temp3, temp4, temp5, temp6, temp7, temp8, temp9, temp10, empty, NULL); return 1; } } } } } char residues[30] = {1}; for(int i=0; i<p.len; i++) { if((unsigned char)p.digit[i]!=255) { char newresidues[30] = {0}; for(int j=0; j<30; j++) { if(residues[j]==1) newresidues[(j*base+p.digit[i])%30] = 1; } memcpy(residues, newresidues, 30); } int haschanged = 1; while(haschanged) { haschanged = 0; for(int j=0; j<p.numrepeats[i]; j++) { for(int l=0; l<30; l++) { if(residues[l]==1 && residues[(l*base+p.repeats[i][j])%30]==0) { residues[(l*base+p.repeats[i][j])%30] = 1; haschanged = 1; } } } } } int coprimeres = 0; for(int i=0; i<30; i++) { if(residues[i]==1) { mpz_set_ui(temp, i); mpz_gcd_ui(temp, temp, 30); if(mpz_cmp_ui(temp, 1)==0) coprimeres = 1; } } if(!coprimeres) { #ifdef PRINTDIVISOREXT familystring(str, p); gmp_printf("\nevery number in %s is divisible by one of 2, 3, or 5\n", str); #endif mpz_clears(gcd, temp, gcd1, gcd2, x, y, z, temp2, temp3, temp4, temp5, temp6, temp7, temp8, temp9, temp10, empty, NULL); return 1; } char residues[42] = {1}; for(int i=0; i<p.len; i++) { if((unsigned char)p.digit[i]!=255) { char newresidues[42] = {0}; for(int j=0; j<42; j++) { if(residues[j]==1) newresidues[(j*base+p.digit[i])%42] = 1; } memcpy(residues, newresidues, 42); } int haschanged = 1; while(haschanged) { haschanged = 0; for(int j=0; j<p.numrepeats[i]; j++) { for(int l=0; l<42; l++) { if(residues[l]==1 && residues[(l*base+p.repeats[i][j])%42]==0) { residues[(l*base+p.repeats[i][j])%42] = 1; haschanged = 1; } } } } } int coprimeres = 0; for(int i=0; i<42; i++) { if(residues[i]==1) { mpz_set_ui(temp, i); mpz_gcd_ui(temp, temp, 42); if(mpz_cmp_ui(temp, 1)==0) coprimeres = 1; } } if(!coprimeres) { #ifdef PRINTDIVISOREXT familystring(str, p); gmp_printf("\nevery number in %s is divisible by one of 2, 3, or 7\n", str); #endif mpz_clears(gcd, temp, gcd1, gcd2, x, y, z, temp2, temp3, temp4, temp5, temp6, temp7, temp8, temp9, temp10, empty, NULL); return 1; } mpz_clears(gcd, temp, gcd1, gcd2, x, y, z, temp2, temp3, temp4, temp5, temp6, temp7, temp8, temp9, temp10, empty, NULL); return 0; } void instancefamily(family* newf, family f, int side, int pos) { for(int i=0; i<f.len; i++) { char* repeatscopy = malloc(f.numrepeats[i]*sizeof(char)); memcpy(repeatscopy, f.repeats[i], f.numrepeats[i]*sizeof(char)); if(i==pos) { if(side==1) { adddigit(newf, f.digit[i], NULL, 0); adddigit(newf, 0, repeatscopy, f.numrepeats[i]); } else if(side==0) { adddigit(newf, f.digit[i], repeatscopy, f.numrepeats[i]); adddigit(newf, 0, NULL, 0); } } else adddigit(newf, f.digit[i], repeatscopy, f.numrepeats[i]); } } int examine(family* f) { char* str = malloc(MAXSTRING); char tempstr[MAXSTRING]; emptyinstancestring(str, *f); if(!nosubword(str)) { free(str); #ifdef PRINTSUBWORD familystring(tempstr, *f); printf("%s has a subword in kernel\n", tempstr); #endif return 0; } else if(isprime(str)) { addtokernel(str); #ifdef PRINTPRIMES familystring(tempstr, *f); printf("%s has a prime\n", tempstr); #endif return 0; } free(str); int trivial = 1; for(int i=0; i<f->len; i++) { int newnumrepeat = 0; for(int j=0; j<f->numrepeats[i]; j++) { char tempstr[MAXSTRING]; instancestring(tempstr, *f, i, j); if(nosubword(tempstr)) f->repeats[i][newnumrepeat++] = f->repeats[i][j]; } f->numrepeats[i] = newnumrepeat; if(newnumrepeat>0) trivial = 0; } // simplify y*y^ny* char lastdigit = 0; int dosimplify = 0; for(int i=0; i<f->len; i++) { if(dosimplify==1 && f->numrepeats[i]==1 && f->repeats[i][0]==lastdigit && (lastdigit==f->digit[i] || (unsigned char)f->digit[i]==255)) { f->repeats[i] = NULL; f->numrepeats[i] = 0; } if((unsigned char)f->digit[i]!=255) { if(f->digit[i] != lastdigit) dosimplify = 0; lastdigit = f->digit[i]; } if(f->numrepeats[i]==1) { dosimplify = 1; lastdigit = f->repeats[i][0]; } else if(f->numrepeats[i]>1) dosimplify = 0; } if(trivial) { #ifdef PRINTTRIVIAL familystring(tempstr, *f); printf("%s is trivial\n", tempstr); #endif return 0; } if(hasdivisor(*f)) { return 0; } return 1; } int split(family* f, list* unsolved, char insplit) { if(insplit==0) { addtolist(unsolved, *f, 0); return 0; } for(int i=0; i<f->len; i++) { for(int j=0; j<f->numrepeats[i]; j++) { char str[MAXSTRING]; doubleinstancestring(str, *f, i, j, i, j); if(!nosubword(str)) { family copyf; familyinit(©f); copyfamily(©f, *f); int newnumrepeats = 0; int removeddigit = copyf.repeats[i][j]; for(int k=0; k<copyf.numrepeats[i]; k++) { if(k!=j) copyf.repeats[i][newnumrepeats++] = copyf.repeats[i][k]; } copyf.numrepeats[i] = newnumrepeats; family newf; familyinit(&newf); for(int k=0; k<copyf.len; k++) { char* newrepeats = malloc(copyf.numrepeats[k]*sizeof(char)); memcpy(newrepeats, copyf.repeats[k], copyf.numrepeats[k]*sizeof(char)); adddigit(&newf, copyf.digit[k], newrepeats, copyf.numrepeats[k]); if(k==i) { newrepeats = malloc(copyf.numrepeats[k]*sizeof(char)); memcpy(newrepeats, copyf.repeats[k], copyf.numrepeats[k]*sizeof(char)); adddigit(&newf, removeddigit, newrepeats, copyf.numrepeats[k]); } } addtolist(unsolved, copyf, 2); addtolist(unsolved, newf, 2); #ifdef PRINTSPLITDOUBLE char str[MAXSTRING]; familystring(str, *f); printf("%s splits into ", str); familystring(str, copyf); printf("%s and ", str); familystring(str, newf); printf("%s\n", str); #endif clearfamily(©f); clearfamily(&newf); return 1; } tripleinstancestring(str, *f, i, j, i, j, i, j); if(!nosubword(str)) { family copyf; familyinit(©f); copyfamily(©f, *f); int newnumrepeats = 0; int removeddigit = copyf.repeats[i][j]; for(int k=0; k<copyf.numrepeats[i]; k++) { if(k!=j) copyf.repeats[i][newnumrepeats++] = copyf.repeats[i][k]; } copyf.numrepeats[i] = newnumrepeats; family newf; familyinit(&newf); for(int k=0; k<copyf.len; k++) { char* newrepeats = malloc(copyf.numrepeats[k]*sizeof(char)); memcpy(newrepeats, copyf.repeats[k], copyf.numrepeats[k]*sizeof(char)); adddigit(&newf, copyf.digit[k], newrepeats, copyf.numrepeats[k]); if(k==i) { newrepeats = malloc(copyf.numrepeats[k]*sizeof(char)); memcpy(newrepeats, copyf.repeats[k], copyf.numrepeats[k]*sizeof(char)); adddigit(&newf, removeddigit, newrepeats, copyf.numrepeats[k]); } } family newf2; familyinit(&newf2); for(int k=0; k<copyf.len; k++) { char* newrepeats = malloc(copyf.numrepeats[k]*sizeof(char)); memcpy(newrepeats, copyf.repeats[k], copyf.numrepeats[k]*sizeof(char)); adddigit(&newf2, copyf.digit[k], newrepeats, copyf.numrepeats[k]); if(k==i) { newrepeats = malloc(copyf.numrepeats[k]*sizeof(char)); memcpy(newrepeats, copyf.repeats[k], copyf.numrepeats[k]*sizeof(char)); adddigit(&newf2, removeddigit, newrepeats, copyf.numrepeats[k]); newrepeats = malloc(copyf.numrepeats[k]*sizeof(char)); memcpy(newrepeats, copyf.repeats[k], copyf.numrepeats[k]*sizeof(char)); adddigit(&newf2, removeddigit, newrepeats, copyf.numrepeats[k]); } } addtolist(unsolved, copyf, 2); addtolist(unsolved, newf, 2); addtolist(unsolved, newf2, 2); #ifdef PRINTSPLITTRIPLE char str[MAXSTRING]; familystring(str, *f); printf("%s splits into ", str); familystring(str, copyf); printf("%s and ", str); familystring(str, newf); printf("%s and ", str); familystring(str, newf2); printf("%s\n", str); #endif clearfamily(©f); clearfamily(&newf); clearfamily(&newf2); return 1; } quadinstancestring(str, *f, i, j, i, j, i, j, i, j); if(!nosubword(str)) { family copyf; familyinit(©f); copyfamily(©f, *f); int newnumrepeats = 0; int removeddigit = copyf.repeats[i][j]; for(int k=0; k<copyf.numrepeats[i]; k++) { if(k!=j) copyf.repeats[i][newnumrepeats++] = copyf.repeats[i][k]; } copyf.numrepeats[i] = newnumrepeats; family newf; familyinit(&newf); for(int k=0; k<copyf.len; k++) { char* newrepeats = malloc(copyf.numrepeats[k]*sizeof(char)); memcpy(newrepeats, copyf.repeats[k], copyf.numrepeats[k]*sizeof(char)); adddigit(&newf, copyf.digit[k], newrepeats, copyf.numrepeats[k]); if(k==i) { newrepeats = malloc(copyf.numrepeats[k]*sizeof(char)); memcpy(newrepeats, copyf.repeats[k], copyf.numrepeats[k]*sizeof(char)); adddigit(&newf, removeddigit, newrepeats, copyf.numrepeats[k]); } } family newf2; familyinit(&newf2); for(int k=0; k<copyf.len; k++) { char* newrepeats = malloc(copyf.numrepeats[k]*sizeof(char)); memcpy(newrepeats, copyf.repeats[k], copyf.numrepeats[k]*sizeof(char)); adddigit(&newf2, copyf.digit[k], newrepeats, copyf.numrepeats[k]); if(k==i) { newrepeats = malloc(copyf.numrepeats[k]*sizeof(char)); memcpy(newrepeats, copyf.repeats[k], copyf.numrepeats[k]*sizeof(char)); adddigit(&newf2, removeddigit, newrepeats, copyf.numrepeats[k]); newrepeats = malloc(copyf.numrepeats[k]*sizeof(char)); memcpy(newrepeats, copyf.repeats[k], copyf.numrepeats[k]*sizeof(char)); adddigit(&newf2, removeddigit, newrepeats, copyf.numrepeats[k]); } } family newf3; familyinit(&newf3); for(int k=0; k<copyf.len; k++) { char* newrepeats = malloc(copyf.numrepeats[k]*sizeof(char)); memcpy(newrepeats, copyf.repeats[k], copyf.numrepeats[k]*sizeof(char)); adddigit(&newf3, copyf.digit[k], newrepeats, copyf.numrepeats[k]); if(k==i) { newrepeats = malloc(copyf.numrepeats[k]*sizeof(char)); memcpy(newrepeats, copyf.repeats[k], copyf.numrepeats[k]*sizeof(char)); adddigit(&newf3, removeddigit, newrepeats, copyf.numrepeats[k]); newrepeats = malloc(copyf.numrepeats[k]*sizeof(char)); memcpy(newrepeats, copyf.repeats[k], copyf.numrepeats[k]*sizeof(char)); adddigit(&newf3, removeddigit, newrepeats, copyf.numrepeats[k]); newrepeats = malloc(copyf.numrepeats[k]*sizeof(char)); memcpy(newrepeats, copyf.repeats[k], copyf.numrepeats[k]*sizeof(char)); adddigit(&newf3, removeddigit, newrepeats, copyf.numrepeats[k]); } } addtolist(unsolved, copyf, 2); addtolist(unsolved, newf, 2); addtolist(unsolved, newf2, 2); addtolist(unsolved, newf3, 2); #ifdef PRINTSPLITQUAD char str[MAXSTRING]; familystring(str, *f); printf("%s splits into ", str); familystring(str, copyf); printf("%s and ", str); familystring(str, newf); printf("%s and ", str); familystring(str, newf2); printf("%s and ", str); familystring(str, newf3); printf("%s\n", str); #endif clearfamily(©f); clearfamily(&newf); clearfamily(&newf2); clearfamily(&newf3); return 1; } quintinstancestring(str, *f, i, j); if(!nosubword(str)) { family copyf; familyinit(©f); copyfamily(©f, *f); int newnumrepeats = 0; int removeddigit = copyf.repeats[i][j]; for(int k=0; k<copyf.numrepeats[i]; k++) { if(k!=j) copyf.repeats[i][newnumrepeats++] = copyf.repeats[i][k]; } copyf.numrepeats[i] = newnumrepeats; addtolist(unsolved, copyf, 2); for(int l=1; l<=9; l++) { family newf; familyinit(&newf); for(int k=0; k<copyf.len; k++) { char* newrepeats = malloc(copyf.numrepeats[k]*sizeof(char)); memcpy(newrepeats, copyf.repeats[k], copyf.numrepeats[k]*sizeof(char)); adddigit(&newf, copyf.digit[k], newrepeats, copyf.numrepeats[k]); if(k==i) { for(int m=0; m<l; m++) { newrepeats = malloc(copyf.numrepeats[k]*sizeof(char)); memcpy(newrepeats, copyf.repeats[k], copyf.numrepeats[k]*sizeof(char)); adddigit(&newf, removeddigit, newrepeats, copyf.numrepeats[k]); } } } addtolist(unsolved, newf, 2); clearfamily(&newf); } clearfamily(©f); #ifdef PRINTSPLITQUINT char str[MAXSTRING]; familystring(str, *f); printf("%s splits ten ways\n", str); #endif return 1; } if(iter>5) { mpz_t gcd, empty, temp; mpz_inits(gcd, empty, temp, NULL); emptyinstancestring(str, *f); mpz_set_str(empty, str, base); mpz_set_str(gcd, str, base); for(int ii=0; ii<f->len; ii++) { for(int jj=0; jj<f->numrepeats[ii]; jj++) { instancestring(str, *f, ii, jj); mpz_set_str(temp, str, base); if(i!=ii || j!=jj) mpz_gcd(gcd, gcd, temp); } } if(mpz_cmp_ui(gcd, 1)>0 && mpz_cmp(empty, gcd)>0) { mpz_clears(gcd, empty, temp, NULL); family copyf; familyinit(©f); for(int ii=0; ii<f->len; ii++) { char* repeatscopy = malloc(f->numrepeats[ii]*sizeof(char)); memcpy(repeatscopy, f->repeats[ii], f->numrepeats[ii]*sizeof(char)); adddigit(©f, f->digit[ii], repeatscopy, f->numrepeats[ii]); if(i==ii) { repeatscopy = malloc(f->numrepeats[ii]*sizeof(char)); memcpy(repeatscopy, f->repeats[ii], f->numrepeats[ii]*sizeof(char)); adddigit(©f, f->repeats[i][j], repeatscopy, f->numrepeats[i]); } } addtolist(unsolved, copyf, 2); #ifdef PRINTSPLITEXT familystring(str, *f); printf("%s splits into ", str); familystring(str, copyf); printf("%s\n", str); #endif clearfamily(©f); return 1; } } } } addtolist(unsolved, *f, 1); return 0; } int split2(family* f, list* unsolved, char insplit) { if(insplit==0) { addtolist(unsolved, *f, 0); return 0; } for(int i=0; i<f->len; i++) { for(int j=0; j<f->numrepeats[i]; j++) { for(int m=i; m<f->len; m++) { for(int k=0; k<f->numrepeats[m]; k++) { if(m==i && j<=k) continue; char str1[MAXSTRING]; char str2[MAXSTRING]; char str3[MAXSTRING]; char str4[MAXSTRING]; char str5[MAXSTRING]; char str6[MAXSTRING]; doubleinstancestring(str1, *f, i, j, m, k); tripleinstancestring(str3, *f, i, j, m, k, i, j); tripleinstancestring(str4, *f, i, k, m, j, i, k); quadinstancestring(str5, *f, i, j, m, k, i, j, m, k); quadinstancestring(str6, *f, i, k, m, j, i, k, m, j); if(m==i) doubleinstancestring(str2, *f, i, k, m, j); if(m==i && !nosubword(str1) && !nosubword(str2)) { family copyf; familyinit(©f); copyfamily(©f, *f); int newnumrepeats = 0; for(int l=0; l<copyf.numrepeats[i]; l++) { if(l!=j) copyf.repeats[i][newnumrepeats++] = copyf.repeats[i][l]; } copyf.numrepeats[i] = newnumrepeats; addtolist(unsolved, copyf, 1); #ifdef PRINTSPLIT char str[MAXSTRING]; familystring(str, *f); printf("%s splits into ", str); familystring(str, copyf); printf("%s and ", str); #endif clearfamily(©f); familyinit(©f); copyfamily(©f, *f); newnumrepeats = 0; for(int l=0; l<copyf.numrepeats[i]; l++) { if(l!=k) copyf.repeats[i][newnumrepeats++] = copyf.repeats[i][l]; } copyf.numrepeats[i] = newnumrepeats; addtolist(unsolved, copyf, 1); #ifdef PRINTSPLIT familystring(str, copyf); printf("%s [because of %s, %s]\n", str, str1, str2); #endif clearfamily(©f); return 1; } else if(m==i && iter>5 && (!nosubword(str1))) { family newf; familyinit(&newf); for(int l=0; l<f->len; l++) { char* newrepeats = malloc(f->numrepeats[l]*sizeof(char)); memcpy(newrepeats, f->repeats[l], f->numrepeats[l]*sizeof(char)); adddigit(&newf, f->digit[l], newrepeats, f->numrepeats[l]); if(i==l) { int newnumrepeats = 0; int removeddigit = f->repeats[i][j]; for(int m=0; m<f->numrepeats[i]; m++) { if(m!=j) newf.repeats[i][newnumrepeats++] = f->repeats[i][m]; } newf.numrepeats[i] = newnumrepeats; newrepeats = malloc(f->numrepeats[l]*sizeof(char)); memcpy(newrepeats, f->repeats[l], f->numrepeats[l]*sizeof(char)); adddigit(&newf, 255, newrepeats, f->numrepeats[l]); newnumrepeats = 0; removeddigit = f->repeats[i][k]; for(int m=0; m<f->numrepeats[i]; m++) { if(m!=k) newf.repeats[i+1][newnumrepeats++] = f->repeats[i][m]; } newf.numrepeats[i+1] = newnumrepeats; } } addtolist(unsolved, newf, 1); #ifdef PRINTSPLIT char str[MAXSTRING]; familystring(str, *f); printf("%s splits into ", str); familystring(str, newf); printf("%s [because of %s]\n", str, str1); #endif clearfamily(&newf); return 1; } else if(m==i && iter>5 && (!nosubword(str2))) { family newf; familyinit(&newf); for(int l=0; l<f->len; l++) { char* newrepeats = malloc(f->numrepeats[l]*sizeof(char)); memcpy(newrepeats, f->repeats[l], f->numrepeats[l]*sizeof(char)); adddigit(&newf, f->digit[l], newrepeats, f->numrepeats[l]); if(i==l) { int newnumrepeats = 0; int removeddigit = f->repeats[i][k]; for(int m=0; m<f->numrepeats[i]; m++) { if(m!=k) newf.repeats[i][newnumrepeats++] = f->repeats[i][m]; } newf.numrepeats[i] = newnumrepeats; newrepeats = malloc(f->numrepeats[l]*sizeof(char)); memcpy(newrepeats, f->repeats[l], f->numrepeats[l]*sizeof(char)); adddigit(&newf, 255, newrepeats, f->numrepeats[l]); newnumrepeats = 0; removeddigit = f->repeats[i][j]; for(int m=0; m<f->numrepeats[i]; m++) { if(m!=j) newf.repeats[i+1][newnumrepeats++] = f->repeats[i][m]; } newf.numrepeats[i+1] = newnumrepeats; } } addtolist(unsolved, newf, 1); #ifdef PRINTSPLIT char str[MAXSTRING]; familystring(str, *f); printf("%s splits into ", str); familystring(str, newf); printf("%s [because of %s]\n", str, str2); #endif clearfamily(&newf); return 1; } else if(m==i && iter>5 && (!nosubword(str3))) { family newf; familyinit(&newf); for(int l=0; l<f->len; l++) { char* newrepeats = malloc(f->numrepeats[l]*sizeof(char)); memcpy(newrepeats, f->repeats[l], f->numrepeats[l]*sizeof(char)); adddigit(&newf, f->digit[l], newrepeats, f->numrepeats[l]); if(i==l) { int newnumrepeats = 0; int removeddigit = f->repeats[i][j]; for(int m=0; m<f->numrepeats[i]; m++) { if(m!=j) newf.repeats[i][newnumrepeats++] = f->repeats[i][m]; } newf.numrepeats[i] = newnumrepeats; newrepeats = malloc(f->numrepeats[l]*sizeof(char)); memcpy(newrepeats, f->repeats[l], f->numrepeats[l]*sizeof(char)); adddigit(&newf, 255, newrepeats, f->numrepeats[l]); newnumrepeats = 0; removeddigit = f->repeats[i][k]; for(int m=0; m<f->numrepeats[i]; m++) { if(m!=k) newf.repeats[i+1][newnumrepeats++] = f->repeats[i][m]; } newf.numrepeats[i+1] = newnumrepeats; newrepeats = malloc(f->numrepeats[l]*sizeof(char)); memcpy(newrepeats, f->repeats[l], f->numrepeats[l]*sizeof(char)); adddigit(&newf, 255, newrepeats, f->numrepeats[l]); newnumrepeats = 0; removeddigit = f->repeats[i][j]; for(int m=0; m<f->numrepeats[i]; m++) { if(m!=j) newf.repeats[i+2][newnumrepeats++] = f->repeats[i][m]; } newf.numrepeats[i+2] = newnumrepeats; } } addtolist(unsolved, newf, 1); #ifdef PRINTSPLITEXT char str[MAXSTRING]; familystring(str, *f); printf("%s splits into ", str); familystring(str, newf); printf("%s [because of %s]\n", str, str3); #endif clearfamily(&newf); return 1; } else if(m==i && iter>5 && (!nosubword(str4))) { family newf; familyinit(&newf); for(int l=0; l<f->len; l++) { char* newrepeats = malloc(f->numrepeats[l]*sizeof(char)); memcpy(newrepeats, f->repeats[l], f->numrepeats[l]*sizeof(char)); adddigit(&newf, f->digit[l], newrepeats, f->numrepeats[l]); if(i==l) { int newnumrepeats = 0; int removeddigit = f->repeats[i][k]; for(int m=0; m<f->numrepeats[i]; m++) { if(m!=k) newf.repeats[i][newnumrepeats++] = f->repeats[i][m]; } newf.numrepeats[i] = newnumrepeats; newrepeats = malloc(f->numrepeats[l]*sizeof(char)); memcpy(newrepeats, f->repeats[l], f->numrepeats[l]*sizeof(char)); adddigit(&newf, 255, newrepeats, f->numrepeats[l]); newnumrepeats = 0; removeddigit = f->repeats[i][j]; for(int m=0; m<f->numrepeats[i]; m++) { if(m!=j) newf.repeats[i+1][newnumrepeats++] = f->repeats[i][m]; } newf.numrepeats[i+1] = newnumrepeats; newrepeats = malloc(f->numrepeats[l]*sizeof(char)); memcpy(newrepeats, f->repeats[l], f->numrepeats[l]*sizeof(char)); adddigit(&newf, 255, newrepeats, f->numrepeats[l]); newnumrepeats = 0; removeddigit = f->repeats[i][k]; for(int m=0; m<f->numrepeats[i]; m++) { if(m!=k) newf.repeats[i+2][newnumrepeats++] = f->repeats[i][m]; } newf.numrepeats[i+2] = newnumrepeats; } } addtolist(unsolved, newf, 1); #ifdef PRINTSPLITEXT char str[MAXSTRING]; familystring(str, *f); printf("%s splits into ", str); familystring(str, newf); printf("%s [because of %s]\n", str, str4); #endif clearfamily(&newf); return 1; } else if(m==i && iter>5 && (!nosubword(str5))) { family newf; familyinit(&newf); for(int l=0; l<f->len; l++) { char* newrepeats = malloc(f->numrepeats[l]*sizeof(char)); memcpy(newrepeats, f->repeats[l], f->numrepeats[l]*sizeof(char)); adddigit(&newf, f->digit[l], newrepeats, f->numrepeats[l]); if(i==l) { int newnumrepeats = 0; int removeddigit = f->repeats[i][j]; for(int m=0; m<f->numrepeats[i]; m++) { if(m!=j) newf.repeats[i][newnumrepeats++] = f->repeats[i][m]; } newf.numrepeats[i] = newnumrepeats; newrepeats = malloc(f->numrepeats[l]*sizeof(char)); memcpy(newrepeats, f->repeats[l], f->numrepeats[l]*sizeof(char)); adddigit(&newf, 255, newrepeats, f->numrepeats[l]); newnumrepeats = 0; removeddigit = f->repeats[i][k]; for(int m=0; m<f->numrepeats[i]; m++) { if(m!=k) newf.repeats[i+1][newnumrepeats++] = f->repeats[i][m]; } newf.numrepeats[i+1] = newnumrepeats; newrepeats = malloc(f->numrepeats[l]*sizeof(char)); memcpy(newrepeats, f->repeats[l], f->numrepeats[l]*sizeof(char)); adddigit(&newf, 255, newrepeats, f->numrepeats[l]); newnumrepeats = 0; removeddigit = f->repeats[i][j]; for(int m=0; m<f->numrepeats[i]; m++) { if(m!=j) newf.repeats[i+2][newnumrepeats++] = f->repeats[i][m]; } newf.numrepeats[i+2] = newnumrepeats; newrepeats = malloc(f->numrepeats[l]*sizeof(char)); memcpy(newrepeats, f->repeats[l], f->numrepeats[l]*sizeof(char)); adddigit(&newf, 255, newrepeats, f->numrepeats[l]); newnumrepeats = 0; removeddigit = f->repeats[i][k]; for(int m=0; m<f->numrepeats[i]; m++) { if(m!=k) newf.repeats[i+3][newnumrepeats++] = f->repeats[i][m]; } newf.numrepeats[i+3] = newnumrepeats; } } addtolist(unsolved, newf, 1); #ifdef PRINTSPLITEXT char str[MAXSTRING]; familystring(str, *f); printf("%s splits into ", str); familystring(str, newf); printf("%s [because of %s]\n", str, str5); #endif clearfamily(&newf); return 1; } else if(m==i && iter>5 && (!nosubword(str6))) { family newf; familyinit(&newf); for(int l=0; l<f->len; l++) { char* newrepeats = malloc(f->numrepeats[l]*sizeof(char)); memcpy(newrepeats, f->repeats[l], f->numrepeats[l]*sizeof(char)); adddigit(&newf, f->digit[l], newrepeats, f->numrepeats[l]); if(i==l) { int newnumrepeats = 0; int removeddigit = f->repeats[i][k]; for(int m=0; m<f->numrepeats[i]; m++) { if(m!=k) newf.repeats[i][newnumrepeats++] = f->repeats[i][m]; } newf.numrepeats[i] = newnumrepeats; newrepeats = malloc(f->numrepeats[l]*sizeof(char)); memcpy(newrepeats, f->repeats[l], f->numrepeats[l]*sizeof(char)); adddigit(&newf, 255, newrepeats, f->numrepeats[l]); newnumrepeats = 0; removeddigit = f->repeats[i][j]; for(int m=0; m<f->numrepeats[i]; m++) { if(m!=j) newf.repeats[i+1][newnumrepeats++] = f->repeats[i][m]; } newf.numrepeats[i+1] = newnumrepeats; newrepeats = malloc(f->numrepeats[l]*sizeof(char)); memcpy(newrepeats, f->repeats[l], f->numrepeats[l]*sizeof(char)); adddigit(&newf, 255, newrepeats, f->numrepeats[l]); newnumrepeats = 0; removeddigit = f->repeats[i][k]; for(int m=0; m<f->numrepeats[i]; m++) { if(m!=k) newf.repeats[i+2][newnumrepeats++] = f->repeats[i][m]; } newf.numrepeats[i+2] = newnumrepeats; newrepeats = malloc(f->numrepeats[l]*sizeof(char)); memcpy(newrepeats, f->repeats[l], f->numrepeats[l]*sizeof(char)); adddigit(&newf, 255, newrepeats, f->numrepeats[l]); newnumrepeats = 0; removeddigit = f->repeats[i][j]; for(int m=0; m<f->numrepeats[i]; m++) { if(m!=j) newf.repeats[i+3][newnumrepeats++] = f->repeats[i][m]; } newf.numrepeats[i+3] = newnumrepeats; } } addtolist(unsolved, newf, 1); #ifdef PRINTSPLITEXT char str[MAXSTRING]; familystring(str, *f); printf("%s splits into ", str); familystring(str, newf); printf("%s [because of %s]\n", str, str6); #endif clearfamily(&newf); return 1; } else if(m>i && !nosubword(str1)) { family copyf; familyinit(©f); copyfamily(©f, *f); int newnumrepeats = 0; for(int l=0; l<copyf.numrepeats[i]; l++) { if(l!=j) copyf.repeats[i][newnumrepeats++] = copyf.repeats[i][l]; } copyf.numrepeats[i] = newnumrepeats; addtolist(unsolved, copyf, 1); #ifdef PRINTSPLIT char str[MAXSTRING]; familystring(str, *f); printf("%s splits into ", str); familystring(str, copyf); printf("%s and ", str); #endif clearfamily(©f); familyinit(©f); copyfamily(©f, *f); newnumrepeats = 0; for(int l=0; l<copyf.numrepeats[m]; l++) { if(l!=k) copyf.repeats[m][newnumrepeats++] = copyf.repeats[m][l]; } copyf.numrepeats[m] = newnumrepeats; addtolist(unsolved, copyf, 1); #ifdef PRINTSPLIT familystring(str, copyf); printf("%s [because of %s]\n", str, str1); #endif clearfamily(©f); return 1; } } } } } addtolist(unsolved, *f, insplit-1); return 0; } void explore(family f, int side, int pos, list* unsolved) { int count = 0; for(int i=0; i<f.len; i++) if(f.numrepeats[i]>0) count++; pos = pos % count; count = 0; for(int i=0; i<f.len; i++) { if(f.numrepeats[i]>0) { if(pos==count) { char str[MAXSTRING]; familystring(str, f); #ifdef PRINTEXPLORE printf("exploring %s as ", str); #endif for(int j=0; j<f.numrepeats[i]; j++) { family newf; familyinit(&newf); instancefamily(&newf, f, side, i); newf.digit[i+1] = f.repeats[i][j]; if(examine(&newf)) addtolist(unsolved, newf, 1); #ifdef PRINTEXPLORE familystring(str, newf); printf("%s, ", str); #endif clearfamily(&newf); } family copyf; familyinit(©f); copyfamily(©f, f); copyf.repeats[i] = NULL; copyf.numrepeats[i] = 0; if(examine(©f)) addtolist(unsolved, copyf, 1); #ifdef PRINTEXPLORE familystring(str, copyf); printf("%s\n", str); #endif clearfamily(©f); break; } count++; } } } int main(int argc, char** argv) { char filename[100]; sprintf(filename, "summary.txt"); FILE* summaryfile; #ifdef CLEARSUMMARY summaryfile = fopen(filename, "w"); fclose(summaryfile); #endif family f; familyinit(&f); int l, h, resume = 0; if(argc==1) { printf("Computes minimal primes for bases between l and h,\n"); printf("possibly along with a set of unsolved families\n"); printf("Usage: ./minimal l h\n"); printf("To resume base b from iteration i: ./minimal resume b i\n"); return 0; } else if(strcmp(argv[1], "resume")==0) { l = h = atoi(argv[2]); resume = 1; iter = atoi(argv[3]); } else if(argc==2) l = h = atoi(argv[1]); else { l = atoi(argv[1]); h = atoi(argv[2]); } #ifdef PRINTDATA mkdir("data", S_IRWXU); #endif #ifdef PRINTITER mkdir("iter", S_IRWXU); #endif for(base=l; base<=h; base++) { #ifdef PRINTSTATS printf("base %d...\n", base); #endif kernelinit(); list unsolved; listinit(&unsolved); if(!resume) { for(int i=0; i<base; i++) for(int j=0; j<base; j++) for(int k=0; k<base; k++) { char str[4]; if(i==0 && j==0) sprintf(str, "%c", digitchar(k)); else if(i==0) sprintf(str, "%c%c", digitchar(j), digitchar(k)); else sprintf(str, "%c%c%c", digitchar(i), digitchar(j), digitchar(k)); if(newminimal(str)) { char* newstr = malloc(4); memcpy(newstr, str, 4); addtokernel(newstr); } } for(int i=1; i<base; i++) for(int j=0; j<base; j++) { char* middles = calloc(base, sizeof(char)); int middlesize = 0; for(int k=0; k<base; k++) { char str[4]; sprintf(str, "%c%c%c", digitchar(i), digitchar(k), digitchar(j)); if(nosubword(str)) middles[middlesize++] = k; } if(middlesize>0) { family f; familyinit(&f); adddigit(&f, i, middles, middlesize); adddigit(&f, j, NULL, 0); if(!hasdivisor(f)) { explore(f, 1, 0, &unsolved); } } else free(middles); } iter = 0; } else { char str[100]; sprintf(str, "iter/minimal-base%d-iter%d.txt", base, iter); FILE* in = fopen(str, "r"); char line[MAXSTRING]; while(fgets(line, MAXSTRING, in)!=NULL) { line[strlen(line)-1] = '\0'; char* newstr = malloc(strlen(line)+1); strcpy(newstr, line); addtokernel(newstr); #ifdef PRINTRESUME printf("added %s to kernel\n", line); #endif } fclose(in); sprintf(str, "iter/unsolved-base%d-iter%d.txt", base, iter); FILE* out = fopen(str, "r"); while(fgets(line, MAXSTRING, in)!=NULL) { family f; familyinit(&f); for(int i=0; i<strlen(line)-1; i++) { int digit; if(line[i]=='{') digit = 255; else digit = invdigitchar(line[i]); if(line[i]!='{' && line[i+1]!='{') { adddigit(&f, digit, NULL, 0); } else { int k = strchr(line+i+1, '}')-(line+i+1)+(line[i]=='{'?1:0)-1; char* middles = calloc(k, sizeof(char)); for(int j=i+2-(line[i]=='{'?1:0); j<k+i+2-(line[i]=='{'?1:0); j++) { middles[j-(i+2-(line[i]=='{'?1:0))] = invdigitchar(line[j]); } adddigit(&f, digit, middles, k); i = k+i+2-(line[i]=='{'?1:0)+1; } } addtolist(&unsolved, f, 2); #ifdef PRINTRESUME familystring(str, f); printf("added %s to unknown list\n", str); #endif clearfamily(&f); } fclose(out); iter++; } for(;; iter++) { if(!onlysimple(unsolved)) { int didsplit = 1; int splititer = 0; while(didsplit) { didsplit = 0; list oldlist; copylist(&oldlist, unsolved); clearlist(&unsolved); for(int j=0; j<oldlist.size; j++) didsplit |= split(&(oldlist.fam[j]), &unsolved, oldlist.split[j]); clearlist(&oldlist); removedupes(&unsolved); copylist(&oldlist, unsolved); clearlist(&unsolved); for(int j=0; j<oldlist.size; j++) if(oldlist.split[j]==0 || examine(&(oldlist.fam[j]))) addtolist(&unsolved, oldlist.fam[j], oldlist.split[j]); clearlist(&oldlist); removedupes(&unsolved); copylist(&oldlist, unsolved); clearlist(&unsolved); for(int j=0; j<oldlist.size; j++) didsplit |= split2(&(oldlist.fam[j]), &unsolved, oldlist.split[j]); clearlist(&oldlist); removedupes(&unsolved); copylist(&oldlist, unsolved); clearlist(&unsolved); for(int j=0; j<oldlist.size; j++) if(oldlist.split[j]==0 || examine(&(oldlist.fam[j]))) addtolist(&unsolved, oldlist.fam[j], oldlist.split[j]); clearlist(&oldlist); removedupes(&unsolved); splititer++; #ifdef PRINTSTATS printf("base %d\titeration %d\tsplit %d\tsize %d\tremain %d\n", base, iter, splititer, K.size, unsolved.size); #endif } } else break; list oldlist; copylist(&oldlist, unsolved); clearlist(&unsolved); for(int j=0; j<oldlist.size; j++) explore(oldlist.fam[j], iter%2, iter, &unsolved); clearlist(&oldlist); removedupes(&unsolved); #ifdef PRINTUNSOLVED printf("Unsolved families after explore:\n"); printlist(unsolved); #endif #ifdef PRINTITER char filename[100]; sprintf(filename, "iter/unsolved-base%d-iter%d.txt", base, iter); FILE* out = fopen(filename, "w"); for(int j=0; j<unsolved.size; j++) { char str[MAXSTRING]; familystring(str, unsolved.fam[j]); fprintf(out, "%s\n", str); } fclose(out); filename[100]; sprintf(filename, "iter/minimal-base%d-iter%d.txt", base, iter); out = fopen(filename, "w"); for(int j=0; j<K.size; j++) { fprintf(out, "%s\n", K.primes[j]); } fclose(out); #endif if(unsolved.size==0) break; } kernel temp; temp.size = 0; temp.primes = NULL; for(int i=0; i<K.size; i++) if(nosubwordskip(K.primes[i], i)) { int size = ++temp.size; temp.primes = realloc(temp.primes, size*sizeof(char*)); temp.primes[size-1] = malloc(strlen(K.primes[i])+1); strcpy(temp.primes[size-1], K.primes[i]); } clearkernel(); K = temp; #ifdef PRINTDATA sprintf(filename, "data/minimal.%d.txt", base); FILE* kernelfile = fopen(filename, "w"); for(int i=0; i<K.size; i++) fprintf(kernelfile, "%s\n", K.primes[i]); fclose(kernelfile); #endif #ifdef PRINTSUMMARY sprintf(filename, "summary.txt"); summaryfile = fopen(filename, "a"); fprintf(summaryfile, "BASE %d:\n", base); fprintf(summaryfile, "\tSize:\t%d\n", K.size); int width = strlen(K.primes[0]); for(int i=1; i<K.size; i++) if(width<strlen(K.primes[i])) width = strlen(K.primes[i]); fprintf(summaryfile, "\tWidth:\t%d\n", width); fprintf(summaryfile, "\tRemain:\t%d\n", unsolved.size); fclose(summaryfile); #endif #ifdef PRINTDATA if(unsolved.size>0) { sprintf(filename, "data/unsolved.%d.txt", base); FILE* unsolvedfile = fopen(filename, "w"); for(int i=0; i<unsolved.size; i++) { char str[MAXSTRING]; if(issimple(unsolved.fam[i])) simplefamilystring(str, unsolved.fam[i]); else familystring(str, unsolved.fam[i]); fprintf(unsolvedfile, "%s\n", str); } fclose(unsolvedfile); } #endif clearkernel(); clearlist(&unsolved); } free(pr); return 0; } [/CODE] (we should first make data up to simple families (i.e. only simple families remain), see [URL="https://github.com/curtisbright/mepn-data/commit/7acfa0656d3c6b759f95a031f475a30f7664a122"]https://github.com/curtisbright/mepn-data/commit/7acfa0656d3c6b759f95a031f475a30f7664a122[/URL] for the example of the original minimal prime problem (i.e. prime > base is not required), and find the smallest prime in the remain simple families, see [URL="https://github.com/curtisbright/mepn-data/commit/4e524f26e39cc3df98f017e8106720ba4588e981"]https://github.com/curtisbright/mepn-data/commit/4e524f26e39cc3df98f017e8106720ba4588e981[/URL] and [URL="https://github.com/curtisbright/mepn-data/commit/f238288fac40d97a85d7cc707367cc91cdf75ec9"]https://github.com/curtisbright/mepn-data/commit/f238288fac40d97a85d7cc707367cc91cdf75ec9[/URL] and [URL="https://github.com/curtisbright/mepn-data/commit/e6b2b806f341e9dc5b96662edba2caf3220c98b7"]https://github.com/curtisbright/mepn-data/commit/e6b2b806f341e9dc5b96662edba2caf3220c98b7[/URL] for the example of the original minimal prime problem (i.e. prime > base is not required), base 29 required some additional strategies, see [URL="https://raw.githubusercontent.com/curtisbright/mepn-data/master/report/report.tex"]https://raw.githubusercontent.com/curtisbright/mepn-data/master/report/report.tex[/URL] and [URL="https://github.com/curtisbright/mepn-data/commits/master?after=dfd73217eb03e6889e63769eda77bcf739922ef3+244&branch=master"]https://github.com/curtisbright/mepn-data/commits/master?after=dfd73217eb03e6889e63769eda77bcf739922ef3+244&branch=master[/URL] for the example of the original minimal prime problem (i.e. prime > base is not required), if b+1 has >=3 distinct prime factors (i.e. [URL="https://oeis.org/A001221"]A001221[/URL](b+1) >= 3), then base b require these additional strategies, the smallest such base b is exactly 29) |
[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) |
* Case (5,1):
** [B]51[/B] is prime, and thus the only minimal prime in this family. * Case (5,3): ** [B]53[/B] is prime, and thus the only minimal prime in this family. * Case (5,5): ** Since 51, 53, 57, 15, 35, 45, 65, 75 are primes, we only need to consider the family 5{0,2,5}5 (since any digits 1, 3, 4, 6, 7 between them will produce smaller primes) *** Since 225, 255, [B]5205[/B] are primes, we only need to consider the families 5{0,5}5 and 5{0,5}25 (since any digits combo 20, 22, 25 between them will produce smaller primes) **** All numbers of the form 5{0,5}5 are divisible by 5, thus cannot be prime. **** For the 5{0,5}25 family, since [B]500025[/B] and [B]505525[/B] are primes, we only need to consider the number 500525 the families 5{5}25, 5{5}025, 5{5}0025, 5{5}0525, 5{5}00525, 5{5}05025 (since any digits combo 000, 055 between (5,25) will produce smaller primes) ***** 500525 is not prime. ***** The smallest prime of the form 5{5}25 is [B]555555555555525[/B] ***** The smallest prime of the form 5{5}025 is [B]55555025[/B] ***** The smallest prime of the form 5{5}0025 is 5555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555550025 (not minimal prime, since 55555025 and 555555555555525 are primes) ***** The smallest prime of the form 5{5}0525 is [B]5550525[/B] ***** The smallest prime of the form 5{5}00525 is [B]5500525[/B] ***** The smallest prime of the form 5{5}05025 is 5555555555555555555555505025 (not minimal prime, since 5550525, 55555025, and 555555555555525 are primes) * Case (5,7): ** [B]57[/B] is prime, and thus the only minimal prime in this family. |
* Case (6,1):
** Since 65, 21, 51, [B]631[/B], [B]661[/B] are primes, we only need to consider the family 6{0,1,4,7}1 (since any digits 2, 3, 5, 6 between them will produce smaller primes) *** Since 111, 141, 401, 471, 701, 711, [B]6101[/B], [B]6441[/B] are primes, we only need to consider the families 6{0}0{0,1,4,7}1, 6{0,4}1{7}1, 6{0,7}4{1}1, 6{0,1,7}7{4,7}1 (since any digits combo 11, 14, 40, 47, 70, 71, 10, 44 between them will produce smaller primes) **** For the 6{0}0{0,1,4,7}1 family, since 6007 is prime, we only need to consider the families 6{0}0{0,1,4}1 and 60{1,4,7}7{0,1,4,7}1 (since any digits combo 1007 between (6,1) will produce smaller primes) ***** For the 6{0}0{0,1,4}1 family, since 111, 141, 401, 6101, 6441, [B]60411[/B] are primes, we only need to consider the families 6{0}1, 6{0}11, 6{0}41 (since any digits combo 10, 11, 14, 40, 41, 44 between (6{0}0,1) will produce smaller primes) ****** All numbers of the form 6{0}1 are divisible by 7, thus cannot be prime. ****** All numbers of the form 6{0}11 are divisible by 3, thus cannot be prime. ****** All numbers of the form 6{0}41 are divisible by 3, thus cannot be prime. ***** For the 60{1,4,7}7{0,1,4,7}1 family, since 701, 711, [B]60741[/B] are primes, we only need to consider the family 60{1,4,7}7{7}1 (since any digits 0, 1, 4 between (60{1,4,7}7,1) will produce smaller primes) ***** Since 471, [B]60171[/B] is prime, we only need to consider the family 60{7}1 (since any digits 1, 4 between (60,7{7}1) will produce smaller primes) ****** All numbers of the form 60{7}1 are divisible by 7, thus cannot be prime. **** For the 6{0,4}1{7}1 family, since 417, 471 are primes, we only need to consider the families 6{0}1{7}1 and 6{0,4}11 ***** For the 6{0}1{7}1 family, since [B]60171[/B] is prime, and thus the only minimal prime in the family 6{0}1{7}1. ***** For the 6{0,4}11 family, since 401, 6441, [B]60411[/B] are primes, we only need to consider the number 6411 and the family 6{0}11 ****** 6411 is not prime. ****** All numbers of the form 6{0}11 are divisible by 3, thus cannot be prime. **** For the 6{0,7}4{1}1 family, since [B]60411[/B] is prime, we only need to consider the families 6{7}4{1}1 and 6{0,7}41 ***** For the 6{7}4{1}1 family, since 111, 6777 are primes, we only need to consider the numbers 641, 6411, 6741, 67411, 67741, 677411 ****** None of 641, 6411, 6741, 67411, 67741, 677411 are primes. ***** For the 6{0,7}41 family, since 701, 6777, [B]60741[/B] are primes, we only need to consider the families 6{0}41 and the numbers 6741, 67741 (since any digits combo 07, 70, 777 between (6,41) will produce smaller primes) ****** All numbers of the form 6{0}41 are divisible by 3, thus cannot be prime. ****** Neither of 6741, 67741 are primes. ***** For the 6{0,1,7}7{4,7}1 family, since 747 is prime, we only need to consider the families 6{0,1,7}7{4}1, 6{0,1,7}7{7}1, 6{0,1,7}7{7}{4}1 (since any digits combo 47 between (6{0,1,7}7,1) will produce smaller primes) ****** For the 6{0,1,7}7{4}1 family, since 6441 is prime, we only need to consider the families 6{0,1,7}71 and 6{0,1,7}741 (since any digits combo 44 between (6{0,1,7}7,1) will produce smaller primes) ******* For the 6{0,1,7}71 family, since all numbers of the form 6{0,7}71 are divisible by 7 and cannot be prime, and 111 is prime (thus, any digits combo 11 between (6,71) will produce smaller primes), we only need to consider the family 6{0,7}1{0,7}71 ******** Since 717 and [B]60171[/B] are primes, we only need to consider the family 61{0,7}71 (since any digit combo 0, 7 between (6,1{0,7}71) will produce smaller primes) ********* Since 177 and 6101 are primes, we only need to consider the number 6171 (since any digit combo 0, 7 between (61,71) will produce smaller primes) ********** 6171 is not prime. ****** All numbers in the 6{0,1,7}7{7}1 or 6{0,1,7}7{7}{4}1 families are also in the 6{0,1,7}7{4}1 family, thus these two families cannot have more minimal primes. |
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Upload past file, the set is not complete for bases >=7, I want to complete them.
I know some primes in the set which is not listed: * base 7: 33333333333333331 * base 8: 77774444441, 7777777777771, 555555555555525, 4[SUB]220[/SUB]7 * base 10: 555555555551 * base 11: A0000000001, A0014444444, 40000000A0041, A000000014444, A044444444441, A144444444411, 40000000000401, A0000044444441, A00000000444441, 11111111111111111, 14444444444441111, 44444444444444111, A1444444444444444, A9999999999999996, 1444444444444444444, 4000000000000000A041, A999999999999999999999, A44444444444444444444444441, 40000000000000000000000000041, 440000000000000000000000000001, 999999999999999999999999999999991, 444444444444444444444444444444444444444444441 (they are in the searching where single-digit primes are included, but this puzzle does not include single-digit primes) * base 13: 80[SUB]32017[/SUB]111 * base 14: 4D[SUB]19698[/SUB] * base 15: DE[SUB]14[/SUB] * base 17: 74[SUB]4904[/SUB] (this problem is much harder than the original minimal prime (where single-digit primes are included), see post [URL="https://mersenneforum.org/showpost.php?p=564315&postcount=57"]https://mersenneforum.org/showpost.php?p=564315&postcount=57[/URL]) |
Found a minimal prime (start with 2 digits) in base 13: 7[SUB]1504[/SUB]1, which equals (7*13^1505-79)/12
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Consider the "simplest" families x{y} and {x}y, where x,y are base b digits
Necessary conditions are gcd(x,y) = 1, gcd(y,b) = 1 [CODE] b, x, y, smallest prime 2, {1}, 1: 3 2, 1, {1}: 3 3, {1}, 1: 13 3, 1, {1}: 13 3, {1}, 2: 5 3, 1, {2}: 5 3, {2}, 1: 7 3, 2, {1}: 7 4, {1}, 1: 5 4, 1, {1}: 5 4, {1}, 3: 7 4, 1, {3}: 7 4, {2}, 1: 41 4, 2, {1}: 37 4, {2}, 3: 11 4, 2, {3}: 11 4, {3}, 1: 13 4, 3, {1}: 13 5, {1}, 1: 31 5, 1, {1}: 31 5, {1}, 2: 7 5, 1, {2}: 7 5, {1}, 3: 0 5, 1, {3}: 43 5, {1}, 4: 0 5, 1, {4}: 1249 5, {2}, 1: 11 5, 2, {1}: 11 5, {2}, 3: 13 5, 2, {3}: 13 5, {3}, 1: 2341 5, 3, {1}: 0 5, {3}, 2: 17 5, 3, {2}: 17 5, {3}, 4: 19 5, 3, {4}: 19 5, {4}, 1: 3121 5, 4, {1}: 0 5, {4}, 3: 23 5, 4, {3}: 23 6, {1}, 1: 7 6, 1, {1}: 7 6, {1}, 5: 11 6, 1, {5}: 11 6, {2}, 1: 13 6, 2, {1}: 13 6, {2}, 5: 17 6, 2, {5}: 17 6, {3}, 1: 19 6, 3, {1}: 19 6, {3}, 5: 23 6, 3, {5}: 23 6, {4}, 1: 1033 6, 4, {1}: 151 6, {4}, 5: 29 6, 4, {5}: 29 6, {5}, 1: 31 6, 5, {1}: 31 7, {1}, 1: 2801 7, 1, {1}: 2801 7, {1}, 2: 401 7, 1, {2}: 457 7, {1}, 3: 59 7, 1, {3}: 73 7, {1}, 4: 11 7, 1, {4}: 11 7, {1}, 5: 61 7, 1, {5}: 89 7, {1}, 6: 13 7, 1, {6}: 13 7, {2}, 1: 113 7, 2, {1}: 743 7, {2}, 3: 17 7, 2, {3}: 17 7, {2}, 5: 19 7, 2, {5}: 19 7, {3}, 1: 116315256993601 7, 3, {1}: 7603 7, {3}, 2: 23 7, 3, {2}: 23 7, {3}, 4: 1201 7, 3, {4}: 179 7, {3}, 5: 173 7, 3, {5}: 9203 7, {4}, 1: 29 7, 4, {1}: 29 7, {4}, 3: 31 7, 4, {3}: 31 7, {4}, 5: 229 7, 4, {5}: 1657 7, {5}, 1: 281 7, 5, {1}: 8413470255870653 7, {5}, 2: 37 7, 5, {2}: 37 7, {5}, 3: 283 7, 5, {3}: 269 7, {5}, 4: 1999 7, 5, {4}: 277 7, {5}, 6: 41 7, 5, {6}: 41 7, {6}, 1: 43 7, 6, {1}: 43 7, {6}, 5: 47 7, 6, {5}: 47 8, {1}, 1: 73 8, 1, {1}: 73 8, {1}, 3: 11 8, 1, {3}: 11 8, {1}, 5: 13 8, 1, {5}: 13 8, {1}, 7: 79 8, 1, {7}: 127 8, {2}, 1: 17 8, 2, {1}: 17 8, {2}, 3: 19 8, 2, {3}: 19 8, {2}, 5: 149 8, 2, {5}: 173 8, {2}, 7: 23 8, 2, {7}: 23 8, {3}, 1: 1753 8, 3, {1}: 1609 8, {3}, 5: 29 8, 3, {5}: 29 8, {3}, 7: 31 8, 3, {7}: 31 8, {4}, 1: 76695841 8, 4, {1}: 284694975049 8, {4}, 3: 2339 8, 4, {3}: 283 8, {4}, 5: 37 8, 4, {5}: 37 8, {4}, 7: 21870014779720278736374332149114462520188534743847615898363462279537144492484599310778624146468224150373895489844303219383829573677353011540369291867378470695590964880740521967077028064041941947533607 8, 4, {7}: 20479 8, {5}, 1: 41 8, 5, {1}: 41 8, {5}, 3: 43 8, 5, {3}: 43 8, {5}, 7: 47 8, 5, {7}: 47 8, {6}, 1: 433 8, 6, {1}: 56657856797822194249 8, {6}, 5: 53 8, 6, {5}: 53 8, {6}, 7: 439 8, 6, {7}: 3583 8, {7}, 1: 549755813881 8, 7, {1}: 457 8, {7}, 3: 59 8, 7, {3}: 59 8, {7}, 5: 61 8, 7, {5}: 61 9, {1}, 1: 0 9, 1, {1}: 0 9, {1}, 2: 11 9, 1, {2}: 11 9, {1}, 4: 13 9, 1, {4}: 13 9, {1}, 5: 0 9, 1, {5}: 131 9, {1}, 7: 97 9, 1, {7}: 151 9, {1}, 8: 17 9, 1, {8}: 17 9, {2}, 1: 19 9, 2, {1}: 19 9, {2}, 5: 23 9, 2, {5}: 23 9, {2}, 7: 14767 9, 2, {7}: 0 9, {3}, 1: 271 9, 3, {1}: 0 9, {3}, 2: 29 9, 3, {2}: 29 9, {3}, 4: 31 9, 3, {4}: 31 9, {3}, 5: 0 9, 3, {5}: 293 9, {3}, 7: 277 9, 3, {7}: 313 9, {3}, 8: 0 9, 3, {8}: 0 9, {4}, 1: 37 9, 4, {1}: 37 9, {4}, 5: 41 9, 4, {5}: 41 9, {4}, 7: 43 9, 4, {7}: 43 9, {5}, 1: 36901 9, 5, {1}: 0 9, {5}, 2: 47 9, 5, {2}: 47 9, {5}, 4: 4099 9, 5, {4}: 172595827849 9, {5}, 7: 457 9, 5, {7}: 0 9, {5}, 8: 53 9, 5, {8}: 53 9, {6}, 1: 541 9, 6, {1}: 0 9, {6}, 5: 59 9, 6, {5}: 59 9, {6}, 7: 61 9, 6, {7}: 61 9, {7}, 1: 631 9, 7, {1}: 577 9, {7}, 2: 0 9, 7, {2}: 587 9, {7}, 4: 67 9, 7, {4}: 67 9, {7}, 5: 0 9, 7, {5}: 617 9, {7}, 8: 71 9, 7, {8}: 71 9, {8}, 1: 73 9, 8, {1}: 73 9, {8}, 5: 0 9, 8, {5}: 6287 9, {8}, 7: 79 9, 8, {7}: 79 10, {1}, 1: 11 10, 1, {1}: 11 10, {1}, 3: 13 10, 1, {3}: 13 10, {1}, 7: 17 10, 1, {7}: 17 10, {1}, 9: 19 10, 1, {9}: 19 10, {2}, 1: 2221 10, 2, {1}: 211 10, {2}, 3: 23 10, 2, {3}: 23 10, {2}, 7: 227 10, 2, {7}: 277 10, {2}, 9: 29 10, 2, {9}: 29 10, {3}, 1: 31 10, 3, {1}: 31 10, {3}, 7: 37 10, 3, {7}: 37 10, {4}, 1: 41 10, 4, {1}: 41 10, {4}, 3: 43 10, 4, {3}: 43 10, {4}, 7: 47 10, 4, {7}: 47 10, {4}, 9: 449 10, 4, {9}: 499 10, {5}, 1: 555555555551 10, 5, {1}: 511111 10, {5}, 3: 53 10, 5, {3}: 53 10, {5}, 7: 557 10, 5, {7}: 577 10, {5}, 9: 59 10, 5, {9}: 59 10, {6}, 1: 61 10, 6, {1}: 61 10, {6}, 7: 67 10, 6, {7}: 67 10, {7}, 1: 71 10, 7, {1}: 71 10, {7}, 3: 73 10, 7, {3}: 73 10, {7}, 9: 79 10, 7, {9}: 79 10, {8}, 1: 881 10, 8, {1}: 811 10, {8}, 3: 83 10, 8, {3}: 83 10, {8}, 7: 887 10, 8, {7}: 877 10, {8}, 9: 89 10, 8, {9}: 89 10, {9}, 1: 991 10, 9, {1}: 911 10, {9}, 7: 97 10, 9, {7}: 97 11, {1}, 1: 50544702849929377 11, 1, {1}: 50544702849929377 11, {1}, 2: 13 11, 1, {2}: 13 11, {1}, 3: 0 11, 1, {3}: 157 11, {1}, 4: 0 11, 1, {4}: 7783884238889124073 11, {1}, 5: 137 11, 1, {5}: 181 11, {1}, 6: 17 11, 1, {6}: 17 11, {1}, 7: 139 11, 1, {7}: 24889 11, {1}, 8: 19 11, 1, {8}: 19 11, {1}, 9: 0 11, 1, {9}: 229 11, {1}, 10: 0 11, 1, {10}: 241 11, {2}, 1: 23 11, 2, {1}: 23 11, {2}, 3: 354313 11, 2, {3}: 3061 11, {2}, 5: 269 11, 2, {5}: 0 11, {2}, 7: 29 11, 2, {7}: 29 11, {2}, 9: 31 11, 2, {9}: 31 11, {3}, 1: 397 11, 3, {1}: 0 11, {3}, 2: 4391 11, 3, {2}: 4259 11, {3}, 4: 37 11, 3, {4}: 37 11, {3}, 5: 401 11, 3, {5}: 0 11, {3}, 7: 85593501187 11, 3, {7}: 0 11, {3}, 8: 41 11, 3, {8}: 41 11, {3}, 10: 43 11, 3, {10}: 43 11, {4}, 1: 29156193474041220857161146715104735751776055777 11, 4, {1}: 0 11, {4}, 3: 47 11, 4, {3}: 47 11, {4}, 5: 5857 11, 4, {5}: 724729 11, {4}, 7: 114124668247 11, 4, {7}: 0 11, {4}, 9: 53 11, 4, {9}: 53 11, {5}, 1: 661 11, 5, {1}: 617 11, {5}, 2: 0 11, 5, {2}: 9212117 11, {5}, 3: 0 11, 5, {3}: 641 11, {5}, 4: 59 11, 5, {4}: 59 11, {5}, 6: 61 11, 5, {6}: 61 11, {5}, 7: 80527 11, 5, {7}: 0 11, {5}, 8: 0 11, 5, {8}: 701 11, {5}, 9: 0 11, 5, {9}: 86381 11, {6}, 1: 67 11, 6, {1}: 67 11, {6}, 5: 71 11, 6, {5}: 71 11, {6}, 7: 73 11, 6, {7}: 73 11, {7}, 1: 42811363313890182397 11, 7, {1}: 859 11, {7}, 2: 79 11, 7, {2}: 79 11, {7}, 3: 0 11, 7, {3}: 883 11, {7}, 4: 0 11, 7, {4}: 108343 11, {7}, 5: 929 11, 7, {5}: 907 11, {7}, 6: 83 11, 7, {6}: 83 11, {7}, 8: 150051217 11, 7, {8}: 114199 11, {7}, 9: 0 11, 7, {9}: 115663 11, {7}, 10: 0 11, 7, {10}: 967 11, {8}, 1: 89 11, 8, {1}: 89 11, {8}, 3: 4447933850793785179 11, 8, {3}: 11047 11, {8}, 5: 1061 11, 8, {5}: 0 11, {8}, 7: 1063 11, 8, {7}: 11579 11, {8}, 9: 97 11, 8, {9}: 97 11, {9}, 1: 20902638977899027326901591016678209 11, 9, {1}: 0 11, {9}, 2: 101 11, 9, {2}: 101 11, {9}, 4: 103 11, 9, {4}: 103 11, {9}, 5: 1193 11, 9, {5}: 0 11, {9}, 7: 2122152919 11, 9, {7}: 0 11, {9}, 8: 107 11, 9, {8}: 107 11, {9}, 10: 109 11, 9, {10}: 109 11, {10}, 1: 1321 11, 10, {1}: 0 11, {10}, 3: 113 11, 10, {3}: 113 11, {10}, 7: 1327 11, 10, {7}: 0 11, {10}, 9: 14639 11, 10, {9}: 80662724392413945103199 12, {1}, 1: 13 12, 1, {1}: 13 12, {1}, 5: 17 12, 1, {5}: 17 12, {1}, 7: 19 12, 1, {7}: 19 12, {1}, 11: 23 12, 1, {11}: 23 12, {2}, 1: 313 12, 2, {1}: 3613 12, {2}, 5: 29 12, 2, {5}: 29 12, {2}, 7: 31 12, 2, {7}: 31 12, {2}, 11: 3779 12, 2, {11}: 431 12, {3}, 1: 37 12, 3, {1}: 37 12, {3}, 5: 41 12, 3, {5}: 41 12, {3}, 7: 43 12, 3, {7}: 43 12, {3}, 11: 47 12, 3, {11}: 47 12, {4}, 1: 7537 12, 4, {1}: 7069 12, {4}, 5: 53 12, 4, 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{6}: 193 17, {11}, 7: 3373 17, 11, {7}: 955271 17, {11}, 8: 0 17, 11, {8}: 3323 17, {11}, 9: 0 17, 11, {9}: 965711 17, {11}, 10: 197 17, 11, {10}: 197 17, {11}, 12: 199 17, 11, {12}: 199 17, {11}, 13: 9394230696635382053176380469368734655867242678435691492562299088334773 17, 11, {13}: 3413 17, {11}, 14: 0 17, 11, {14}: 991811 17, {11}, 15: 0 17, 11, {15}: 3449 17, {11}, 16: 0 17, 11, {16}: 3467 17, {12}, 1: 3673 17, 12, {1}: 59263 17, {12}, 5: 3677 17, 12, {5}: 1378486138632359758323050626992747918650304829615263354995388341922232251095972398990848507942018139080087311 17, {12}, 7: 211 17, 12, {7}: 211 17, {12}, 11: 62639 17, 12, {11}: 434824684403093 17, {12}, 13: 88940907373 17, 12, {13}: 18191917 17, {13}, 1: 3768651696722334407412704432886748501027917638216745188121713071850567124051602405518201798458848401 17, 13, {1}: 635636818875898469533 17, {13}, 2: 223 17, 13, {2}: 223 17, {13}, 3: 0 17, 13, {3}: 1101433 17, {13}, 4: 0 17, 13, {4}: 1106653 17, {13}, 5: 27845915749943 17, 13, {5}: 3847 17, {13}, 6: 227 17, 13, {6}: 227 17, {13}, 7: 228154556301155739164141873957905004400241046704207 17, 13, {7}: 93736740613 17, {13}, 8: 229 17, 13, {8}: 229 17, {13}, 9: 0 17, 13, {9}: 3919 17, {13}, 10: 0 17, 13, {10}: 1137973 17, {13}, 11: 3989 17, 13, {11}: 1143193 17, {13}, 12: 233 17, 13, {12}: 233 17, {13}, 14: 160688404748616050182618301672566324918805941 17, 13, {14}: 5693449087 17, {13}, 15: 0 17, 13, {15}: 4027 17, {13}, 16: 0 17, 13, {16}: 1169293 17, {14}, 1: 239 17, 14, {1}: 239 17, {14}, 3: 241 17, 14, {3}: 241 17, {14}, 5: 4289 17, 14, {5}: 0 17, {14}, 9: 21120367 17, 14, {9}: 39353705070153506531713748224825668451495187254508108657151444786252940850766831171765242341910761541585017932281107 17, {14}, 11: 103764391931 17, 14, {11}: 0 17, {14}, 13: 251 17, 14, {13}: 251 17, {14}, 15: 533707265356695216704103124038332368542873946283142922292003783538887189897335797029414828786775114395339696927112171531204140647898504891296804256353517276617145553123762661125816798731498754318189024538693935214938297276615812780293 17, 14, {15}: 73387 17, {15}, 1: 4591 17, 15, {1}: 0 17, {15}, 2: 257 17, 15, {2}: 257 17, {15}, 4: 546208347402889 17, 15, {4}: 74923 17, {15}, 7: 4597 17, 15, {7}: 0 17, {15}, 8: 263 17, 15, {8}: 263 17, {15}, 11: 111176134211 17, 15, {11}: 0 17, {15}, 13: 4603 17, 15, {13}: 0 17, {15}, 14: 269 17, 15, {14}: 269 17, {15}, 16: 271 17, 15, {16}: 271 17, {16}, 1: 34271896307617 17, 16, {1}: 0 17, {16}, 3: 74443609190419550764562450397778200846849192983001551466849044370008879517232307105675227070196683723355515193456559323901778769141226118951876996802487398051974943265833071289084569071200666892787 17, 16, {3}: 3110633786280773828357619125469664392231273829727161657043436458261702025678677414669396669249 17, {16}, 5: 277 17, 16, {5}: 277 17, {16}, 7: 4903 17, 16, {7}: 0 17, {16}, 9: 281 17, 16, {9}: 281 17, {16}, 11: 283 17, 16, {11}: 283 17, {16}, 13: 4909 17, 16, {13}: 0 17, {16}, 15: 24137567 17, 16, {15}: 66886068539071498820247358361862720864806052666582265636907882027208271253 18, {1}, 1: 19 18, 1, {1}: 19 18, {1}, 5: 23 18, 1, {5}: 23 18, {1}, 7: 349 18, 1, {7}: 457 18, {1}, 11: 29 18, 1, {11}: 29 18, {1}, 13: 31 18, 1, {13}: 31 18, {1}, 17: 359 18, 1, {17}: 647 18, {2}, 1: 37 18, 2, {1}: 37 18, {2}, 5: 41 18, 2, {5}: 41 18, {2}, 7: 43 18, 2, {7}: 43 18, {2}, 11: 47 18, 2, {11}: 47 18, {2}, 13: 72025897 18, 2, {13}: 3198298525119427 18, {2}, 17: 53 18, 2, {17}: 53 18, {3}, 1: 18523 18, 3, {1}: 991 18, {3}, 5: 59 18, 3, {5}: 59 18, {3}, 7: 61 18, 3, {7}: 61 18, {3}, 11: 108038837 18, 3, {11}: 1181 18, {3}, 13: 67 18, 3, {13}: 67 18, {3}, 17: 71 18, 3, {17}: 71 18, {4}, 1: 73 18, 4, {1}: 73 18, {4}, 5: 1373 18, 4, {5}: 47321007179 18, {4}, 7: 79 18, 4, {7}: 79 18, {4}, 11: 83 18, 4, {11}: 83 18, {4}, 13: 1381 18, 4, {13}: 1543 18, {4}, 17: 89 18, 4, {17}: 89 18, {5}, 1: 30871 18, 5, {1}: 34130064295121260303 18, {5}, 7: 97 18, 5, {7}: 97 18, {5}, 11: 101 18, 5, {11}: 101 18, {5}, 13: 103 18, 5, {13}: 103 18, {5}, 17: 107 18, 5, {17}: 107 18, {6}, 1: 109 18, 6, {1}: 109 18, {6}, 5: 113 18, 6, {5}: 113 18, {6}, 7: 3889397851 18, 6, {7}: 412073923449193 18, {6}, 11: 2063 18, 6, {11}: 2153 18, {6}, 13: 37057 18, 6, {13}: 39451 18, {6}, 17: 2069 18, 6, {17}: 2267 18, {7}, 1: 127 18, 7, {1}: 127 18, {7}, 5: 131 18, 7, {5}: 131 18, {7}, 11: 137 18, 7, {11}: 137 18, {7}, 13: 139 18, 7, {13}: 139 18, {7}, 17: 2411 18, 7, {17}: 2591 18, {8}, 1: 49393 18, 8, {1}: 845983 18, {8}, 5: 149 18, 8, {5}: 149 18, {8}, 7: 151 18, 8, {7}: 151 18, {8}, 11: 889211 18, 8, {11}: 2801 18, {8}, 13: 157 18, 8, {13}: 157 18, {8}, 17: 2753 18, 8, {17}: 578415690713087 18, {9}, 1: 163 18, 9, {1}: 163 18, {9}, 5: 167 18, 9, {5}: 167 18, {9}, 7: 1000357 18, 9, {7}: 3049 18, {9}, 11: 173 18, 9, {11}: 173 18, {9}, 13: 55579 18, 9, {13}: 3163 18, {9}, 17: 179 18, 9, {17}: 179 18, {10}, 1: 181 18, 10, {1}: 181 18, {10}, 7: 3968612127339681427 18, 10, {7}: 3373 18, {10}, 11: 191 18, 10, {11}: 191 18, {10}, 13: 193 18, 10, {13}: 193 18, {10}, 17: 197 18, 10, {17}: 197 18, {11}, 1: 199 18, 11, {1}: 199 18, {11}, 5: 3767 18, 11, {5}: 3659 18, {11}, 7: 3769 18, 11, {7}: 3697 18, {11}, 13: 211 18, 11, {13}: 211 18, {11}, 17: 3779 18, 11, {17}: 132239526911 18, {12}, 1: 140018322601 18, 12, {1}: 3907 18, {12}, 5: 74093 18, 12, {5}: 71699 18, {12}, 7: 223 18, 12, {7}: 223 18, {12}, 11: 227 18, 12, {11}: 227 18, {12}, 13: 229 18, 12, {13}: 229 18, {12}, 17: 233 18, 12, {17}: 233 18, {13}, 1: 4447 18, 13, {1}: 4231 18, {13}, 5: 239 18, 13, {5}: 239 18, {13}, 7: 241 18, 13, {7}: 241 18, {13}, 11: 4457 18, 13, {11}: 4421 18, {13}, 17: 251 18, 13, {17}: 251 18, {14}, 1: 4789 18, 14, {1}: 26565103 18, {14}, 5: 257 18, 14, {5}: 257 18, {14}, 11: 263 18, 14, {11}: 263 18, {14}, 13: 4801 18, 14, {13}: 4783 18, {14}, 17: 269 18, 14, {17}: 269 18, {15}, 1: 271 18, 15, {1}: 271 18, {15}, 7: 277 18, 15, {7}: 277 18, {15}, 11: 281 18, 15, {11}: 281 18, {15}, 13: 283 18, 15, {13}: 283 18, {15}, 17: 5147 18, 15, {17}: 30233087 18, {16}, 1: 32011489 18, 16, {1}: 30344239 18, {16}, 5: 293 18, 16, {5}: 293 18, {16}, 7: 5479 18, 16, {7}: 95713 18, {16}, 11: 5483 18, 16, {11}: 5393 18, {16}, 13: 32011501 18, 16, {13}: 5431 18, {16}, 17: 98801 18, 16, {17}: 5507 18, {17}, 1: 307 18, 17, {1}: 307 18, {17}, 5: 311 18, 17, {5}: 311 18, {17}, 7: 313 18, 17, {7}: 313 18, {17}, 11: 317 18, 17, {11}: 317 18, {17}, 13: 5827 18, 17, {13}: 9770144707511081415118442597789015238720654947319882836100223544506052645981243442054558121499672250712069138857313219 [/CODE] |
[QUOTE=sweety439;566066]Consider the "simplest" families x{y} and {x}y, where x,y are base b digits
Necessary conditions are gcd(x,y) = 1, gcd(y,b) = 1 Zeros means there are no such prime of this family with <= 3000 digits (include the case that this family is ruled out to contain only composites)[/QUOTE] For the zeros in the table: (for bases <=16) Base 5: {1}3: covering set {2,3} {1}4: covering set {2,3} 3{1}: covering set {2,3} 4{1}: covering set {2,3} Base 9: {1}1: full algebra factors: (9^n-1)/8 = (3^n-1)*(3^n+1)/8 1{1}: full algebra factors: (9^n-1)/8 = (3^n-1)*(3^n+1)/8 {1}5: covering set {2,5} 2{7}: covering set {2,5} 3{1}: full algebra factors: (25*9^n-1)/8 = (5*3^n-1)*(5*3^n+1)/8 {3}5: covering set {2,5} {3}8: covering set {2,5} 3{8}: full algebra factors: 4*9^n-1 = (2*9^n-1)*(2*9^n+1) 5{1}: covering set {2,5} 5{7}: covering set {2,5} 6{1}: covering set {2,5} {7}2: covering set {2,5} {7}5: covering set {2,5} {8}5: full algebra factors: 9^n-4 = (3^n-2)*(3^n+2) Base 11: {1}3: covering set {2,3} {1}4: covering set {2,3} {1}9: covering set {2,3} {1}A: covering set {2,3} 2{5}: covering set {2,3} 3{1}: covering set {2,3} 3{5}: covering set {2,3} 3{7}: covering set {2,3} 4{1}: covering set {2,3} 4{7}: covering set {2,3} {5}2: covering set {2,3} {5}3: covering set {2,3} 5{7}: (unsolved family) ------------------------------------------------------- {5}8: covering set {2,3} {5}9: covering set {2,3} {7}3: covering set {2,3} {7}4: covering set {2,3} {7}9: covering set {2,3} {7}A: covering set {2,3} 8{5}: covering set {2,3} 9{1}: covering set {2,3} 9{5}: covering set {2,3} 9{7}: covering set {2,3} A{1}: covering set {2,3} A{7}: covering set {2,3} Base 13: {1}5: covering set {2,5,17} {1}7: covering set {2,7} {1}8: covering set {2,7} 2{9}: covering set {2,7} {3}7: covering set {2,7} {3}A: covering set {2,7} 4{B}: covering set {2,7} {5}7: covering set {2,7} {5}C: covering set {2,7} 7{1}: covering set {2,7} 7{3}: covering set {2,7} 7{5}: covering set {2,7} 7{9}: covering set {2,7} 7{B}: covering set {2,7} 8{1}: covering set {2,7} {9}2: covering set {2,7} 9{5}: (unsolved family) ------------------------------------------------------- {9}7: covering set {2,7} A{3}: covering set {2,7} {B}4: covering set {2,7} {B}7: covering set {2,7} C{5}: covering set {2,7} Base 14: {2}5: covering set {3,5} 3{D}: covering set {3,5} {4}9: covering set {3,5} 4{D}: (smallest prime is 4D[SUB]19698[/SUB] = 5*14^19698-1) 5{B}: covering set {3,5} 6{1}: covering set {3,5} 6{B}: covering set {3,5} {8}3: covering set {3,5} {8}5: covering set {3,5} 8{D}: partial algebra factors: 9*14^n-1 = (3*14^(n/2)-1)*(3*14^(n/2)+1) for even n, divisible by 5 for odd n A{1}: covering set {3,5} A{D}: covering set {3,5} B{1}: partial algebra factors: (144*14^n-1)/13 = (12*14^(n/2)-1)*(12*14^(n/2)+1)/13 for even n, divisible by 5 for odd n {B}5: covering set {3,5} {D}3: covering set {3,5} {D}5: partial algebra factors: 14^n-9 = (14^(n/2)-3)*(14^(n/2)+3) for even n, divisible by 5 for odd n Base 16: 1{5}: full algebra factors: (4*16^n-1)/3 = (2*4^n-1)*(2*4^n+1)/3 {4}1: full algebra factors: (4*16^n-49)/15 = (2*4^n-7)*(2*4^n+7)/15 {4}D: covering set {3,7,13} 7{3}: full algebra factors: (36*16^n-1)/5 = (6*4^n-1)*(6*4^n+1)/5 8{1}: full algebra factors: (121*16^n-1)/15 = (11*4^n-1)*(11*4^n+1)/15 8{5}: full algebra factors: (25*16^n-1)/3 = (5*4^n-1)*(5*4^n+1)/3 {8}F: covering set {3,7,13} 8{F}: full algebra factors: 9*16^n-1 = (3*4^n-1)*(3*4^n+1) {C}B: full algebra factors: (4*16^n-9)/5 = (2*4^n-3)*(2*4^n+3)/5 {C}D: full algebra factors: (4*16^n+1)/5 = (2*4^n-2*2^n+1)*(2*4^n+2*2^n+1) D{1}: full algebra factors: (196*16^n-1)/15 = (14*4^n-1)*(14*4^n+1)/15 D{B}: (unsolved family) ------------------------------------------------------- E{1}: covering set {3,7,13} {F}7: full algebra factors: 16^n-9 = (4^n-3)*(4^n+3) |
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This file is the smallest prime (not include x or y themselves) in given simple family x{y} or {x}y (where x,y are base b digits) in given base 2<=b<=24, where gcd(x,y) = 1, gcd(y,b) = 1 (searched up to 5000 base b digits, 0 if no such prime found (include the case such that x{y} or {x}y proven composite by all or partial algebra factors)
format of file: b,x,{y}: smallest prime of the form x{y} in base b b,{x},y: smallest prime of the form {x}y in base b such primes are generalized near-repdigit primes base b already excluded families x{y} and {x}y with NUMERICAL covering set (e.g. {1}3, {1}4, 3{1}, 4{1} in base 5) Such primes are ALWAYS minimal prime (start with 2 digits) in base b, except when the repeating digit (i.e. y in x{y}, or x in {x}y) is 1 and base b has generalized repunit primes (i.e. all digits are 1) smaller than the prime (in base b = 9, 25, 32, 49, 64, 81, 121, 125, 144, 169, 216, 225, 243, 289, 324, 343, 361, 441, 484, 529, 625, ..., no generalized repunit primes exist, thus in these bases b, such primes are always minimal primes (start with 2 digits) in base b) Also, * in base 35, all such primes with <= 313 digits are minimal primes (start with 2 digits) * in base 39, all such primes with <= 349 digits are minimal primes (start with 2 digits) * in base 47, all such primes with <= 127 digits are minimal primes (start with 2 digits) * in base 51, all such primes with <= 4229 digits are minimal primes (start with 2 digits) * in base 91, all such primes with <= 4421 digits are minimal primes (start with 2 digits) * in base 92, all such primes with <= 439 digits are minimal primes (start with 2 digits) * in base 124, all such primes with <= 599 digits are minimal primes (start with 2 digits) * in base 135, all such primes with <= 1171 digits are minimal primes (start with 2 digits) * in base 139, all such primes with <= 163 digits are minimal primes (start with 2 digits) * in base 142, all such primes with <= 1231 digits are minimal primes (start with 2 digits) * in base 152, all such primes with <= 270217 digits are minimal primes (start with 2 digits) * in base 171, all such primes with <= 181 digits are minimal primes (start with 2 digits) * in base 174, all such primes with <= 3251 digits are minimal primes (start with 2 digits) * in base 182, all such primes with <= 167 digits are minimal primes (start with 2 digits) * in base 183, all such primes with <= 223 digits are minimal primes (start with 2 digits) * in base 184, all such primes with <= 16703 digits are minimal primes (start with 2 digits) * in base 185, all such primes with <= 66337 (at least) digits are minimal primes (start with 2 digits) * in base 199, all such primes with <= 577 digits are minimal primes (start with 2 digits) * in base 200, all such primes with <= 17807 digits are minimal primes (start with 2 digits) * in base 201, all such primes with <= 271 digits are minimal primes (start with 2 digits) |
In fact, all these primes are minimal primes (start with 2 digits) base b:
* Smallest generalized repunit prime base b (if exists) * The primes for all k<b for [URL="http://www.noprimeleftbehind.net/crus/Sierp-conjectures.htm"]CRUS Sierpinski conjecture[/URL] base b * The primes for all k<b for [URL="http://www.noprimeleftbehind.net/crus/Riesel-conjectures.htm"]CRUS Riesel conjecture[/URL] base b * Smallest generalized near-repdigit primes base b of the form x{y} or {x}y for all (x,y) digit pair (if exists) Since .... * Generalized repunit numbers base b are 111...111 in base b * k<b for [URL="http://www.noprimeleftbehind.net/crus/Sierp-conjectures.htm"]CRUS Sierpinski conjecture[/URL] base b are [k]000...0001 in base b * k<b for [URL="http://www.noprimeleftbehind.net/crus/Riesel-conjectures.htm"]CRUS Riesel conjecture[/URL] base b are [k-1][b-1][b-1][b-1]...[b-1][b-1][b-1] in base b * Generalized near-repdigit numbers base b of the form x{y} or {x}y are [x][y][y][y]...[y][y][y] or [x][x][x]...[x][x][x][y] in base b |
I think that the "minimal primes (start with 2 digits) problem" for all bases b>90 will never be proven (when searched to 1M or 1G or even 1T base b digits, and even when we allow probable primes in place of proven primes.), like CRUS S/R280, S/R511, S/R855, and S/R910 problems and the "Sierpinski/Riesel twin prime conjecture" (the conjecture that 237 is the smallest k divisible by 3 such that k*2^n+-1 are not twin primes for all n>=1), for more information, see [URL="https://mersenneforum.org/showpost.php?p=564315&postcount=57"]this post[/URL]
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[QUOTE=sweety439;566267]This file is the smallest prime (not include x or y themselves) in given simple family x{y} or {x}y (where x,y are base b digits) in given base 2<=b<=24, where gcd(x,y) = 1, gcd(y,b) = 1 (searched up to 5000 base b digits, 0 if no such prime found (include the case such that x{y} or {x}y proven composite by all or partial algebra factors)
format of file: b,x,{y}: smallest prime of the form x{y} in base b b,{x},y: smallest prime of the form {x}y in base b such primes are generalized near-repdigit primes base b already excluded families x{y} and {x}y with NUMERICAL covering set (e.g. {1}3, {1}4, 3{1}, 4{1} in base 5)[/QUOTE] extended data to base 36 |
[QUOTE=sweety439;566014]* Case (5,1):
** [B]51[/B] is prime, and thus the only minimal prime in this family. * Case (5,3): ** [B]53[/B] is prime, and thus the only minimal prime in this family. * Case (5,5): ** Since 51, 53, 57, 15, 35, 45, 65, 75 are primes, we only need to consider the family 5{0,2,5}5 (since any digits 1, 3, 4, 6, 7 between them will produce smaller primes) *** Since 225, 255, [B]5205[/B] are primes, we only need to consider the families 5{0,5}5 and 5{0,5}25 (since any digits combo 20, 22, 25 between them will produce smaller primes) **** However, all numbers of the form 5{0,5}5 are divisible by 5, thus cannot be prime, therefore, there is no minimal primes in this family. **** For the 5{0,5}25 family, since [B]500025[/B] and [B]505525[/B] are primes, we only need to consider the number 500525 the families 5{5}25, 5{5}025, 5{5}0025, 5{5}0525, 5{5}00525, 5{5}05025 (since any digits combo 000, 055 between (5,25) will produce smaller primes) ***** However, 500525 is not prime, therefore, there is no minimal primes in this family. ***** The smallest prime of the form 5{5}25 is [B]555555555555525[/B] ***** The smallest prime of the form 5{5}025 is [B]55555025[/B] ***** For the 5{5}0025 family, since 55555025 is prime, we only need to consider the numbers 50025, 550025, 5550025, 55550025 (since any digit combo 5555 between (5,0025) will produce smaller primes) ****** However, none of them are primes, therefore, there is no minimal primes in this family. ***** The smallest prime of the form 5{5}0525 is [B]5550525[/B] ***** The smallest prime of the form 5{5}00525 is [B]5500525[/B] ***** For the 5{5}05025 family, since 55555025 is prime, we only need to consider the numbers 505025, 5505025, 55505025 (since any digit combo 555 between (5,05025) will produce smaller primes) ****** However, none of them are primes, therefore, there is no minimal primes in this family. * Case (5,7): ** [B]57[/B] is prime, and thus the only minimal prime in this family.[/QUOTE] * Case (6,3): ** Since 65, 13, 23, 53, 73, [B]643[/B] are primes, we only need to consider the family 6{0,3,6}3 (since any digits 1, 2, 4, 5, 7 between them will produce smaller primes) *** All numbers of the form 6{0,3,6}3 are divisible by 3, thus cannot be prime. * Case (6,5): ** [B]65[/B] is prime, and thus the only minimal prime in this family. * Case (6,7): ** Since 65, 27, 37, 57, [B]667[/B] are primes, we only need to consider the family 6{0,1,4,7}7 (since any digits 2, 3, 5, 6 between them will produce smaller primes) *** Since 107, 117, 147, 177, 407, 417, 717, 747, [B]6007[/B], [B]6477[/B], [B]6707[/B], [B]6777[/B] are primes, we only need to consider the families 60{1,4,7}7, 6{0}17, 6{0,4}4{4}7, 6{0}77 (since any digits combo 00, 10, 11, 14, 17, 40, 41, 47, 70, 71, 74, 77 between them will produce smaller primes) **** All numbers of the form 6{0}17 or 6{0}77 are divisible by 3, thus cannot be prime. **** For the 60{1,4,7}7 family, since 117, 147, 177, 417, 6477, 717, 747, 6777 are primes, we only need to consider the numbers 6017, 6047, 6077 and the family 60{4}7 (since any digit combo 11, 14, 17, 41, 47, 71, 74, 77 between (60,7) will produce smaller primes) ***** None of 6017, 6047, 6077 are primes. **** All numbers of the form 60{4}7 are divisible by 21, thus cannot be prime. **** For the 6{0,4}4{4}7 family, since 6007 and 407 are primes, we only need to consider the families 6{4}7 and 60{4}7 (since any digit combo 00, 40 between (6,4{4}7) will produce smaller primes) ***** All numbers of the form 6{4}7 are divisible by 3, 5, or 15, thus cannot be prime. ***** All numbers of the form 60{4}7 are divisible by 21, thus cannot be prime. |
[QUOTE=sweety439;566014]* Case (5,1):
** [B]51[/B] is prime, and thus the only minimal prime in this family. * Case (5,3): ** [B]53[/B] is prime, and thus the only minimal prime in this family. * Case (5,5): ** Since 51, 53, 57, 15, 35, 45, 65, 75 are primes, we only need to consider the family 5{0,2,5}5 (since any digits 1, 3, 4, 6, 7 between them will produce smaller primes) *** Since 225, 255, [B]5205[/B] are primes, we only need to consider the families 5{0,5}5 and 5{0,5}25 (since any digits combo 20, 22, 25 between them will produce smaller primes) **** However, all numbers of the form 5{0,5}5 are divisible by 5, thus cannot be prime, therefore, there is no minimal primes in this family. **** For the 5{0,5}25 family, since [B]500025[/B] and [B]505525[/B] are primes, we only need to consider the number 500525 the families 5{5}25, 5{5}025, 5{5}0025, 5{5}0525, 5{5}00525, 5{5}05025 (since any digits combo 000, 055 between (5,25) will produce smaller primes) ***** However, 500525 is not prime, therefore, there is no minimal primes in this family. ***** The smallest prime of the form 5{5}25 is [B]555555555555525[/B] ***** The smallest prime of the form 5{5}025 is [B]55555025[/B] ***** For the 5{5}0025 family, since 55555025 is prime, we only need to consider the numbers 50025, 550025, 5550025, 55550025 (since any digit combo 5555 between (5,0025) will produce smaller primes) ****** However, none of them are primes, therefore, there is no minimal primes in this family. ***** The smallest prime of the form 5{5}0525 is [B]5550525[/B] ***** The smallest prime of the form 5{5}00525 is [B]5500525[/B] ***** For the 5{5}05025 family, since 55555025 is prime, we only need to consider the numbers 505025, 5505025, 55505025 (since any digit combo 555 between (5,05025) will produce smaller primes) ****** However, none of them are primes, therefore, there is no minimal primes in this family. * Case (5,7): ** [B]57[/B] is prime, and thus the only minimal prime in this family.[/QUOTE] In fact, * the smallest prime in the 5{5}0025 family is 5555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555550025, which can be written as 5[SUB]183[/SUB]025 and equal the prime (5*8^187-20333)/7, but this prime is not minimal prime. * the smallest prime in the 5{5}05025 family is 5555555555555555555555505025, but this prime is not minimal prime. |
[QUOTE=sweety439;566013]* 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) *** All minimal primes in the family 4{1,4,6}1 are [B]4611[/B], [B]444641[/B], [B]444444441[/B], see [URL="https://scholar.colorado.edu/downloads/hh63sw661"]https://scholar.colorado.edu/downloads/hh63sw661[/URL] (the base 8 section) * 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) **** However, all numbers of the form 4{0}3 are divisible by 7, and all numbers of the form 44{0}3 are divisible by 3, thus cannot be prime, therefore, there is no minimal primes in this family. * 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] **** For the 44{7}7 family, since 47777 is prime, we only need to consider the numbers 447, 4477, 44777 ***** However, none of them are primes, therefore, there is no minimal primes in this family.[/QUOTE] In fact, * the smallest prime in the 44{7}7 family is 4477777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777, which can be written as 447[SUB]851[/SUB] and equal the prime 37*8^851-1, but this prime is not minimal prime. |
[QUOTE=sweety439;567354]* Case (6,3):
** Since 65, 13, 23, 53, 73, [B]643[/B] are primes, we only need to consider the family 6{0,3,6}3 (since any digits 1, 2, 4, 5, 7 between them will produce smaller primes) *** However, all numbers of the form 6{0,3,6}3 are divisible by 3, thus cannot be prime, therefore, there is no minimal primes in this family. * Case (6,5): ** [B]65[/B] is prime, and thus the only minimal prime in this family. * Case (6,7): ** Since 65, 27, 37, 57, [B]667[/B] are primes, we only need to consider the family 6{0,1,4,7}7 (since any digits 2, 3, 5, 6 between them will produce smaller primes) *** Since 107, 117, 147, 177, 407, 417, 717, 747, [B]6007[/B], [B]6477[/B], [B]6707[/B], [B]6777[/B] are primes, we only need to consider the families 60{1,4,7}7, 6{0}17, 6{0,4}4{4}7, 6{0}77 (since any digits combo 00, 10, 11, 14, 17, 40, 41, 47, 70, 71, 74, 77 between them will produce smaller primes) **** For the 60{1,4,7}7 family, since 117, 147, 177, 417, 6477, 717, 747, 6777 are primes, we only need to consider the numbers 6017, 6047, 6077 and the family 60{4}7 (since any digit combo 11, 14, 17, 41, 47, 71, 74, 77 between (60,7) will produce smaller primes) ***** However, none of 6017, 6047, 6077 are primes, and all numbers of the form 60{4}7 are divisible by 21 (octal 21, decimal 17), therefore, there is no minimal primes in this family. **** For the 6{0}17 family, since 6007 is prime, we only need to consider the number 6017 (since any digit combo 00 between (6,17) will produce smaller primes) ***** However, 6017 is not prime, therefore, there is no minimal primes in this family. **** For the 6{0,4}4{4}7 family, since 6007 and 407 are primes, we only need to consider the families 6{4}7 and 60{4}7 (since any digit combo 00, 40 between (6,4{4}7) will produce smaller primes) ***** However, all numbers of the form 6{4}7 are divisible by 3, 5, or 15 (octal 15, decimal 13), and all numbers of the form 60{4}7 are divisible by 21 (octal 21, decimal 17), therefore, there is no minimal primes in this family. **** For the 6{0}77 family, since 6007 is prime, we only need to consider the number 6077 (since any digit combo 00 between (6,77) will produce smaller primes) ***** However, 6077 is not prime, therefore, there is no minimal primes in this family.[/QUOTE] * Case (7,1): ** Since 73, 75, 21, 51, [B]701[/B], [B]711[/B] are primes, we only need to consider the family 7{4,6,7}1 (since any digits 0, 1, 2, 3, 5 between them will produce smaller primes) *** Since 747, 767, 471, 661, [B]7461[/B], [B]7641[/B] are primes, we only need to consider the families 7{4,7}4{4}1, 7{7}61, 7{7}7{4,6,7}1 (since any digits combo 46, 47, 64, 66, 67 between them will produce smaller primes) **** For the 7{4,7}4{4}1 family, since 747, 471 are primes, we only need to consider the family 7{7}{4}1 (since any digits combo 47 between (7,4{4}1) will produce smaller primes) ***** The smallest prime of the form 7{7}1 is [B]7777777777771[/B] ***** The smallest prime of the form 7{7}41 is 777777777777777777777777777777777777777777777777777777777777777777777777777777741 (not minimal prime, since 7777777777771 is prime) ***** The smallest prime of the form 7{7}441 is 777777777777777777777777777777777777777777777777777777777777777777777777777777777777441 (not minimal prime, since 7777777777771 is prime) ***** The smallest prime of the form 7{7}4441 is 777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777774441 (not minimal prime, since 7777777777771 is prime) ***** The smallest prime of the form 7{7}44441 is 7777777777777777777777777777777777777777777777777777777744441 (not minimal prime, since 7777777777771 is prime) ***** All numbers of the form 7{7}444441 are divisible by 7, thus cannot be prime. ***** The smallest prime of the form 7{7}4444441 is [B]77774444441[/B] ****** Since this prime has just 4 7's, we only need to consider the families with <=3 7's ******* The smallest prime of the form 7{4}1 is [B]744444441[/B] ******* All numbers of the form 77{4}1 are divisible by 5, thus cannot be prime. ******* The smallest prime of the form 777{4}1 is 777444444444441 (not minimal prime, since 444444441 and 744444441 are primes) |
* Case (7,3):
** [B]73[/B] is prime, and thus the only minimal prime in this family. * Case (7,5): ** [B]75[/B] is prime, and thus the only minimal prime in this family. * Case (7,7): ** Since 73, 75, 27, 37, 57, [B]717[/B], [B]747[/B], [B]767[/B] are primes, we only need to consider the family 7{0,7}7 (since any digits 1, 2, 3, 4, 5, 6 between them will produce smaller primes) *** All numbers of the form 7{0,7}7 are divisible by 7, thus cannot be prime. |
[QUOTE=sweety439;566429]This file is the smallest prime (not include x or y themselves) in given simple family x{y} or {x}y (where x,y are base b digits) in given base 2<=b<=24, where gcd(x,y) = 1, gcd(y,b) = 1 (searched up to 5000 base b digits, 0 if no such prime found (include the case such that x{y} or {x}y proven composite by all or partial algebra factors)
format of file: b,x,{y}: smallest prime of the form x{y} in base b b,{x},y: smallest prime of the form {x}y in base b such primes are generalized near-repdigit primes base b already excluded families x{y} and {x}y with NUMERICAL covering set (e.g. {1}3, {1}4, 3{1}, 4{1} in base 5) Such primes are ALWAYS minimal prime (start with 2 digits) in base b, except when the repeating digit (i.e. y in x{y}, or x in {x}y) is 1 and base b has generalized repunit primes (i.e. all digits are 1) smaller than the prime (in base b = 9, 25, 32, 49, 64, 81, 121, 125, 144, 169, 216, 225, 243, 289, 324, 343, 361, 441, 484, 529, 625, ..., no generalized repunit primes exist, thus in these bases b, such primes are always minimal primes (start with 2 digits) in base b) extended data to base 36[/QUOTE] search the simple families x{0}y with gcd(x,y) = 1, gcd(y,b) = 1, gcd(x+y,b-1) = 1 Such primes are ALWAYS minimal prime (start with 2 digits) in base b, except when the base (b) is prime, and x = 1 (while 10 is prime and a subsequence of the prime, but with [URL="https://mersenneforum.org/showpost.php?p=531632&postcount=7"]LaurV's suggestion[/URL], the prime 10 (i.e. the prime = base) is also not counted just as the primes < base, all such primes (i.e. all smallest primes of the form x{0}y) is ALWAYS minimal prime (start with b+1) in base b) |
Base b minimal primes (start with 2 digits) includes:
* The smallest repunit prime base b if exists * The smallest generalized Fermat prime base b for even b if exists * The smallest generalized half Fermat prime (> (b+1)/2) base b for odd b if exists * The smallest [URL="https://www.rieselprime.de/ziki/Williams_prime_MM_table"]Williams prime with 1st kind[/URL] base b if exists * The smallest [URL="https://www.rieselprime.de/ziki/Williams_prime_MP_table"]Williams prime with 2nd kind[/URL] base b if exists * The smallest [URL="https://www.rieselprime.de/ziki/Williams_prime_PP_table"]Williams prime with 4th kind[/URL] base b [B]for bases b which no generalized Fermat primes exist (this includes all odd bases) and b is not prime (this condition is not needed if as LaurV's suggestion, the prime 10 (=b) is also excluded)[/B] if exists * The smallest dual Williams prime with 1st kind base b if exists * The smallest dual Williams prime with 2nd kind base b [B]for composite bases b (this condition is not needed if as LaurV's suggestion, the prime 10 (=b) is also excluded)[/B] if exists * The smallest dual Williams prime with 4th kind base b [B]for bases b which no generalized Fermat primes exist (this includes all odd bases) and b is not prime (this condition is not needed if as LaurV's suggestion, the prime 10 (=b) is also excluded)[/B] if exists * The smallest prime of the form 2*b^n+1 for bases b>2 if exists * The smallest prime of the form 2*b^n-1 for bases b>2 if exists * The smallest prime of the form b^n+2 for bases b>2 with gcd(b,2)=1 [B]for composite bases b (this condition is not needed if as LaurV's suggestion, the prime 10 (=b) is also excluded)[/B] if exists * The smallest prime of the form b^n-2 for bases b>2 with gcd(b,2)=1 if exists * The smallest prime of the form 3*b^n+1 for bases b>3 if exists * The smallest prime of the form 3*b^n-1 for bases b>3 if exists * The smallest prime of the form b^n+3 for bases b>3 with gcd(b,3)=1 [B]for composite bases b (this condition is not needed if as LaurV's suggestion, the prime 10 (=b) is also excluded)[/B] if exists * The smallest prime of the form b^n-3 for bases b>3 with gcd(b,3)=1 if exists * The smallest prime of the form 4*b^n+1 for bases b>4 if exists * The smallest prime of the form 4*b^n-1 for bases b>4 if exists * The smallest prime of the form b^n+4 for bases b>4 with gcd(b,4)=1 [B]for composite bases b (this condition is not needed if as LaurV's suggestion, the prime 10 (=b) is also excluded)[/B] if exists * The smallest prime of the form b^n-4 for bases b>4 with gcd(b,4)=1 if exists ... * The smallest prime of the form k*b^n+1 for fixed 1<=k<=b-1 (i.e. the prime for the [URL="http://www.noprimeleftbehind.net/crus/Sierp-conjectures.htm"]CRUS Sierpinski conjecture[/URL] for fixed 1<=k<=b-1) if exists * The smallest prime of the form k*b^n-1 for fixed 1<=k<=b-1 (i.e. the prime for the [URL="http://www.noprimeleftbehind.net/crus/Riesel-conjectures.htm"]CRUS Riesel conjecture[/URL] for fixed 1<=k<=b-1) if exists * The smallest prime of the form b^n+k for fixed 1<=k<=b-1 if exists * The smallest prime of the form b^n-k for fixed 1<=k<=b-1 if exists * The smallest prime of the form (k*b^n-1)/gcd(k-1,b-1) for fixed k with 0<=(k-1)/gcd(k-1,b-1)<=b-1 and [B]gcd(k-1,b-1) < b-1 (this reason is because if the repeating digit is 1, then this prime may not be minimal prime (start with 2 digits), unless there are no repunit primes base b (e.g. b = 9, 25, 32, 49, 64, 81, ...)[/B] (i.e. the prime for the [URL="https://mersenneforum.org/attachment.php?attachmentid=24053&d=1609098432"]extended Riesel conjecture[/URL] for fixed k satisfying these two conditions) if exists * The smallest prime of the form (b^n-k)/gcd(k-1,b-1) for fixed k with gcd(b,k) = 1 and 0<=k<=b-1 (see post [URL="https://mersenneforum.org/showpost.php?p=571731&postcount=140"]#140[/URL] for references of these families) |
[QUOTE=LaurV;531632]I found an easy way to generate those sets, and to prove that they are complete.
For the "starting from two digits" version, neither one of the exposed sets for 7 and 8 are complete. Some larger primes are still lurking in the dark there. I have the complete sets for both 8, and 7 for the both cases when the base itself is included in the set or not*, but I don't want to spoil the puzzle, this is an interesting little problem... hehe... Hint: [CODE] gp > a=(7^17-5)/2 %1 = 116315256993601 gp > isprime(a) %2 = 1 gp > digits(a,7) %3 = [3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 1] gp > [/CODE]--------- *when the base is prime, like for 5 and 7, the sets are different; including the base results in automatic elimination of all possible extension numbers with "0 after 1" from the set, which is quite restrictive, so I also calculated the lists for the "base is not included" version, i.e. base-5 starting from 6, and base-7 starting from 8; in this case, for example, base-5 will include numbers like 104 and 10103 which are prime, and base-7 list will include 1022, 1051, 1202, .... 1100021 ... etc, they are "enriched" compared with the case when the first "10" is included. So I have the complete list for 8, and the complete two lists for 7, the normal one, and the "enriched" one. Base-5 is easy, in any case.[/QUOTE] If even the prime "10" (i.e. prime = the base (b)) is excluded, the the minimal primes will be: (only listed prime bases, since for composite bases the set of these primes is completely the same as the set of minimal primes with >=2 digits): [CODE] 2: {11} 3: {12, 21, 111} 5: {12, 21, 23, 32, 34, 43, 104, 111, 131, 133, 313, 401, 414, 3101, 10103, 14444, 30301, 33001, 33331, 44441, 300031, 100000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000013} 7: {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, 1022, 1051, 1112, 1202, 1211, 1222, 2111, 3031, 3055, 3334, 3503, 3505, 3545, 4504, 4555, 5011, 5455, 5545, 5554, 6034, 6634, 11111, 11201, 30011, 30101, 31001, 31111, 33001, 33311, 35555, 40054, 100121, 150001, 300053, 351101, 531101, 1100021, 33333301, 5100000001, ..., 33333333333333331, ...} 11: {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, 10A, 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, 1101, 11A9, 1305, 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, 5105, 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, 9101, 9107, 9305, 9505, 9703, A052, A119, A151, A175, A515, A517, A575, A577, A5A8, A719, A779, A911, AAA9, 10011, 10075, 10091, 10109, 10411, 10444, 10705, 10709, 10774, 10901, 11104, 11131, 11144, 11191, 1141A, 114A1, 13757, 1411A, 14477, 144A4, 14A04, 14A11, 17045, 17704, 1774A, 17777, 177A4, 17A47, 1A091, 1A109, 1A114, 1A404, 1A411, 1A709, 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, 70103, 70471, 70583, 70714, 71474, 717A4, 71A09, 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, A1404, A4177, A4401, A4717, A5228, A52AA, A5558, A580A, A5822, A58AA, A5A59, A5AA2, A6096, A6966, A6999, A7051, A7778, A7808, A9055, A9091, A9699, A9969, AA52A, AA58A, 100019, 100079, 101113, 101119, 101911, 107003, 140004, 144011, 144404, 1A0019, 1A0141, 1A5001, 1A7005, 1A9001, 222223, 222823, 300107, 300202, 300323, 303203, 307577, 310007, 332003, 370777, 400555, 401A11, 404001, 404111, 405AAA, 41A011, 440A41, 441011, 451777, 455555, 470051, 470444, 474404, 4A0401, 4A4041, 500015, 500053, 500077, 500507, 505577, 522A2A, 525223, 528A2A, 531707, 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, 701074, 701777, 704408, 704417, 704457, 704484, 707041, 707441, 707708, 707744, 707784, 710777, 717044, 717077, 740008, 74484A, 770441, 770744, 770748, 770771, 777017, 777071, 777448, 777484, 777701, 7778A8, 777A19, 777A48, 778883, 78A808, 790003, 7A1009, 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, A01901, A05509, A0A058, A0A955, A10114, A555A2, A55999, A59991, A5A222, A5A22A, A60609, A66069, A66906, A69006, A79005, A87888, A90099, A90996, A96006, A96666, A97177, A97771, AA0A58, AA5A22, AAA522, 1000501, 1011141, 1030007, 1070047, 111114A, 1111A14, 1111A41, 1144441, 14A4444, 1700005, 1700474, 1A44444, ...} [/CODE] |
See posts [URL="https://mersenneforum.org/showpost.php?p=524766&postcount=306"]https://mersenneforum.org/showpost.php?p=524766&postcount=306[/URL], [URL="https://mersenneforum.org/showpost.php?p=531332&postcount=325"]https://mersenneforum.org/showpost.php?p=531332&postcount=325[/URL], [URL="https://mersenneforum.org/showpost.php?p=531333&postcount=326"]https://mersenneforum.org/showpost.php?p=531333&postcount=326[/URL] for the proof for base 5 (when single-digit primes are excluded but 10 (i.e. base) is included)
If 10 (i.e. base) is excluded, then for the primes containing 10: any digits before 10 cannot be 2 (because of 21) any digits after 10 cannot be 2 (because of 12) And we have the prime [B]104[/B], and for other prime numbers, any digits after 10 cannot be 4 |
1 Attachment(s)
[QUOTE=sweety439;567578]search the simple families x{0}y with gcd(x,y) = 1, gcd(y,b) = 1, gcd(x+y,b-1) = 1
Such primes are ALWAYS minimal prime (start with 2 digits) in base b, except when the base (b) is prime, and x = 1 (while 10 is prime and a subsequence of the prime, but with [URL="https://mersenneforum.org/showpost.php?p=531632&postcount=7"]LaurV's suggestion[/URL], the prime 10 (i.e. the prime = base) is also not counted just as the primes > base, all such primes (i.e. all smallest primes of the form x{0}y) is ALWAYS minimal prime (start with b+1) in base b)[/QUOTE] Update the file for the smallest primes in these families for bases up to 36 |
Minimal set of prime-strings with ≥2 digits in bases 2 to 12 (only bases 2 to 8 are proved to be complete)
[CODE] 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, 33333333333333331} 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, 77774444441, 7777777777771, 555555555555525, 44444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444447} 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, 5161111111, 8333333335, ..., 300000000035, ..., 544444444444, ..., 2000000000007, ..., 56111111111111111111111111111111111111, ..., 7666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666662, ...} 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, ..., 555555555551, ..., 5000000000000000000000000000027, ...} 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, 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, ..., 600000A999, ..., A000144444, ..., A900000066, ..., 33333333337, ..., 44444444447, ..., A0000000001, ..., A0014444444, ..., 40000000A0041, ..., A000000014444, ..., A044444444441, ..., A144444444411, ..., 40000000000401, ..., A0000044444441, ..., A00000000444441, ..., 11111111111111111, ..., 14444444444441111, ..., 44444444444444111, ..., A1444444444444444, ..., A9999999999999996, ..., 888888888888888883, ..., 1444444444444444444, ..., 7777777777777777771, ..., 4000000000000000A041, ..., A999999999999999999999, ..., A44444444444444444444444441, ..., 40000000000000000000000000041, ..., 440000000000000000000000000001, ..., 999999999999999999999999999999991, ..., 444444444444444444444444444444444444444444441, ...} 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, ..., B0000000000000000000000000009B, ...} [/CODE] |
[QUOTE=sweety439;567578]search the simple families x{0}y with gcd(x,y) = 1, gcd(y,b) = 1, gcd(x+y,b-1) = 1
Such primes are ALWAYS minimal prime (start with 2 digits) in base b, except when the base (b) is prime, and x = 1 (while 10 is prime and a subsequence of the prime, but with [URL="https://mersenneforum.org/showpost.php?p=531632&postcount=7"]LaurV's suggestion[/URL], the prime 10 (i.e. the prime = base) is also not counted just as the primes < base, all such primes (i.e. all smallest primes of the form x{0}y) is ALWAYS minimal prime (start with b+1) in base b)[/QUOTE] If as LaurV's suggestion, the prime 10 (i.e. the prime = base) is also not counted just as the primes < base, then the last digit of all primes in the set must be coprime to the base (since the last digit of all primes which do not divide the base are coprime to the base, and all primes > base do not divide the base), also, the first digit of all primes in the set (in fact, for all numbers) cannot be 0, thus, 0 can be neither the first digit nor the last digit, for the primes in this set, and 0 can only be just the middle digits, and the simple family x{0}y (where x,y are any nonzero digits in this base) ALWAYS need to test (unless this family is ruled out to only contain composites), like the simple families x{y} (y != 1) and {x}y (x != 1), which are also ALWAYS need to test (unless they are ruled out to only contain composites). |
In base 9, family {3}{0}5 does not need to be tested because....
* If the number of digits 3 is even, then the number is divisible by 5. * If the number of digits 3 is odd, then the number is divisible by 2. Thus, this family has a numerical covering set {2,5} and is ruled out to only contain composites. Note that {3}{0}5 is not simple family (simple families are x{d}y with d digit, x,y strings of digits (can be empty string)) In base 9, many such non-simple families exist, e.g. {1}6{1}, see page 13 of [URL="https://cs.uwaterloo.ca/~cbright/reports/mepn.pdf"]https://cs.uwaterloo.ca/~cbright/reports/mepn.pdf[/URL] |
Some simple families which are ruled out to only contain composites: (all substrings with length >=2 of all numbers in these families are not primes, except base 8 6{4}7 and 60{4}7 families, they are listed here because all substrings with length >=2 of all numbers with <220 4's in these two families are not primes)
Base 5: 3{0}1 (divisible by 2) Base 6: 4{0}1 (divisible by 5) Base 8: 1{0}1 (sum of cubes) 2{0}5 (divisible by 7) 4{0}3 (divisible by 7) 44{0}3 (divisible by 3) 6{0}1 (divisible by 7) 6{4}7 (divisible by 3, 5, or 13) 60{4}7 (divisible by 17) Base 9: {1} (difference of squares) {1}5 (divisible by 2 or 5) 2{7} (divisible by 2 or 5) 3{1} (difference of squares) {3}5 (divisible by 2 or 5) {3}8 (divisible by 2 or 5) 3{8} (difference of squares) 5{1} (divisible by 2 or 5) 5{7} (divisible by 2 or 5) 6{1} (divisible by 2 or 5) {7}2 (divisible by 2 or 5) {7}5 (divisible by 2 or 5) {8}5 (difference of squares) Base 10: 4{6}9 (divisible by 7) Base 12: A{0}1 (divisible by 11) {B}9B (even number of B's is difference of squares, odd number of B's is divisible by 13) Base 14: 3{D} (divisible by 3 or 5) 4{0}1 (divisible by 3 or 5) 8{D} (even number of D's is difference of squares, odd number of D's is divisible by 5) A{D} (divisible by 3 or 5) B{0}1 (divisible by 3 or 5) {D}3 (divisible by 3 or 5) {D}5 (even number of D's is divisible by 5, odd number of D's is difference of squares) Base 16: 1{5} (difference of squares) 8{F} (difference of squares) {C}D (x^4+4*y^4) {F}7 (difference of squares) Base 17: 1{9} (even number of 9's is difference of squares, odd number of 9's is divisible by 2) Base 20: 7{J} (divisible by 3 or 7) 8{0}1 (divisible by 3 or 7) C{J} (divisible by 3 or 7) D{0}1 (divisible by 3 or 7) Base 24: 3{N} (even number of N's is difference of squares, odd number of N's is divisible by 5) 5{N} (even number of N's is divisible by 5, odd number of N's is difference of squares) {6}1 (even number of 6's is difference of squares, odd number of 6's is divisible by 5) 8{N} (even number of N's is difference of squares, odd number of N's is divisible by 5) Base 25: {1} (difference of squares) 1{3} (difference of squares) 1{8} (difference of squares) D{1} (divisible by 2 or 13) Base 27: 8{0}1 (sum of cubes) 9{G} (difference of cubes) {D}E (sum of cubes) Base 32: 1{0}1 (sum of 5th powers) {1} (difference of 5th powers, the only trivial is 11111, but 11111 is not prime) Base 38: C{b} (divisible by 3, 5, or 17) G{0}1 (divisible by 3, 5, or 17) Base 47: 8{0}1 (divisible by 3, 5, or 13) D{k} (divisible by 3, 5, or 13) G{0}1 (divisible by 3, 5, or 17) |
[QUOTE=sweety439;567582]Base b minimal primes (start with 2 digits) includes:
* The smallest repunit prime base b if exists * The smallest generalized Fermat prime base b for even b if exists * The smallest generalized half Fermat prime (> (b+1)/2) base b for odd b if exists * The smallest [URL="https://www.rieselprime.de/ziki/Williams_prime_MM_table"]Williams prime with 1st kind[/URL] base b if exists * The smallest [URL="https://www.rieselprime.de/ziki/Williams_prime_MP_table"]Williams prime with 2nd kind[/URL] base b if exists * The smallest [URL="https://www.rieselprime.de/ziki/Williams_prime_PP_table"]Williams prime with 4th kind[/URL] base b [B]for bases b which no generalized Fermat primes exist (this includes all odd bases) and b is not prime (this condition is not needed if as LaurV's suggestion, the prime 10 (=b) is also excluded)[/B] if exists * The smallest dual Williams prime with 1st kind base b if exists * The smallest dual Williams prime with 2nd kind base b [B]for composite bases b (this condition is not needed if as LaurV's suggestion, the prime 10 (=b) is also excluded)[/B] if exists * The smallest dual Williams prime with 4th kind base b [B]for bases b which no generalized Fermat primes exist (this includes all odd bases) and b is not prime (this condition is not needed if as LaurV's suggestion, the prime 10 (=b) is also excluded)[/B] if exists * The smallest prime of the form 2*b^n+1 for bases b>2 if exists * The smallest prime of the form 2*b^n-1 for bases b>2 if exists * The smallest prime of the form b^n+2 for bases b>2 with gcd(b,2)=1 [B]for composite bases b (this condition is not needed if as LaurV's suggestion, the prime 10 (=b) is also excluded)[/B] if exists * The smallest prime of the form b^n-2 for bases b>2 with gcd(b,2)=1 if exists * The smallest prime of the form 3*b^n+1 for bases b>3 if exists * The smallest prime of the form 3*b^n-1 for bases b>3 if exists * The smallest prime of the form b^n+3 for bases b>3 with gcd(b,3)=1 [B]for composite bases b (this condition is not needed if as LaurV's suggestion, the prime 10 (=b) is also excluded)[/B] if exists * The smallest prime of the form b^n-3 for bases b>3 with gcd(b,3)=1 if exists * The smallest prime of the form 4*b^n+1 for bases b>4 if exists * The smallest prime of the form 4*b^n-1 for bases b>4 if exists * The smallest prime of the form b^n+4 for bases b>4 with gcd(b,4)=1 [B]for composite bases b (this condition is not needed if as LaurV's suggestion, the prime 10 (=b) is also excluded)[/B] if exists * The smallest prime of the form b^n-4 for bases b>4 with gcd(b,4)=1 if exists ... * The smallest prime of the form k*b^n+1 for fixed 1<=k<=b-1 (i.e. the prime for the [URL="http://www.noprimeleftbehind.net/crus/Sierp-conjectures.htm"]CRUS Sierpinski conjecture[/URL] for fixed 1<=k<=b-1) if exists * The smallest prime of the form k*b^n-1 for fixed 1<=k<=b-1 (i.e. the prime for the [URL="http://www.noprimeleftbehind.net/crus/Riesel-conjectures.htm"]CRUS Riesel conjecture[/URL] for fixed 1<=k<=b-1) if exists * The smallest prime of the form b^n+k for fixed 1<=k<=b-1 if exists * The smallest prime of the form b^n-k for fixed 1<=k<=b-1 if exists * The smallest prime of the form (k*b^n-1)/gcd(k-1,b-1) for fixed k with 0<=(k-1)/gcd(k-1,b-1)<=b-1 and [B]gcd(k-1,b-1) < b-1 (this reason is because if the repeating digit is 1, then this prime may not be minimal prime (start with 2 digits), unless there are no repunit primes base b (e.g. b = 9, 25, 32, 49, 64, 81, ...)[/B] (i.e. the prime for the [URL="https://mersenneforum.org/attachment.php?attachmentid=24053&d=1609098432"]extended Riesel conjecture[/URL] for fixed k satisfying these two conditions) if exists * The smallest prime of the form (b^n-k)/gcd(k-1,b-1) for fixed k with gcd(b,k) = 1 and 0<=k<=b-1[/QUOTE] These are the minimal primes (start with 2 digits) in base b for 2<=b<=64: * The smallest repunit prime base b: 3, 13, 5, 31, 7, 2801, 73, (not exist), 11, 50544702849929377, 13, 30941, 211, 241, 17, 307, 19, 109912203092239643840221, 421, 463, 23, 292561, 601, (not exist), 321272407, 757, 29, 732541, 31, 917087137, (not exist), 1123, 2458736461986831391, 57785511861854089559605684285384747472873075954938549266821996762520614682090417010479587236790517340193840109863642510356045237096340500854836834673594590986502765133399405931445515950293723048093118292954035082630781507315268041070570042738804650015484793905221070413101021864355439951875266340353210153398276807146377561258956649022201316646128234211457693681312361704211065831222237374054447781785197765525068555496240581389620280398439369732560881984414556748507653965669519761, 37, 6765811783780036261, 1483, 50322737201397037309232643922534935391510719645806123027236338191297773996287037475296763303738120063947710508065397284342914454082093489188333435337356055506801965232663559960802538728833796534291601599594460801094950652338245308336678262969650945954793076571188076285097774508994928135805851461589780682301845186135236651321513610921527111042159747801555758087206120961819012271336066498881331471146011538796171976969227414611180652471781807744608704658147356974307300714996994451224795449952999716213239830256631836859640201416928684063130279139058641, 41, 1723, 43, 3500201, 3835261, 585578449280908796570517800071, 47, 4939353696332137648660158610486273245800498531219046056285398249895046060595791007616253627660064463584012737427605759732894439061580553419678353685587762357233722998146101218334328347614340561470069315963989297, 1868467947605686541562499217713, (not exist), 2551, (51^4229-1)/50, 53, 178250690949465223, 2971, 7141212583461249612878870081, 31401724537, 3307, 59, 3541, 61, 52379047267, 3907, 16007041, (not exist) * The smallest Williams prime with 1st kind base b: 3, 5, 11, 19, 29, 41, 3583, 71, 89, 109, 131, 2027, 181, 408700964355468749, 239, 271, 5507, 846825857, 379, 419, 461, 17276416353328819798072137388863592892072278184923153720493777138850572564953, 839967991029301247, 599, 3885038158778096269468893991882380063764065770433606110283149695964997245520484669311748838825973451239771955518933348332721403496018696846203290707966794803507099534240007184258836096614399, 701, 2368778164222232774191928573951, 811, 26099, 929, 991, 34847, 3095263992211830248865791, 1457749, 1259, 3417547576787, 37*38^136211-1, 1481, 1559, 7790170955239, 1721, 11416381666493, 13728945815551, 1979, 2069, 2161, 108287, 2351, 146031379699707031249999999999999999999999999, 2549, 19390405631, 7741603, 2861, 2969, 3079, 3191, 191747, 11911981, 3539, 3659, 139784395906071076766586020581268962303747288598567336951484722224451313085811771730116807299236427117842661797319704284843879372078242712851621827298082457986459462052386395974649371235894861683121487306574721459683501339777513560734278325650981380266285435618548139498448328096596807841457960474839935725016673894773768583677720662043161779832373490477494167310555190935956550302093326676711884265997512315330943608474964176395642249725224549579353670398605084716869153595460341758191671267601756450678548385413476307461538457595934906420684517898691543670687505315965265329598657370673413817965286377458043419147312784602056733277747759104526431673619974552441432434037533316091375742425601781283038065394099553927737205886353765152865548598839643177897844563113475601473328953408482456049673131973616976185783209288099190146109681267522543094039170871215260791829073435382418155171778782135316645578882955339664308529274906792023008178996964036098381727671689194723738996872705045605820037786049396276334730253078530300611653046136767706617293109455653251330444209017346414173013914938198059188297805588034443087345116883221415800773297093648337901984538392154514037514582882439555394954054641625018652439839301610996424664396983974323085517222193755969542276935638297070776680160676683227405039735827499267662993946306495247080085431407385375097236369554025762623869932975200275621419639360304816808572748496017393168318232665972570446440075892500649906727508939244057537554716592457169274187691031345168964785229426983746654648388804888079257248797899967499299858456238588374112649305901600140655206101680127, 15502913, 4648579506574807007231 * The smallest Williams prime with 2nd kind base b: 3, 7, 13, 101, 31, 43, 449, 73, 9001, 259374246011, 19009, 157, 2549, 211, 241, 1336337, 307, 218336795902605993201009018384568383223, 31129600000000000001, 421, 463, 255042399139852495799, 13249, 601, 16901, 13817467, 757, 23549, 23490001, 858874531, 35740566642812256257, 34849, 1123, 41651, 45361, 4678622632622773, 699421969744001971270254593, 1483, 62401, 42147180671470348388835886625344411346196083191529631288482561146240326026998221506440783978133517939838164995885825751477859968041, 1723, 77659, 161168129, 89101, 4380121, 26080959134473636132102571567, 108289, 115249, 122501, 2551, 2802982140528952023258759169, 10491513900891286499026738735135091160586124006333470053075798452149811519270024300012619626518018615344041193898353540073317686478669982334935722735157214044438750121387067129353403131395924739885547787541632580071542418404213751970278909499759877550283845617177464886504494817733888728186342721678314630157112704845506395528141971726016370931785914435731423398869999159630585365445775982492048522620099890853763237006462647136796940109416875810919060529600308292936053534656732307484233137785394600768454409163465028938105210676278897166335101081590402220699091389419760947664521051089812248945464579938920557862949540927479290799182068014020779544988135009962261241327200782535102674719135072434562508040789141676449576125079219727758870356884247118065440773617950056021530846873049589550019133078080663722176789313870897580396810177755781909618527239261721244480572786387976447602293449132613742946319671833328964972697119246892682336265152730921237713665354507537137443702829445952017487340491300683675985895629691084321187366733688850419017738601345840255501180273454585732065945743750216650542146084610529712530337630563500776116033357996283312548212598145790910607713043929753354827632869448608011951580713339509715548347910626001375466250058120039571083029723284705185200451797478498912933282140762819306704853286212160517919160926804950671170197329021477887747077647785029849474482618823322434015887000629289614915885122710872705478908601275767330249679458511998686296432297421162041793449010008226217042869255891247535123750707697346124793337408964749519851092431624559732210808836121020390168896446735325790252142542398429166658420843624167654453, 5130766694717659087768092673, 2971, 540897281, 10370809, 3307, 359216400347725176472840139, 3541, 3091222461661, 42969828958366879401068146141598580737, 3907, 16515073 |
* The smallest Williams prime with 4th kind base b (for b != 1 mod 3): [not minimal prime (start with 2 digits) if either b is prime or base b has smaller generalized Fermat prime, but for the case that b is prime, it is still minimal prime (start with 2 digits) if we use LaurV's suggestion, i.e. start with b+1 instead of b]
7, 13, 31, 43, 73, 811, 1453, 157, 211, 241, 307, 3768826516993, 421, 463, 12697, 601, 18253, 757, 615334471, 27901, 1107296257, 1123, 44101, 1726273, 1483, 2372761, 1723, 75853, 87121, 93151, 106033, 599298932737, 2551, 158981126352779044590102826209115342318059775372698133871491241388097301966680877821738760704616125782843491355455960710073030287313404870590681666644752545879191893959727029866211537628677981607279205572507381073830401006677162824033234341436459420880686565908174585159142942438136179315586329074318947952541865853, 151687, 2971, 178753, 3307, 3541, 1338153989063049216000000000000000000001, 3907, 48326086052867645032352571108528903615254734667108057821332757600957454538355546211631290156513879123036351230974951391062798157776810891656336682957284917485088940693788242992185798654992956966627018064055387274320725152943868432582696386314597516885379356294528772183874293272350708412107233383892387582454781698467578958840732553153 * The smallest generalized Fermat prime base b (for even b): 3, 5, 7, (not exist), 11, 13, 197, 17, 19, 401, 23, 577, 677, 29, 31, (not exist), 1336337, 37, [unknown], 41, 43, 197352587024076973231046657, 47, 5308417, [unknown], 53, 2917, 3137, 59, 61, [unknown], (not exist) * The smallest generalized half Fermat prime (> (b+1)/2) base b (for odd b): 5, 13, 1201, 41, 61, 14281, 113, 41761, 181, 97241, 139921, 313, (not exist), 421, [unknown], 703204309121, 613, [unknown], 761, 31879515457326527173216321, 5844100138801, 1013, 11905643330881, 1201, 1301, 31129845205681, [unknown], 5278001, 1741, 1861, [unknown] * The smallest dual Williams prime with 1st kind base b: 3, 7, 13, 3121, 31, 43, 549755813881, 73, 991, 1321, 248821, 157, 2731, 211, 241, 34271896307617, 307, 6841, 13107199999999999999981, 421, 463, 141050039560662968926081, 331753, 601, 17551, 7625597484961, 757, 1816075630094014572464024421543167816955354437761, 21869999971, 29761, 34359738337, 1185889, 1123, 42841, 60466141, 1173587600912967505181585220815870451386152316472799938266409866089889961869797411886878993830039201370297, 79235131, 1483, 262143999999961, 68881, 1723, 3418759, 121987944123281928470243645070631579418581, 91081, 4477411, 229344961, 254803921, 36703368217294125441230211032033660188753, 124951, 2551, 140557, 1621038246414954860589967996431649201, 157411, 2971, 5416169448144841, 185137, 3307, 30155888444737842601, 3541, 844596241, 238267, 3907, 16777153 * The smallest dual Williams prime with 2nd kind base b: [not minimal prime (start with 2 digits) if b is prime, but still minimal prime (start with 2 digits) if we use LaurV's suggestion, i.e. start with b+1 instead of b] 3, 5, 7, 29, 11, 13, 71, 17, 19, 131, 23, 181, 2177953337809371149, 29, 31, 83537, 5849, 37, 419, 41, 43, 279863, 47, 15649, 701, 53, 811, 420707233300229, 59, 61, 3599131035634557106248430806148785487095757694641533306480604458089470064537190296255232548883112685719936728506816716098566612844395439751206812144692131084107807, 35969, 67, 1259, 71, 73, 1481, 1559, 79, 1721, 83, 79549, 1979, 89, 2161, 2766668711962335809450748011342447, 2351, 97, 2549, 101, 103, 2861, 107, 109, 3191, 113, 11316553, 3539, 3659, 3142742836081, 218340105584957, 250109, 127 * The smallest dual Williams prime with 4th kind base b (for b != 1 mod 3): [not minimal prime (start with 2 digits) if either b is prime or base b has smaller generalized Fermat prime, but for the case that b is prime, it is still minimal prime (start with 2 digits) if we use LaurV's suggestion, i.e. start with b+1 instead of b] 5, 7, 11, 13, 17, 19, 23, 157, 29, 31, 307, 37, 41, 43, 47, 601, 53, 757, 59, 61, 32801, 67, 71, 73, 1483, 79, 83, 74131, 89, 8303765671, 4879729, 97, 101, 103, 107, 109, 113, 3307, 3541, 216061, 3907, 127 |
[QUOTE=LaurV;531632]I found an easy way to generate those sets, and to prove that they are complete.
For the "starting from two digits" version, neither one of the exposed sets for 7 and 8 are complete. Some larger primes are still lurking in the dark there. I have the complete sets for both 8, and 7 for the both cases when the base itself is included in the set or not*, but I don't want to spoil the puzzle, this is an interesting little problem... hehe... Hint: [CODE] gp > a=(7^17-5)/2 %1 = 116315256993601 gp > isprime(a) %2 = 1 gp > digits(a,7) %3 = [3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 1] gp > [/CODE]--------- *when the base is prime, like for 5 and 7, the sets are different; including the base results in automatic elimination of all possible extension numbers with "0 after 1" from the set, which is quite restrictive, so I also calculated the lists for the "base is not included" version, i.e. base-5 starting from 6, and base-7 starting from 8; in this case, for example, base-5 will include numbers like 104 and 10103 which are prime, and base-7 list will include 1022, 1051, 1202, .... 1100021 ... etc, they are "enriched" compared with the case when the first "10" is included. So I have the complete list for 8, and the complete two lists for 7, the normal one, and the "enriched" one. Base-5 is easy, in any case.[/QUOTE] Proof of base 5 in the case which the prime 10 (i.e. the prime = base) is also not counted just as the primes < base: 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), (1,2), (1,3), (1,4), (2,1), (2,2), (2,3), (2,4), (3,1), (3,2), (3,3), (3,4), (4,1), (4,2), (4,3), (4,4) * Case (1,1): ** Since 12, 21, [B]111[/B], [B]131[/B] are primes, we only need to consider the family 1{0,4}1 (since any digits 1, 2, 3 between them will produce smaller primes) *** All numbers of the form 1{0,4}1 are divisible by 2, thus cannot be prime. * Case (1,2): ** [B]12[/B] is prime, and thus the only minimal prime in this family. * Case (1,3): ** Since 12, 23, 43, [B]133[/B] are primes, we only need to consider the family 1{0,1}3 (since any digits 2, 3, 4 between them will produce smaller primes) *** Since 111 is prime, we only need to consider the families 1{0}3 and 1{0}1{0}3 (since any digit combo 11 between (1,3) will produce smaller primes) **** All numbers of the form 1{0}3 are divisible by 2, thus cannot be prime. **** For the 1{0}1{0}3 family, since [B]10103[/B] is prime, we only need to consider the families 1{0}13 and 11{0}3 (since any digit combo 010 between (1,3) will produce smaller primes) ***** The smallest prime of the form 1{0}13 is [B]100000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000013[/B], which can be written as 10[SUB]93[/SUB]13 and equal the prime 5^95+8 ***** All numbers of the form 11{0}3 are divisible by 3, thus cannot be prime. * Case (1,4): ** Since 12, 34, [B]104[/B] are primes, we only need to consider the families 1{1,4}4 (since any digits 0, 2, 3 between them will produce smaller primes) *** Since 111, 414 are primes, we only need to consider the family 1{4}4 and 11{4}4 (since any digit combo 11 or 41 between them will produce smaller primes) **** The smallest prime of the form 1{4}4 is [B]14444[/B]. **** All numbers of the form 11{4}4 are divisible by 2, thus cannot be prime. |
* Case (2,1):
** [B]21[/B] is prime, and thus the only minimal prime in this family. * Case (2,2): ** Since 21, 23, 12, 32 are primes, we only need to consider the family 2{0,2,4}2 (since any digits 1, 3 between them will produce smaller primes) *** All numbers of the form 2{0,2,4}2 are divisible by 2, thus cannot be prime. * Case (2,3): ** [B]23[/B] is prime, and thus the only minimal prime in this family. * Case (2,4): ** Since 21, 23, 34 are primes, we only need to consider the family 2{0,2,4}4 (since any digits 1, 3 between them will produce smaller primes) *** All numbers of the form 2{0,2,4}4 are divisible by 2, thus cannot be prime. * Case (3,1): ** Since 32, 34, 21 are primes, we only need to consider the family 3{0,1,3}1 (since any digits 2, 4 between them will produce smaller primes) *** Since 313, 111, 131, [B]3101[/B] are primes, we only need to consider the families 3{0,3}1 and 3{0,3}11 (since any digit combo 10, 11, 13 between (3,1) will produce smaller primes) **** For the 3{0,3}1 family, we can separate this family to four families: ***** For the 30{0,3}01 family, we have the prime [B]30301[/B], and the remain case is the family 30{0}01. ****** All numbers of the form 30{0}01 are divisible by 2, thus cannot be prime. ***** For the 30{0,3}31 family, note that there must be an even number of 3's between (30,31), or the result number will be divisible by 2 and cannot be prime. ****** Since 33331 is prime, any digit combo 33 between (30,31) will produce smaller primes. ******* Thus, the only possible prime is the smallest prime in the family 30{0}31, and this prime is [B]300031[/B]. ***** For the 33{0,3}01 family, note that there must be an even number of 3's between (33,01), or the result number will be divisible by 2 and cannot be prime. ****** Since 33331 is prime, any digit combo 33 between (33,01) will produce smaller primes. ******* Thus, the only possible prime is the smallest prime in the family 33{0}01, and this prime is [B]33001[/B]. ***** For the 33{0,3}31 family, we have the prime [B]33331[/B], and the remain case is the family 33{0}31. ****** All numbers of the form 33{0}31 are divisible by 2, thus cannot be prime. * Case (3,2): ** [B]32[/B] is prime, and thus the only minimal prime in this family. * Case (3,3): ** Since 32, 34, 23, 43, [B]313[/B] are primes, we only need to consider the family 3{0,3}3 (since any digits 1, 2, 4 between them will produce smaller primes) *** All numbers of the form 3{0,3}3 are divisible by 3, thus cannot be prime. * Case (3,4): ** [B]34[/B] is prime, and thus the only minimal prime in this family. |
* Case (4,1):
** Since 43, 21, [B]401[/B] are primes, we only need to consider the family 4{1,4}1 (since any digits 0, 2, 3 between them will produce smaller primes) *** Since 414, 111 are primes, we only need to consider the family 4{4}1 and 4{4}11 (since any digit combo 14 or 11 between them will produce smaller primes) **** The smallest prime of the form 4{4}1 is [B]44441[/B]. **** All numbers of the form 4{4}11 are divisible by 2, thus cannot be prime. * Case (4,2): ** Since 43, 12, 32 are primes, we only need to consider the family 4{0,2,4}2 (since any digits 1, 3 between them will produce smaller primes) *** All numbers of the form 4{0,2,4}2 are divisible by 2, thus cannot be prime. * Case (4,3): ** [B]43[/B] is prime, and thus the only minimal prime in this family. * Case (4,4): ** Since 43, 34, [B]414[/B] are primes, we only need to consider the family 4{0,2,4}4 (since any digits 1, 3 between them will produce smaller primes) *** All numbers of the form 4{0,2,4}4 are divisible by 2, thus cannot be prime. |
Thus, we completed and proved the set of minimal primes (start with b+1, instead of b or 2) of base b=5:
[CODE] 12 21 23 32 34 43 104 111 131 133 313 401 414 3101 10103 14444 30301 33001 33331 44441 300031 100000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000013 [/CODE] |
Proof of base 2 in the case which the prime 10 (i.e. the prime = base) is also not counted just as the primes < base:
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) * Case (1,1): ** [B]11[/B] is prime, and thus the only minimal prime in this family. |
Proof of base 3 in the case which the prime 10 (i.e. the prime = base) is also not counted just as the primes < base:
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), (1,2), (2,1), (2,2) * Case (1,1): ** Since 12, 21, [B]111[/B] are primes, we only need to consider the family 1{0}1 (since any digits 1, 2 between them will produce smaller primes) *** All numbers of the form 1{0}1 are divisible by 2, thus cannot be prime. * Case (1,2): ** [B]12[/B] is prime, and thus the only minimal prime in this family. * Case (2,1): ** [B]21[/B] is prime, and thus the only minimal prime in this family. * Case (2,2): ** Since 21, 12 are primes, we only need to consider the family 2{0,2}2 (since any digits 1 between them will produce smaller primes) *** All numbers of the form 2{0,2}2 are divisible by 2, thus cannot be prime. |
Proof of base 4:
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), (2,1), (2,3), (3,1), (3,3) * Case (1,1): ** [B]11[/B] is prime, and thus the only minimal prime in this family. * Case (1,3): ** [B]13[/B] is prime, and thus the only minimal prime in this family. * Case (2,1): ** Since 23, 11, 31, [B]221[/B] are primes, we only need to consider the family 2{0}1 (since any digits 1, 2, 3 between them will produce smaller primes) *** All numbers of the form 2{0}1 are divisible by 3, thus cannot be prime. * Case (2,3): ** [B]23[/B] is prime, and thus the only minimal prime in this family. * Case (3,1): ** [B]31[/B] is prime, and thus the only minimal prime in this family. * Case (3,3): ** Since 31, 13, 23 are primes, we only need to consider the family 3{0,3}3 (since any digits 1, 2 between them will produce smaller primes) *** All numbers of the form 3{0,3}3 are divisible by 3, thus cannot be prime. |
Minimal set of prime-strings (> base, as LaurV's suggestion) in bases 2 to 12 (only bases 2 to 8 are proved to be complete)
[CODE] 2: {11} 3: {12, 21, 111} 4: {11, 13, 23, 31, 221} 5: {12, 21, 23, 32, 34, 43, 104, 111, 131, 133, 313, 401, 414, 3101, 10103, 14444, 30301, 33001, 33331, 44441, 300031, 100000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000013} 6: {11, 15, 21, 25, 31, 35, 45, 51, 4401, 4441, 40041} 7: {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, 1022, 1051, 1112, 1202, 1211, 1222, 2111, 3031, 3055, 3334, 3503, 3505, 3545, 4504, 4555, 5011, 5455, 5545, 5554, 6034, 6634, 11111, 11201, 30011, 30101, 31001, 31111, 33001, 33311, 35555, 40054, 100121, 150001, 300053, 351101, 531101, 1100021, 33333301, 5100000001, ..., 33333333333333331, ...} 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, 77774444441, 7777777777771, 555555555555525, 44444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444447} 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, 5161111111, 8333333335, ..., 300000000035, ..., 544444444444, ..., 2000000000007, ..., 56111111111111111111111111111111111111, ..., 7666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666662, ...} 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, ..., 555555555551, ..., 5000000000000000000000000000027, ...} 11: {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, 10A, 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, 1101, 11A9, 1305, 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, 5105, 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, 9101, 9107, 9305, 9505, 9703, A052, A119, A151, A175, A515, A517, A575, A577, A5A8, A719, A779, A911, AAA9, 10011, 10075, 10091, 10109, 10411, 10444, 10705, 10709, 10774, 10901, 11104, 11131, 11144, 11191, 1141A, 114A1, 13757, 1411A, 14477, 144A4, 14A04, 14A11, 17045, 17704, 1774A, 17777, 177A4, 17A47, 1A091, 1A109, 1A114, 1A404, 1A411, 1A709, 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, 70103, 70471, 70583, 70714, 71474, 717A4, 71A09, 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, A1404, A4177, A4401, A4717, A5228, A52AA, A5558, A580A, A5822, A58AA, A5A59, A5AA2, A6096, A6966, A6999, A7051, A7778, A7808, A9055, A9091, A9699, A9969, AA52A, AA58A, 100019, 100079, 101113, 101119, 101911, 107003, 140004, 144011, 144404, 1A0019, 1A0141, 1A5001, 1A7005, 1A9001, 222223, 222823, 300107, 300202, 300323, 303203, 307577, 310007, 332003, 370777, 400555, 401A11, 404001, 404111, 405AAA, 41A011, 440A41, 441011, 451777, 455555, 470051, 470444, 474404, 4A0401, 4A4041, 500015, 500053, 500077, 500507, 505577, 522A2A, 525223, 528A2A, 531707, 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, 701074, 701777, 704408, 704417, 704457, 704484, 707041, 707441, 707708, 707744, 707784, 710777, 717044, 717077, 740008, 74484A, 770441, 770744, 770748, 770771, 777017, 777071, 777448, 777484, 777701, 7778A8, 777A19, 777A48, 778883, 78A808, 790003, 7A1009, 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, A01901, A05509, A0A058, A0A955, A10114, A555A2, A55999, A59991, A5A222, A5A22A, A60609, A66069, A66906, A69006, A79005, A87888, A90099, A90996, A96006, A96666, A97177, A97771, AA0A58, AA5A22, AAA522, 1000501, 1011141, 1030007, 1070047, 111114A, 1111A14, 1111A41, 1144441, 14A4444, 1700005, 1700474, 1A44444, 2555505, 2845055, 3030023, 3100003, 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, 7100447, 7174404, 717444A, 7400404, 7700717, 7701077, 7701707, 7707778, 7774004, 7777104, 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, A001091, A006906, A010044, A040041, A0AAA58, A141111, A5222A2, A600A69, A906606, A909009, A990009, A997701, AA55A52, AAA5552, AAAAA52, 10004747, 10005007, 17000744, 22888823, 28888223, 30010111, 30555777, 31011111, 33000023, 40A00041, 45000055, 47040004, 50377777, 50555553, 5282AAA2, 55505003, 555A5A52, 555AAA2A, 55A5A552, 5AAAAA2A, 60A99999, 70000057, 70070474, 70074704, 70174004, 70700078, 70700474, 70704704, 70710707, 70771007, 70777177, 71074004, 74470001, 77000177, 77070477, 77100077, 77470004, 77700404, 77710007, 77717707, 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, A1900001, A5555009, A5A55552, A9700001, A9909006, A9990006, A9990606, A9999917, A9999966, 100000507, 100035077, 100050777, 100057707, 101111114, 15A000001, 170000447, 300577777, 40000A401, 447771777, 44A444441, 474000004, 477700004, 477777774, 505000003, 55555AA2A, 5555A5A2A, 700000147, 700017004, 700044004, 700077774, 700170004, 701000047, 701700004, 704000044, 704040004, 707070774, 707077704, 707770704, 707777004, 717000004, 717700007, 770000078, 770004704, 770070747, 770070774, 770700008, 770700084, 770707074, 777000044, 777000774, 777717007, 777770477, 777770747, 7777777A4, 77A700008, 888888302, 900000091, 900090799, 970009099, 990990007, 997000077, 999999997, A0000AA58, A00990001, A05555559, A44444111, A44444777, A44477777, A66666669, A90000606, A99999006, A99999099, 1000007447, 1005000007, 1500000001, ..., A000144444, ..., A900000066, ..., 33333333337, ..., 44444444447, ..., A0000000001, ..., A0014444444, ..., 40000000A0041, ..., A000000014444, ..., A044444444441, ..., A144444444411, ..., 40000000000401, ..., A0000044444441, ..., A00000000444441, ..., 11111111111111111, ..., 14444444444441111, ..., 44444444444444111, ..., A1444444444444444, ..., A9999999999999996, ..., 888888888888888883, ..., 1444444444444444444, ..., 7777777777777777771, ..., 4000000000000000A041, ..., A999999999999999999999, ..., A44444444444444444444444441, ..., 1500000000000000000000000007, ..., 40000000000000000000000000041, ..., 440000000000000000000000000001, ..., 999999999999999999999999999999991, ..., A477777777777777777777777777777777777777777, ..., 444444444444444444444444444444444444444444441, ...} 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, BBBBBB99B, ..., B0000000000000000000000000009B, ...} [/CODE] |
Base 11 5{7} family searched to around 15000 digits, without finding any (probable) primes
|
Now, we proved the set of minimal primes (start with b+1, which is equivalent to start with b, if b is composite) of base b=8:
[CODE] 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 77774444441 7777777777771 555555555555525 44444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444447 [/CODE] |
Let L(b) be the minimal set of the strings for the primes >b in base b
[CODE] b |L(b)| largest element in L(b) largest element in L(b) in base b written in decimal 2 1 11 3 3 3 111 13 4 5 221 41 5 22 10[SUB]93[/SUB]13 5^95+8 6 11 40041 5209 8 75 4[SUB]220[/SUB]7 (2^665+17)/7 [/CODE] |
[QUOTE=sweety439;567919]Minimal set of prime-strings (> base, as LaurV's suggestion) in bases 2 to 12 (only bases 2 to 8 are proved to be complete)
[CODE] 2: {11} 3: {12, 21, 111} 4: {11, 13, 23, 31, 221} 5: {12, 21, 23, 32, 34, 43, 104, 111, 131, 133, 313, 401, 414, 3101, 10103, 14444, 30301, 33001, 33331, 44441, 300031, 100000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000013} 6: {11, 15, 21, 25, 31, 35, 45, 51, 4401, 4441, 40041} 7: {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, 1022, 1051, 1112, 1202, 1211, 1222, 2111, 3031, 3055, 3334, 3503, 3505, 3545, 4504, 4555, 5011, 5455, 5545, 5554, 6034, 6634, 11111, 11201, 30011, 30101, 31001, 31111, 33001, 33311, 35555, 40054, 100121, 150001, 300053, 351101, 531101, 1100021, 33333301, 5100000001, ..., 33333333333333331, ...} 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, 77774444441, 7777777777771, 555555555555525, 44444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444447} 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, 5161111111, 8333333335, ..., 300000000035, ..., 544444444444, ..., 2000000000007, ..., 56111111111111111111111111111111111111, ..., 7666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666662, ...} 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, ..., 555555555551, ..., 5000000000000000000000000000027, ...} 11: {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, 10A, 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, 1101, 11A9, 1305, 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, 5105, 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, 9101, 9107, 9305, 9505, 9703, A052, A119, A151, A175, A515, A517, A575, A577, A5A8, A719, A779, A911, AAA9, 10011, 10075, 10091, 10109, 10411, 10444, 10705, 10709, 10774, 10901, 11104, 11131, 11144, 11191, 1141A, 114A1, 13757, 1411A, 14477, 144A4, 14A04, 14A11, 17045, 17704, 1774A, 17777, 177A4, 17A47, 1A091, 1A109, 1A114, 1A404, 1A411, 1A709, 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, 70103, 70471, 70583, 70714, 71474, 717A4, 71A09, 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, A1404, A4177, A4401, A4717, A5228, A52AA, A5558, A580A, A5822, A58AA, A5A59, A5AA2, A6096, A6966, A6999, A7051, A7778, A7808, A9055, A9091, A9699, A9969, AA52A, AA58A, 100019, 100079, 101113, 101119, 101911, 107003, 140004, 144011, 144404, 1A0019, 1A0141, 1A5001, 1A7005, 1A9001, 222223, 222823, 300107, 300202, 300323, 303203, 307577, 310007, 332003, 370777, 400555, 401A11, 404001, 404111, 405AAA, 41A011, 440A41, 441011, 451777, 455555, 470051, 470444, 474404, 4A0401, 4A4041, 500015, 500053, 500077, 500507, 505577, 522A2A, 525223, 528A2A, 531707, 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, 701074, 701777, 704408, 704417, 704457, 704484, 707041, 707441, 707708, 707744, 707784, 710777, 717044, 717077, 740008, 74484A, 770441, 770744, 770748, 770771, 777017, 777071, 777448, 777484, 777701, 7778A8, 777A19, 777A48, 778883, 78A808, 790003, 7A1009, 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, A01901, A05509, A0A058, A0A955, A10114, A555A2, A55999, A59991, A5A222, A5A22A, A60609, A66069, A66906, A69006, A79005, A87888, A90099, A90996, A96006, A96666, A97177, A97771, AA0A58, AA5A22, AAA522, 1000501, 1011141, 1030007, 1070047, 111114A, 1111A14, 1111A41, 1144441, 14A4444, 1700005, 1700474, 1A44444, 2555505, 2845055, 3030023, 3100003, 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, 7100447, 7174404, 717444A, 7400404, 7700717, 7701077, 7701707, 7707778, 7774004, 7777104, 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, A001091, A006906, A010044, A040041, A0AAA58, A141111, A5222A2, A600A69, A906606, A909009, A990009, A997701, AA55A52, AAA5552, AAAAA52, 10004747, 10005007, 17000744, 22888823, 28888223, 30010111, 30555777, 31011111, 33000023, 40A00041, 45000055, 47040004, 50377777, 50555553, 5282AAA2, 55505003, 555A5A52, 555AAA2A, 55A5A552, 5AAAAA2A, 60A99999, 70000057, 70070474, 70074704, 70174004, 70700078, 70700474, 70704704, 70710707, 70771007, 70777177, 71074004, 74470001, 77000177, 77070477, 77100077, 77470004, 77700404, 77710007, 77717707, 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, A1900001, A5555009, A5A55552, A9700001, A9909006, A9990006, A9990606, A9999917, A9999966, 100000507, 100035077, 100050777, 100057707, 101111114, 15A000001, 170000447, 300577777, 40000A401, 447771777, 44A444441, 474000004, 477700004, 477777774, 505000003, 55555AA2A, 5555A5A2A, 700000147, 700017004, 700044004, 700077774, 700170004, 701000047, 701700004, 704000044, 704040004, 707070774, 707077704, 707770704, 707777004, 717000004, 717700007, 770000078, 770004704, 770070747, 770070774, 770700008, 770700084, 770707074, 777000044, 777000774, 777717007, 777770477, 777770747, 7777777A4, 77A700008, 888888302, 900000091, 900090799, 970009099, 990990007, 997000077, 999999997, A0000AA58, A00990001, A05555559, A44444111, A44444777, A44477777, A66666669, A90000606, A99999006, A99999099, 1000007447, 1005000007, ..., A000144444, ..., A900000066, ..., 33333333337, ..., 44444444447, ..., A0000000001, ..., A0014444444, ..., 40000000A0041, ..., A000000014444, ..., A044444444441, ..., A144444444411, ..., 40000000000401, ..., A0000044444441, ..., A00000000444441, ..., 11111111111111111, ..., 14444444444441111, ..., 44444444444444111, ..., A1444444444444444, ..., A9999999999999996, ..., 888888888888888883, ..., 1444444444444444444, ..., 7777777777777777771, ..., 4000000000000000A041, ..., A999999999999999999999, ..., A44444444444444444444444441, ..., 40000000000000000000000000041, ..., 440000000000000000000000000001, ..., 999999999999999999999999999999991, ..., 444444444444444444444444444444444444444444441, ...} 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, ..., B0000000000000000000000000009B, ...} [/CODE][/QUOTE] Large minimal primes (start with b+1) base b written in standard form (a*b^n+c)/gcd(a+c,b-1) with a>=1, c != 0, gcd(a,c) = 1, gcd(b,c) = 1: Base 5: 100000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000013 = 10[SUB]93[/SUB]13 = 5^95+8 Base 7: 33333333333333331 = 3[SUB]16[/SUB]1 = (7^17-5)/2 Base 8: 7777777777771 = 7[SUB]12[/SUB]1 = 8^13-7 555555555555525 = 5[SUB]13[/SUB]25 = (5*8^15-173)/7 44444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444447 = 4[SUB]220[/SUB]7 = (4*8^221+17)/7 Base 9: 300000000035 = 30[SUB]9[/SUB]35 = 3*9^11+32 544444444444 = 54[SUB]11[/SUB] = (11*9^11-1)/2 2000000000007 = 20[SUB]11[/SUB]7 = 2*9^12+7 56111111111111111111111111111111111111 = 561[SUB]36[/SUB] = (409*9^36-1)/8 7666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666662 = 76[SUB]329[/SUB]2 = (31*9^330-19)/4 Base 10: 555555555551 = 5[SUB]11[/SUB]1 = (5*10^12-41)/9 5000000000000000000000000000027 = 50[SUB]28[/SUB]27 = 5*10^30+27 Base 12: B0000000000000000000000000009B = B0[SUB]27[/SUB]9B = 11*12^29+119 |
Note: The length of the repeating digits is always written in decimal, e.g. 9E[SUB]800873[/SUB] (in base 23) for (106*23^800873-7)/11, the largest minimal (probable) prime in base 23, and 4[SUB]220[/SUB]7 (in base 8) for (4*8^221+17)/7, the largest minimal prime (start with 2 digits) in base 8 (also, the value of a,b,c,gcd(a+c,b-1) in the formula (a*b^n+c)/gcd(a+c,b-1) are also written in decimal)
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The total proof for base 8:
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) * Case (5,1): ** [B]51[/B] is prime, and thus the only minimal prime in this family. * Case (5,3): ** [B]53[/B] is prime, and thus the only minimal prime in this family. * Case (5,5): ** Since 51, 53, 57, 15, 35, 45, 65, 75 are primes, we only need to consider the family 5{0,2,5}5 (since any digits 1, 3, 4, 6, 7 between them will produce smaller primes) *** Since 225, 255, [B]5205[/B] are primes, we only need to consider the families 5{0,5}5 and 5{0,5}25 (since any digits combo 20, 22, 25 between them will produce smaller primes) **** All numbers of the form 5{0,5}5 are divisible by 5, thus cannot be prime. **** For the 5{0,5}25 family, since [B]500025[/B] and [B]505525[/B] are primes, we only need to consider the number 500525 the families 5{5}25, 5{5}025, 5{5}0025, 5{5}0525, 5{5}00525, 5{5}05025 (since any digits combo 000, 055 between (5,25) will produce smaller primes) ***** 500525 is not prime. ***** The smallest prime of the form 5{5}25 is [B]555555555555525[/B] ***** The smallest prime of the form 5{5}025 is [B]55555025[/B] ***** The smallest prime of the form 5{5}0025 is 5555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555550025 (not minimal prime, since 55555025 and 555555555555525 are primes) ***** The smallest prime of the form 5{5}0525 is [B]5550525[/B] ***** The smallest prime of the form 5{5}00525 is [B]5500525[/B] ***** The smallest prime of the form 5{5}05025 is 5555555555555555555555505025 (not minimal prime, since 5550525, 55555025, and 555555555555525 are primes) * Case (5,7): ** [B]57[/B] is prime, and thus the only minimal prime in this family. * Case (6,1): ** Since 65, 21, 51, 631, 661 are primes, we only need to consider the family 6{0,1,4,7}1 (since any digits 2, 3, 5, 6 between them will produce smaller primes) *** Since 111, 141, 401, 471, 701, 711, 6101, 6441 are primes, we only need to consider the families 6{0}0{0,1,4,7}1, 6{0,4}1{7}1, 6{0,7}4{1}1, 6{0,1,7}7{4,7}1 (since any digits combo 11, 14, 40, 47, 70, 71, 10, 44 between them will produce smaller primes) **** For the 6{0}0{0,1,4,7}1 family, since 6007 is prime, we only need to consider the families 6{0}0{0,1,4}1 and 60{1,4,7}7{0,1,4,7}1 (since any digits combo 1007 between (6,1) will produce smaller primes) ***** For the 6{0}0{0,1,4}1 family, since 111, 141, 401, 6101, 6441, 60411 are primes, we only need to consider the families 6{0}1, 6{0}11, 6{0}41 (since any digits combo 10, 11, 14, 40, 41, 44 between (6{0}0,1) will produce smaller primes) ****** All numbers of the form 6{0}1 are divisible by 7, thus cannot be prime. ****** All numbers of the form 6{0}11 are divisible by 3, thus cannot be prime. ****** All numbers of the form 6{0}41 are divisible by 3, thus cannot be prime. ***** For the 60{1,4,7}7{0,1,4,7}1 family, since 701, 711, 60741 are primes, we only need to consider the family 60{1,4,7}7{7}1 (since any digits 0, 1, 4 between (60{1,4,7}7,1) will produce smaller primes) ***** Since 471, 60171 is prime, we only need to consider the family 60{7}1 (since any digits 1, 4 between (60,7{7}1) will produce smaller primes) ****** All numbers of the form 60{7}1 are divisible by 7, thus cannot be prime. **** For the 6{0,4}1{7}1 family, since 417, 471 are primes, we only need to consider the families 6{0}1{7}1 and 6{0,4}11 ***** For the 6{0}1{7}1 family, since 60171 is prime, and thus the only minimal prime in the family 6{0}1{7}1. ***** For the 6{0,4}11 family, since 401, 6441, 60411 are primes, we only need to consider the number 6411 and the family 6{0}11 ****** 6411 is not prime. ****** All numbers of the form 6{0}11 are divisible by 3, thus cannot be prime. **** For the 6{0,7}4{1}1 family, since 60411 is prime, we only need to consider the families 6{7}4{1}1 and 6{0,7}41 ***** For the 6{7}4{1}1 family, since 111, 6777 are primes, we only need to consider the numbers 641, 6411, 6741, 67411, 67741, 677411 ****** None of 641, 6411, 6741, 67411, 67741, 677411 are primes. ***** For the 6{0,7}41 family, since 701, 6777, 60741 are primes, we only need to consider the families 6{0}41 and the numbers 6741, 67741 (since any digits combo 07, 70, 777 between (6,41) will produce smaller primes) ****** All numbers of the form 6{0}41 are divisible by 3, thus cannot be prime. ****** Neither of 6741, 67741 are primes. ***** For the 6{0,1,7}7{4,7}1 family, since 747 is prime, we only need to consider the families 6{0,1,7}7{4}1, 6{0,1,7}7{7}1, 6{0,1,7}7{7}{4}1 (since any digits combo 47 between (6{0,1,7}7,1) will produce smaller primes) ****** For the 6{0,1,7}7{4}1 family, since 6441 is prime, we only need to consider the families 6{0,1,7}71 and 6{0,1,7}741 (since any digits combo 44 between (6{0,1,7}7,1) will produce smaller primes) ******* For the 6{0,1,7}71 family, since all numbers of the form 6{0,7}71 are divisible by 7 and cannot be prime, and 111 is prime (thus, any digits combo 11 between (6,71) will produce smaller primes), we only need to consider the family 6{0,7}1{0,7}71 ******** Since 717 and 60171 are primes, we only need to consider the family 61{0,7}71 (since any digit combo 0, 7 between (6,1{0,7}71) will produce smaller primes) ********* Since 177 and 6101 are primes, we only need to consider the number 6171 (since any digit combo 0, 7 between (61,71) will produce smaller primes) ********** 6171 is not prime. ****** All numbers in the 6{0,1,7}7{7}1 or 6{0,1,7}7{7}{4}1 families are also in the 6{0,1,7}7{4}1 family, thus these two families cannot have more minimal primes. * Case (6,3): ** Since 65, 13, 23, 53, 73, [B]643[/B] are primes, we only need to consider the family 6{0,3,6}3 (since any digits 1, 2, 4, 5, 7 between them will produce smaller primes) *** All numbers of the form 6{0,3,6}3 are divisible by 3, thus cannot be prime. * Case (6,5): ** [B]65[/B] is prime, and thus the only minimal prime in this family. * Case (6,7): ** Since 65, 27, 37, 57, [B]667[/B] are primes, we only need to consider the family 6{0,1,4,7}7 (since any digits 2, 3, 5, 6 between them will produce smaller primes) *** Since 107, 117, 147, 177, 407, 417, 717, 747, [B]6007[/B], [B]6477[/B], [B]6707[/B], [B]6777[/B] are primes, we only need to consider the families 60{1,4,7}7, 6{0}17, 6{0,4}4{4}7, 6{0}77 (since any digits combo 00, 10, 11, 14, 17, 40, 41, 47, 70, 71, 74, 77 between them will produce smaller primes) **** All numbers of the form 6{0}17 or 6{0}77 are divisible by 3, thus cannot be prime. **** For the 60{1,4,7}7 family, since 117, 147, 177, 417, 6477, 717, 747, 6777 are primes, we only need to consider the numbers 6017, 6047, 6077 and the family 60{4}7 (since any digit combo 11, 14, 17, 41, 47, 71, 74, 77 between (60,7) will produce smaller primes) ***** None of 6017, 6047, 6077 are primes. **** All numbers of the form 60{4}7 are divisible by 21, thus cannot be prime. **** For the 6{0,4}4{4}7 family, since 6007 and 407 are primes, we only need to consider the families 6{4}7 and 60{4}7 (since any digit combo 00, 40 between (6,4{4}7) will produce smaller primes) ***** All numbers of the form 6{4}7 are divisible by 3, 5, or 15, thus cannot be prime. ***** All numbers of the form 60{4}7 are divisible by 21, thus cannot be prime. * Case (7,1): ** Since 73, 75, 21, 51, [B]701[/B], [B]711[/B] are primes, we only need to consider the family 7{4,6,7}1 (since any digits 0, 1, 2, 3, 5 between them will produce smaller primes) *** Since 747, 767, 471, 661, [B]7461[/B], [B]7641[/B] are primes, we only need to consider the families 7{4,7}4{4}1, 7{7}61, 7{7}7{4,6,7}1 (since any digits combo 46, 47, 64, 66, 67 between them will produce smaller primes) **** For the 7{4,7}4{4}1 family, since 747, 471 are primes, we only need to consider the family 7{7}{4}1 (since any digits combo 47 between (7,4{4}1) will produce smaller primes) ***** The smallest prime of the form 7{7}1 is [B]7777777777771[/B] ***** The smallest prime of the form 7{7}41 is 777777777777777777777777777777777777777777777777777777777777777777777777777777741 (not minimal prime, since 7777777777771 is prime) ***** The smallest prime of the form 7{7}441 is 777777777777777777777777777777777777777777777777777777777777777777777777777777777777441 (not minimal prime, since 7777777777771 is prime) ***** The smallest prime of the form 7{7}4441 is 777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777774441 (not minimal prime, since 7777777777771 is prime) ***** The smallest prime of the form 7{7}44441 is 7777777777777777777777777777777777777777777777777777777744441 (not minimal prime, since 7777777777771 is prime) ***** All numbers of the form 7{7}444441 are divisible by 7, thus cannot be prime. ***** The smallest prime of the form 7{7}4444441 is [B]77774444441[/B] ****** Since this prime has just 4 7's, we only need to consider the families with <=3 7's ******* The smallest prime of the form 7{4}1 is [B]744444441[/B] ******* All numbers of the form 77{4}1 are divisible by 5, thus cannot be prime. ******* The smallest prime of the form 777{4}1 is 777444444444441 (not minimal prime, since 444444441 and 744444441 are primes) * Case (7,3): ** [B]73[/B] is prime, and thus the only minimal prime in this family. * Case (7,5): ** [B]75[/B] is prime, and thus the only minimal prime in this family. * Case (7,7): ** Since 73, 75, 27, 37, 57, [B]717[/B], [B]747[/B], [B]767[/B] are primes, we only need to consider the family 7{0,7}7 (since any digits 1, 2, 3, 4, 5, 6 between them will produce smaller primes) *** All numbers of the form 7{0,7}7 are divisible by 7, thus cannot be prime. |
[QUOTE=sweety439;567582]Base b minimal primes (start with 2 digits) includes:
* The smallest repunit prime base b if exists * The smallest generalized Fermat prime base b for even b if exists * The smallest generalized half Fermat prime (> (b+1)/2) base b for odd b if exists * The smallest [URL="https://www.rieselprime.de/ziki/Williams_prime_MM_table"]Williams prime with 1st kind[/URL] base b if exists * The smallest [URL="https://www.rieselprime.de/ziki/Williams_prime_MP_table"]Williams prime with 2nd kind[/URL] base b if exists * The smallest [URL="https://www.rieselprime.de/ziki/Williams_prime_PP_table"]Williams prime with 4th kind[/URL] base b [B]for bases b which no generalized Fermat primes exist (this includes all odd bases) and b is not prime (this condition is not needed if as LaurV's suggestion, the prime 10 (=b) is also excluded)[/B] if exists * The smallest dual Williams prime with 1st kind base b if exists * The smallest dual Williams prime with 2nd kind base b [B]for composite bases b (this condition is not needed if as LaurV's suggestion, the prime 10 (=b) is also excluded)[/B] if exists * The smallest dual Williams prime with 4th kind base b [B]for bases b which no generalized Fermat primes exist (this includes all odd bases) and b is not prime (this condition is not needed if as LaurV's suggestion, the prime 10 (=b) is also excluded)[/B] if exists * The smallest prime of the form 2*b^n+1 for bases b>2 if exists * The smallest prime of the form 2*b^n-1 for bases b>2 if exists * The smallest prime of the form b^n+2 for bases b>2 with gcd(b,2)=1 [B]for composite bases b (this condition is not needed if as LaurV's suggestion, the prime 10 (=b) is also excluded)[/B] if exists * The smallest prime of the form b^n-2 for bases b>2 with gcd(b,2)=1 if exists * The smallest prime of the form 3*b^n+1 for bases b>3 if exists * The smallest prime of the form 3*b^n-1 for bases b>3 if exists * The smallest prime of the form b^n+3 for bases b>3 with gcd(b,3)=1 [B]for composite bases b (this condition is not needed if as LaurV's suggestion, the prime 10 (=b) is also excluded)[/B] if exists * The smallest prime of the form b^n-3 for bases b>3 with gcd(b,3)=1 if exists * The smallest prime of the form 4*b^n+1 for bases b>4 if exists * The smallest prime of the form 4*b^n-1 for bases b>4 if exists * The smallest prime of the form b^n+4 for bases b>4 with gcd(b,4)=1 [B]for composite bases b (this condition is not needed if as LaurV's suggestion, the prime 10 (=b) is also excluded)[/B] if exists * The smallest prime of the form b^n-4 for bases b>4 with gcd(b,4)=1 if exists ... * The smallest prime of the form k*b^n+1 for fixed 1<=k<=b-1 (i.e. the prime for the [URL="http://www.noprimeleftbehind.net/crus/Sierp-conjectures.htm"]CRUS Sierpinski conjecture[/URL] for fixed 1<=k<=b-1) if exists * The smallest prime of the form k*b^n-1 for fixed 1<=k<=b-1 (i.e. the prime for the [URL="http://www.noprimeleftbehind.net/crus/Riesel-conjectures.htm"]CRUS Riesel conjecture[/URL] for fixed 1<=k<=b-1) if exists * The smallest prime of the form b^n+k for fixed 1<=k<=b-1 if exists * The smallest prime of the form b^n-k for fixed 1<=k<=b-1 if exists * The smallest prime of the form (k*b^n-1)/gcd(k-1,b-1) for fixed k with 0<=(k-1)/gcd(k-1,b-1)<=b-1 and [B]gcd(k-1,b-1) < b-1 (this reason is because if the repeating digit is 1, then this prime may not be minimal prime (start with 2 digits), unless there are no repunit primes base b (e.g. b = 9, 25, 32, 49, 64, 81, ...)[/B] (i.e. the prime for the [URL="https://mersenneforum.org/attachment.php?attachmentid=24053&d=1609098432"]extended Riesel conjecture[/URL] for fixed k satisfying these two conditions) if exists * The smallest prime of the form (b^n-k)/gcd(k-1,b-1) for fixed k with gcd(b,k) = 1 and 0<=k<=b-1[/QUOTE] There are OEIS sequences for these families in various bases: * The smallest repunit prime base b: [URL="https://oeis.org/A084740"]A084740[/URL] (exponent), [URL="https://oeis.org/A084738"]A084738[/URL] (corresponding primes) * The smallest generalized Fermat prime base b for even b: [URL="https://oeis.org/A079706"]A079706[/URL] (exponent), [URL="https://oeis.org/A228101"]A228101[/URL] (exponent of exponent), [URL="https://oeis.org/A084712"]A084712[/URL] (corresponding primes) * The smallest Williams prime with 1st kind base b: [URL="https://oeis.org/A122396"]A122396[/URL] (exponent + 1, for prime b) * The smallest Williams prime with 2nd kind base b: [URL="https://oeis.org/A305531"]A305531[/URL] (exponent), [URL="https://oeis.org/A087139"]A087139[/URL] (exponent + 1, for prime b) * The smallest dual Williams prime with 1st kind base b: [URL="https://oeis.org/A113516"]A113516[/URL] (exponent) * The smallest dual Williams prime with 2nd kind base b: [URL="https://oeis.org/A076845"]A076845[/URL] (exponent), [URL="https://oeis.org/A076846"]A076846[/URL] (corresponding primes) * The smallest prime of the form 2*b^n+1 for bases b: [URL="https://oeis.org/A119624"]A119624[/URL] (exponent), [URL="https://oeis.org/A098872"]A098872[/URL] (exponent, for b divisible by 6) * The smallest prime of the form 2*b^n-1 for bases b: [URL="https://oeis.org/A119591"]A119591[/URL] (exponent), [URL="https://oeis.org/A098873"]A098873[/URL] (exponent, for b divisible by 6) * The smallest prime of the form b^n+2 for bases b with gcd(b,2)=1: [URL="https://oeis.org/A138066"]A138066[/URL] (exponent), [URL="https://oeis.org/A084713"]A084713[/URL] (corresponding primes) * The smallest prime of the form b^n-2 for bases b with gcd(b,2)=1: [URL="https://oeis.org/A255707"]A255707[/URL] (exponent), [URL="https://oeis.org/A084714"]A084714[/URL] (corresponding primes) * The smallest prime of the form 3*b^n+1 for bases b: [URL="https://oeis.org/A098877"]A098877[/URL] (exponent, for b divisible by 6) * The smallest prime of the form 3*b^n-1 for bases b: [URL="https://oeis.org/A098876"]A098876[/URL] (exponent, for b divisible by 6) |
[QUOTE=sweety439;567582]Base b minimal primes (start with 2 digits) includes:
* The smallest repunit prime base b if exists * The smallest generalized Fermat prime base b for even b if exists * The smallest generalized half Fermat prime (> (b+1)/2) base b for odd b if exists * The smallest [URL="https://www.rieselprime.de/ziki/Williams_prime_MM_table"]Williams prime with 1st kind[/URL] base b if exists * The smallest [URL="https://www.rieselprime.de/ziki/Williams_prime_MP_table"]Williams prime with 2nd kind[/URL] base b if exists * The smallest [URL="https://www.rieselprime.de/ziki/Williams_prime_PP_table"]Williams prime with 4th kind[/URL] base b [B]for bases b which no generalized Fermat primes exist (this includes all odd bases) and b is not prime (this condition is not needed if as LaurV's suggestion, the prime 10 (=b) is also excluded)[/B] if exists * The smallest dual Williams prime with 1st kind base b if exists * The smallest dual Williams prime with 2nd kind base b [B]for composite bases b (this condition is not needed if as LaurV's suggestion, the prime 10 (=b) is also excluded)[/B] if exists * The smallest dual Williams prime with 4th kind base b [B]for bases b which no generalized Fermat primes exist (this includes all odd bases) and b is not prime (this condition is not needed if as LaurV's suggestion, the prime 10 (=b) is also excluded)[/B] if exists * The smallest prime of the form 2*b^n+1 for bases b>2 if exists * The smallest prime of the form 2*b^n-1 for bases b>2 if exists * The smallest prime of the form b^n+2 for bases b>2 with gcd(b,2)=1 [B]for composite bases b (this condition is not needed if as LaurV's suggestion, the prime 10 (=b) is also excluded)[/B] if exists * The smallest prime of the form b^n-2 for bases b>2 with gcd(b,2)=1 if exists * The smallest prime of the form 3*b^n+1 for bases b>3 if exists * The smallest prime of the form 3*b^n-1 for bases b>3 if exists * The smallest prime of the form b^n+3 for bases b>3 with gcd(b,3)=1 [B]for composite bases b (this condition is not needed if as LaurV's suggestion, the prime 10 (=b) is also excluded)[/B] if exists * The smallest prime of the form b^n-3 for bases b>3 with gcd(b,3)=1 if exists * The smallest prime of the form 4*b^n+1 for bases b>4 if exists * The smallest prime of the form 4*b^n-1 for bases b>4 if exists * The smallest prime of the form b^n+4 for bases b>4 with gcd(b,4)=1 [B]for composite bases b (this condition is not needed if as LaurV's suggestion, the prime 10 (=b) is also excluded)[/B] if exists * The smallest prime of the form b^n-4 for bases b>4 with gcd(b,4)=1 if exists ... * The smallest prime of the form k*b^n+1 for fixed 1<=k<=b-1 (i.e. the prime for the [URL="http://www.noprimeleftbehind.net/crus/Sierp-conjectures.htm"]CRUS Sierpinski conjecture[/URL] for fixed 1<=k<=b-1) if exists * The smallest prime of the form k*b^n-1 for fixed 1<=k<=b-1 (i.e. the prime for the [URL="http://www.noprimeleftbehind.net/crus/Riesel-conjectures.htm"]CRUS Riesel conjecture[/URL] for fixed 1<=k<=b-1) if exists * The smallest prime of the form b^n+k for fixed 1<=k<=b-1 if exists * The smallest prime of the form b^n-k for fixed 1<=k<=b-1 if exists * The smallest prime of the form (k*b^n-1)/gcd(k-1,b-1) for fixed k with 0<=(k-1)/gcd(k-1,b-1)<=b-1 and [B]gcd(k-1,b-1) < b-1 (this reason is because if the repeating digit is 1, then this prime may not be minimal prime (start with 2 digits), unless there are no repunit primes base b (e.g. b = 9, 25, 32, 49, 64, 81, ...)[/B] (i.e. the prime for the [URL="https://mersenneforum.org/attachment.php?attachmentid=24053&d=1609098432"]extended Riesel conjecture[/URL] for fixed k satisfying these two conditions) if exists * The smallest prime of the form (b^n-k)/gcd(k-1,b-1) for fixed k with gcd(b,k) = 1 and 0<=k<=b-1[/QUOTE] Related project searching for these primes: * repunit primes base b: [URL="http://www.fermatquotient.com/PrimSerien/GenRepu.txt"]http://www.fermatquotient.com/PrimSerien/GenRepu.txt[/URL] [URL="https://www.ams.org/journals/mcom/1993-61-204/S0025-5718-1993-1185243-9/S0025-5718-1993-1185243-9.pdf"]https://www.ams.org/journals/mcom/1993-61-204/S0025-5718-1993-1185243-9/S0025-5718-1993-1185243-9.pdf[/URL] [URL="https://web.archive.org/web/20021111141203/http://www.users.globalnet.co.uk/~aads/primes.html"]https://web.archive.org/web/20021111141203/http://www.users.globalnet.co.uk/~aads/primes.html[/URL] [URL="https://web.archive.org/web/20021114005730/http://www.users.globalnet.co.uk/~aads/titans.html"]https://web.archive.org/web/20021114005730/http://www.users.globalnet.co.uk/~aads/titans.html[/URL] [URL="http://www.primenumbers.net/Henri/us/MersFermus.htm"]http://www.primenumbers.net/Henri/us/MersFermus.htm[/URL] [URL="https://listserv.nodak.edu/cgi-bin/wa.exe?A2=NMBRTHRY;417ab0d6.0906"]https://listserv.nodak.edu/cgi-bin/wa.exe?A2=NMBRTHRY;417ab0d6.0906[/URL] * generalized Fermat prime base b for even b: [URL="http://yves.gallot.pagesperso-orange.fr/primes/index.html"]http://yves.gallot.pagesperso-orange.fr/primes/index.html[/URL] [URL="http://jeppesn.dk/generalized-fermat.html"]http://jeppesn.dk/generalized-fermat.html[/URL] [URL="http://www.noprimeleftbehind.net/crus/GFN-primes.htm"]http://www.noprimeleftbehind.net/crus/GFN-primes.htm[/URL] [URL="http://www.prothsearch.com/"]http://www.prothsearch.com/[/URL] (when b<=12) [URL="http://www.primegrid.com/stats_genefer.php"]http://www.primegrid.com/stats_genefer.php[/URL] * generalized half Fermat prime (> (b+1)/2) base b for odd b: [URL="http://www.fermatquotient.com/PrimSerien/GenFermOdd.txt"]http://www.fermatquotient.com/PrimSerien/GenFermOdd.txt[/URL] [URL="http://www.prothsearch.com/"]http://www.prothsearch.com/[/URL] (when b<=12) * Williams prime with 1st kind base b: [URL="https://harvey563.tripod.com/wills.txt"]https://harvey563.tripod.com/wills.txt[/URL] [URL="https://www.rieselprime.de/ziki/Williams_prime_MM_table"]https://www.rieselprime.de/ziki/Williams_prime_MM_table[/URL] [URL="https://www.rieselprime.de/ziki/Williams_prime_MM_least"]https://www.rieselprime.de/ziki/Williams_prime_MM_least[/URL] [URL="http://www.bitman.name/math/table/484"]http://www.bitman.name/math/table/484[/URL] [URL="http://matwbn.icm.edu.pl/ksiazki/aa/aa39/aa3912.pdf"]http://matwbn.icm.edu.pl/ksiazki/aa/aa39/aa3912.pdf[/URL] [URL="https://www.ams.org/journals/mcom/2000-69-232/S0025-5718-00-01212-6/S0025-5718-00-01212-6.pdf"]https://www.ams.org/journals/mcom/2000-69-232/S0025-5718-00-01212-6/S0025-5718-00-01212-6.pdf[/URL] * Williams prime with 2nd kind base b: [URL="https://www.rieselprime.de/ziki/Williams_prime_MP_table"]https://www.rieselprime.de/ziki/Williams_prime_MP_table[/URL] [URL="https://www.rieselprime.de/ziki/Williams_prime_MP_least"]https://www.rieselprime.de/ziki/Williams_prime_MP_least[/URL] [URL="http://www.bitman.name/math/table/477"]http://www.bitman.name/math/table/477[/URL] * Williams prime with 4th kind base b: [URL="https://www.rieselprime.de/ziki/Williams_prime_PP_table"]https://www.rieselprime.de/ziki/Williams_prime_PP_table[/URL] [URL="https://www.rieselprime.de/ziki/Williams_prime_PP_least"]https://www.rieselprime.de/ziki/Williams_prime_PP_least[/URL] [URL="http://www.bitman.name/math/table/474"]http://www.bitman.name/math/table/474[/URL] * dual Williams prime with 1st kind base b: [URL="http://www.bitman.name/math/table/435"]http://www.bitman.name/math/table/435[/URL] (when b is prime) * prime of the form 2*b^n+1 for bases b: [URL="https://mersenneforum.org/showthread.php?t=19725"]https://mersenneforum.org/showthread.php?t=19725[/URL] (when b is prime and b == 11 mod 12) [URL="https://primes.utm.edu/top20/page.php?id=37"]https://primes.utm.edu/top20/page.php?id=37[/URL] (when b is prime and b == 11 mod 12) * prime of the form b^n-2 for bases b: [URL="https://www.primepuzzles.net/puzzles/puzz_887.htm"]https://www.primepuzzles.net/puzzles/puzz_887.htm[/URL] (when b is prime) * prime of the form k*b^n+1 for base b: (this prime is minimal prime (start with 2 digits) if k<b) [URL="http://www.noprimeleftbehind.net/crus/Sierp-conjectures.htm"]http://www.noprimeleftbehind.net/crus/Sierp-conjectures.htm[/URL] [URL="http://www.noprimeleftbehind.net/crus/Sierp-conjectures-powers2.htm"]http://www.noprimeleftbehind.net/crus/Sierp-conjectures-powers2.htm[/URL] [URL="https://www.utm.edu/staff/caldwell/preprints/2to100.pdf"]https://www.utm.edu/staff/caldwell/preprints/2to100.pdf[/URL] (these three websites do not include the case where k > CK, thus the tables are not complete if the CK of this base b is <b) (these three websites exclude the case where k is rational power of b, e.g. 4*32^n+1, for this case, see the link of generalized Fermat prime base b for even b) [URL="https://www.rieselprime.de/ziki/Proth_prime_small_bases_least_n"]https://www.rieselprime.de/ziki/Proth_prime_small_bases_least_n[/URL] (for k<=12) * prime of the form k*b^n-1 for base b: (this prime is minimal prime (start with 2 digits) if k<b) [URL="http://www.noprimeleftbehind.net/crus/Riesel-conjectures.htm"]http://www.noprimeleftbehind.net/crus/Riesel-conjectures.htm[/URL] [URL="http://www.noprimeleftbehind.net/crus/Riesel-conjectures-powers2.htm"]http://www.noprimeleftbehind.net/crus/Riesel-conjectures-powers2.htm[/URL] (these two websites do not include the case where k > CK, thus the tables are not complete if the CK of this base b is <b) [URL="https://www.rieselprime.de/ziki/Riesel_prime_small_bases_least_n"]https://www.rieselprime.de/ziki/Riesel_prime_small_bases_least_n[/URL] (for k<=12) * prime of the form (k*b^n-1)/gcd(k-1,b-1) in base b (this prime is minimal prime (start with 2 digits) if (k-1)/gcd(k-1,b-1) < b) [URL="https://mersenneforum.org/attachment.php?attachmentid=24053&d=1609098432"]https://mersenneforum.org/attachment.php?attachmentid=24053&d=1609098432[/URL] See also: * [URL="https://primes.utm.edu/primes/lists/all.txt"]Top proven primes[/URL] * [URL="http://www.primenumbers.net/prptop/prptop.php"]Top PRPs[/URL] |
In odd bases, the smallest prime of the form x{0}yz or xy{0}z (where x,y,z are odd digits) is always minimal prime (start with 2 digits), since in odd bases, any number whose digits sum is even are even numbers, thus cannot be prime.
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Extended the search to bases 13 to 16, note that in base 16, family {5}45 is (16^n-49)/3, which can be factored as differences of squares, thus this family need not to be searched.
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[QUOTE=sweety439;566267]This file is the smallest prime (not include x or y themselves) in given simple family x{y} or {x}y (where x,y are base b digits) in given base 2<=b<=24, where gcd(x,y) = 1, gcd(y,b) = 1 (searched up to 5000 base b digits, 0 if no such prime found (include the case such that x{y} or {x}y proven composite by all or partial algebra factors)
format of file: b,x,{y}: smallest prime of the form x{y} in base b b,{x},y: smallest prime of the form {x}y in base b such primes are generalized near-repdigit primes base b already excluded families x{y} and {x}y with NUMERICAL covering set (e.g. {1}3, {1}4, 3{1}, 4{1} in base 5) Such primes are ALWAYS minimal prime (start with 2 digits) in base b, except when the repeating digit (i.e. y in x{y}, or x in {x}y) is 1 and base b has generalized repunit primes (i.e. all digits are 1) smaller than the prime (in base b = 9, 25, 32, 49, 64, 81, 121, 125, 144, 169, 216, 225, 243, 289, 324, 343, 361, 441, 484, 529, 625, ..., no generalized repunit primes exist, thus in these bases b, such primes are always minimal primes (start with 2 digits) in base b) Also, * in base 35, all such primes with <= 313 digits are minimal primes (start with 2 digits) * in base 39, all such primes with <= 349 digits are minimal primes (start with 2 digits) * in base 47, all such primes with <= 127 digits are minimal primes (start with 2 digits) * in base 51, all such primes with <= 4229 digits are minimal primes (start with 2 digits) * in base 91, all such primes with <= 4421 digits are minimal primes (start with 2 digits) * in base 92, all such primes with <= 439 digits are minimal primes (start with 2 digits) * in base 124, all such primes with <= 599 digits are minimal primes (start with 2 digits) * in base 135, all such primes with <= 1171 digits are minimal primes (start with 2 digits) * in base 139, all such primes with <= 163 digits are minimal primes (start with 2 digits) * in base 142, all such primes with <= 1231 digits are minimal primes (start with 2 digits) * in base 152, all such primes with <= 270217 digits are minimal primes (start with 2 digits) * in base 171, all such primes with <= 181 digits are minimal primes (start with 2 digits) * in base 174, all such primes with <= 3251 digits are minimal primes (start with 2 digits) * in base 182, all such primes with <= 167 digits are minimal primes (start with 2 digits) * in base 183, all such primes with <= 223 digits are minimal primes (start with 2 digits) * in base 184, all such primes with <= 16703 digits are minimal primes (start with 2 digits) * in base 185, all such primes with <= 66337 (at least) digits are minimal primes (start with 2 digits) * in base 199, all such primes with <= 577 digits are minimal primes (start with 2 digits) * in base 200, all such primes with <= 17807 digits are minimal primes (start with 2 digits) * in base 201, all such primes with <= 271 digits are minimal primes (start with 2 digits)[/QUOTE] Family x{d}y (where d is base-b digit, x.y are base-b strings (may be empty)) in base b means family xddd...dddy in base b, i.e. means this set {xy, xdy, xddy, xdddy, xddddy, xdddddy, xddddddy, ...} in base b |
[QUOTE=sweety439;567578]search the simple families x{0}y with gcd(x,y) = 1, gcd(y,b) = 1, gcd(x+y,b-1) = 1
Such primes are ALWAYS minimal prime (start with 2 digits) in base b, except when the base (b) is prime, and x = 1 (while 10 is prime and a subsequence of the prime, but with [URL="https://mersenneforum.org/showpost.php?p=531632&postcount=7"]LaurV's suggestion[/URL], the prime 10 (i.e. the prime = base) is also not counted just as the primes < base, all such primes (i.e. all smallest primes of the form x{0}y) is ALWAYS minimal prime (start with b+1) in base b)[/QUOTE] the formula of family x{0}y in base b is very easy, it is just x*b^n+y since any number ending with the digit 0 is divisible by the base (b) and thus cannot be prime >b (our set is the primes >b in base b), and any number cannot begin with the digit 0, therefore, all digits 0 in the numbers in the minimal set of our set (in any base b>=2) are middle digits (i.e. neither the left-most digit nor the right-most digit). |
[QUOTE=sweety439;567919]Minimal set of prime-strings (> base, as LaurV's suggestion) in bases 2 to 12 (only bases 2 to 8 are proved to be complete)[/QUOTE]
start searching bases 13 to 16 the minimal primes (start with b+1) are [CODE] 13: {14, 16, 1A, 23, 25, 2B, 32, 34, 38, 41, 47, 49, 52, 56, 58, 61, 65, 6B, 76, 7A, 7C, 83, 85, 89, 9A, A1, A7, A9, B6, B8, C1, C7, CB, 10C, 119, 11B, 122, 133, 155, 157, 173, 179, 17B, 188, 197, 1B1, 1B5, 1CC, 209, 212, 218, 229, 272, 274, 281, 287, 292, 296, 298, 29C, 2C9, 311, 313, 331, 33B, 355, 371, 373, 379, 397, 3A6, 3AA, 3B3, 3B9, 3BB, 3CA, 43C, 445, 44B, 45A, 463, 4A3, 4A5, 4B2, 4B4, 4BA, 50C, 511, 515, 533, 54A, 551, 559, 571, 575, 57B, 595, 599, 5B3, 5B9, 5CC, 607, 629, 63A, 643, 674, 704, 715, 724, 728, 731, 737, 739, 742, 751, 75B, 773, 775, 779, 782, 784, 791, 793, 797, 7B1, 812, 818, 874, 878, 8AB, 8B4, 902, 919, 922, 926, 92C, 937, 93B, 946, 95B, 962, 968, 971, 977, 979, 982, 98C, 9B3, 9B5, A03, A3C, A45, A4B, A54, AA3, AAB, B02, B0C, B11, B15, B17, B24, B33, B39, B42, B57, B59, B71, B93, B9B, BA4, BAA, BB1, BB9, BC2, BCC, C29, C43, C98, CA3, 1013, 1031, 1037, 105B, 1075, 10B7, 10BB, 1105, 1112, 1121, 1127, 113C, 1172, 1187, 1208, 1211, 1277, 12C8, 1307, 1309, 131C, 139C, 151C, 1721, 1727, 1787, 1901, 1909, 1912, 1918, 193C, 1981, 198B, 199C, 19B2, 19C3, 1B29, 1BB2, 1BBC, 1C28, 1C39, 2021, 2078, 2117, 2201, 2221, 2267, 2278, 2627, 2678, 2711, 2771, 2788, 3037, 3053, 306A, 3077, 3091, 309B, 30AC, 3305, 353C, 35AB, 35BA, 35BC, 3677, 3905, 390B, 39C5, 3A0C, 3AB5, 3B5C, 3C35, 3C59, 3C95, 403A, 40AB, 4333, 435B, 4403, 44C3, 4535, 4544, 454C, 45B5, 45BB, 480B, 4B35, 4B5B, 4C36, 5057, 5077, 509B, 50A4, 5107, 5305, 530B, 539C, 53AB, 53C9, 5444, 5455, 54C4, 5503, 5545, 55AB, 5774, 5794, 590B, 594B, 5974, 59B4, 5A4C, 5A53, 5AA4, 5AB5, 5ABB, 5ACA, 5B4B, 5B5A, 5BA5, 5CA4, 6227, 6278, 6667, 6698, 6733, 6872, 6928, 6944, 694C, 6973, 6986, 6997, 69C8, 6AC3, 6C92, 6C94, 7019, 7057, 70B5, 7103, 710B, 7118, 7127, 7129, 7172, 7178, 7192, 7211, 7217, 7219, 7271, 7303, 7408, 7433, 7444, 7505, 7507, 7574, 770B, 7774, 7778, 7787, 7871, 7877, 7888, 794B, 7994, 79B4, 7B43, 7B74, 7B94, 7BB2, 8027, 8072, 8081, 80BA, 8171, 8207, 821C, 848B, 8687, 8711, 8722, 87BB, 8867, 88B2, 88BA, 8B22, 8B2A, 8BAC, 9004, 9017, 9031, 9053, 9055, 9073, 9091, 90BB, 90C8, 9107, 9118, 913C, 9181, 91C3, 9284, 935C, 93C5, 9424, 9428, 9448, 9509, 959C, 96C4, 9703, 9743, 9745, 974B, 97B2, 9811, 981B, 987B, 98B1, 991C, 9967, 9998, 9B12, 9B74, 9B92, 9BBC, 9C55, 9C86, 9CC4, A0BA, A306, A436, A535, A5B5, A636, A6C3, A80B, AB04, AB22, AB35, AB3B, AB4C, AB55, ABAC, ABB5, AC36, ACA5, B044, B04A, B0B7, B129, B1B2, B219, B222, B291, B299, B2CA, B35A, B3A5, B404, B44C, B45B, B4B3, B501, B51C, B55A, B5A5, B5AB, B5C3, B707, B792, B794, B905, B912, B9C5, BA5B, BAB3, BB03, BB45, BB72, BBA5, BBB2, BC44, BC53, BC95, BC99, C30A, C36A, C395, C454, C535, C553, C593, C944, C953, C964, CC94, 10015, 10051, 10099, 10118, 10291, 10712, 10772, 10811, 10877, 10921, 10B92, 11111, 11135, 11171, 111C8, 11531, 11C03, 13001, 13177, 13777, 13915, 13951, 13991, 159BB, 17018, 17102, 17111, 17117, 17171, 17177, 17708, 17711, 17801, 18071, 18101, 18271, 18B27, 19003, 19153, 19315, 19351, 19591, 19913, 19951, 1C099, 20171, 20177, 20207, 20227, 20777, 21011, 21077, 2111C, 21707, 22207, 30017, 300B5, 301C9, 3033A, 303A3, 303C5, 3050B, 305C9, 3095C, 30B05, 31007, 3159B, 31999, 31C09, 3330A, 33353, 33593, 33757, 33C5C, 33CC5, 35003, 3591B, 39353, 39539, 39935, 39995, 3ACCC, 3C5C3, 3CC53, 40043, 40306, 405C4, 408BC, 40BBB, 40C54, 43066, 4366A, 4443A, 45055, 45505, 45554, 4555C, 455BC, 455C3, 45C04, 488BC, 4B03B, 4B0B5, 4B55C, 4BB0B, 4C003, 4C054, 4C5C4, 50053, 500B1, 5035A, 504B5, 5053A, 50554, 505B4, 50A35, 50B07, 50BBA, 5139B, 519BB, 51BB7, 535AC, 53A5C, 53AC5, 53BAC, 54004, 54035, 5403B, 545C3, 54B05, 54B5C, 54BBC, 54C53, 55357, 5535B, 553AC, 554BC, 55537, 55544, 5554C, 55577, 555A4, 555BB, 55A5C, 55B04, 55B55, 55B77, 55BB5, 55BC4, 55C54, 55C5A, 57403, 591BB, 59443, 59BB7, 5A044, 5AC04, 5AC35, 5B001, 5B007, 5B0AB, 5B0B4, 5B4C5, 5B544, 5B555, 5B5BB, 5B744, 5B777, 5BA0B, 5BB44, 5BB55, 5BBC4, 5BC54, 5C039, 5C35A, 5C53A, 60098, 60964, 60988, 60A63, 66094, 66377, 66692, 66694, 669C2, 669C4, 66A36, 67022, 67099, 67222, 67277, 67772, 68627, 69088, 690C4, 69808, 69994, 6A663, 7007B, 70181, 70222, 70277, 70772, 70808, 70B0B, 70B29, 71113, 71711, 71908, 71999, 7199B, 71BB7, 71BBB, 74035, 74305, 7430B, 74503, 75443, 75454, 75535, 77072, 77108, 77177, 77717, 77BBB, 78011, 79BBB, 7B007, 7B7B7, 7B7BB, 7BBB3, 7BBB7, 80117, 80221, 80771, 80777, 80807, 8084B, 80B7B, 80BBB, 81107, 8400B, 86267, 87107, 87277, 87727, 87B27, 88111, 88201, 88702, 88771, 8888B, 88B77, 88BBB, 8B1BC, 8B727, 90035, 90059, 90088, 90095, 9009B, 90101, 90103, 90305, 90488, 904CC, 90574, 90644, 9064C, 90806, 908B7, 9090B, 90994, 90B09, 90C35, 90C59, 90C64, 91111, 91135, 91315, 9180B, 92008, 92408, 92488, 93359, 93395, 944C2, 944CC, 94505, 9455C, 94804, 94888, 94C0C, 94C33, 94C4C, 95045, 95504, 95573, 955C4, 95C54, 96044, 97BBB, 98066, 98408, 98444, 98804, 98848, 99001, 99005, 9900B, 99074, 990BC, 99113, 99175, 99278, 99335, 99454, 994C3, 99517, 99593, 9984B, 99881, 99904, 99917, 99935, 99955, 99973, 999BB, 999C2, 999C4, 99B99, 9B00B, 9B04B, 9B0B4, 9B1BB, 9BB04, 9C059, 9C244, 9C404, 9C44C, 9C488, 9C503, 9C5C9, 9C644, 9C664, 9CC88, 9CCC2, A00B4, A05BB, A08B2, A08BC, A0BC4, A3336, A3633, A443A, A4443, A50BB, A55C5, A5AAC, A5BBA, A5C53, A5C55, AACC5, AB05B, AB0BB, AB40A, ABBBC, ABC4A, ACC5A, ACCC3, B0053, B0075, B010B, B0455, B0743, B0774, B0909, B0BB4, B2277, B2A2C, B3005, B351B, B37B5, B3A0B, B3ABC, B3B0A, B400A, B4035, B403B, B4053, B4305, B4BC5, B4C0A, B504B, B50BA, B530A, B5454, B54BC, B54C5, B5544, B55B5, B5B44, B5B4C, B5BB5, B7403, B7535, B77BB, B7955, B7B7B, B9207, B9504, B9999, BA055, BA305, BABC5, BAC35, BB054, BB05A, BB207, BB3B5, BB4C3, BB504, BB544, BB54C, BB5B5, BB753, BB7B7, BBABC, BBB04, BBB4C, BBB55, BBBAC, BC035, BC455, C0353, C0359, C03AC, C0904, C0959, C0A5A, C0CC5, C3059, C335C, C5A0A, C5A44, C5AAC, C6692, C69C2, C904C, C9305, C9905, C995C, C99C5, C9C04, C9C59, C9CC2, CA50A, CA5AC, CAA05, CAA5A, CC335, CC544, CC5AA, CC935, CC955, 100039, 100178, 100718, 100903, 101177, 101708, 101711, 101777, 102017, 102071, 103999, 107081, 107777, 108217, 109111, 109151, 110078, 110108, 110717, 111017, 111103, 1111C3, 111301, 111707, 113501, 115103, 117017, 117107, 117181, 117701, 120701, 13C999, 159103, 170717, 177002, 177707, 180002, 187001, 18C002, 19111C, 199903, 1B0007, 1BB077, 1BBB07, 1C0903, 1C8002, 1C9993, 200027, 207107, 217777, 219991, 220027, 222227, 270008, 271007, 277777, 290444, 300059, 300509, 303359, 303995, 309959, 30B50A, 3336AC, 333707, 33395C, 335707, 3360A3, 350009, 36660A, 3666AC, 370007, 377B07, 39001C, 399503, 3BC005, 400366, 400555, 400B3B, 400B53, 400BB5, 400CC3, 4030B5, 40B053, 40B30B, 40B505, 43600A, 450004, 4A088B, 4B0503, 4B5C05, 4BBBB5, 4BC505, 500039, 50045B, 50405B, 504B0B, 50555B, 5055B5, 505B0A, 509003, 50A50B, 50B045, 50B054, 539B01, 550054, 5500BA, 55040B, 553BC5, 5553C5, 55550B, 5555C3, 555C04, 55B00A, 55BB0B, 570007, 5A500B, 5A555B, 5AC505, 5B055B, 5B0B5B, 5B5B5C, 5B5BC5, 5BB05B, 5BBB0B, 5BBB54, 5BBBB4, 5BBC0A, 5BC405, 5C5A5A, 5CA5A5, 600694, 6060A3, 609992, 637777, 6606A3, 6660A3, 667727, 667808, 668777, 669664, 670088, 679988, 696064, 69C064, 6A6333, 700727, 700811, 700909, 70098B, 700B92, 701117, 701171, 701717, 707027, 707111, 707171, 707201, 707801, 70788B, 7080BB, 708101, 70881B, 70887B, 70B227, 710012, 710177, 711002, 711017, 711071, 717707, 718001, 718111, 720077, 722002, 727777, 74BB3B, 74BB53, 770102, 770171, 770801, 777112, 777202, 777727, 777772, 778801, 77B772, 780008, 78087B, 781001, 788B07, 79088B, 794555, 7B000B, 7B0535, 7B077B, 7B2777, 7B4BBB, 7BB4BB, 800021, 800717, 801077, 80BB07, 811117, 870077, 8777B7, 877B77, 880177, 88071B, 88077B, 8808BC, 887017, 88707B, 888227, 88877B, 8887B7, 888821, 888827, 888BB7, 8B001B, 8B00BB, 8BBB77, 8BBBB7, 900097, 900BC9, 901115, 903935, 904033, 90440C, 908008, 908866, 909359, 909C05, 90B944, 90C95C, 90CC95, 91008B, 91115C, 911503, 920888, 930335, 933503, 935903, 940033, 94040C, 940808, 94CCCC, 950005, 950744, 95555C, 9555C5, 95C003, 95C005, 96400C, 96440C, 96664C, 966664, 966994, 969094, 969964, 97008B, 97080B, 975554, 97800B, 97880B, 980006, 980864, 980B07, 984884, 986006, 986606, 986644, 988006, 988088, 988664, 988817, 988886, 988B0B, 98B007, 990115, 990151, 990694, 990B44, 990C5C, 991501, 993059, 99408B, 994555, 995404, 995435, 996694, 9978BB, 998087, 999097, 999103, 99944C, 999503, 9995C3, 999754, 999901, 99990B, 999B09, 99B4C4, 99C0C5, 99C539, 99CC05, 9B9444, 9B9909, 9C0484, 9C0808, 9C2888, 9C400C, 9C4CCC, 9C6994, 9C90C5, 9C9C5C, 9CC008, 9CC5C3, 9CC905, 9CCC08, A0055B, A005AC, A0088B, A00B2C, A00BBB, A0555C, A05CAA, A0A5AC, A0A5CA, A0AC05, A0AC5A, A0B50B, A0BB0B, A0BBB4, A0C5AC, A3660A, A5050B, A555AC, A5B00B, AA0C05, AAA05C, AAA0C5, AAC05C, AB4444, ABB00B, AC050A, AC333A, B0001B, B00099, B0030B, B004B5, B00A35, B00B54, B030BA, B05043, B0555B, B05B0A, B05B5B, B07B53, B09074, B09755, B09975, B09995, B0AB0B, B0B04B, B0B535, B0BB53, B4C055, B50003, B5003A, B500A3, B50504, B50B04, B53BC5, B54BBB, B550BB, B555BC, B55C55, B5B004, B5B0BB, B5B50B, B5B554, B5B55C, B5B5B4, B5BBB4, B5BBBC, B5BC0A, B5C045, B5C054, B70995, B70B3B, B74555, B74B55, B99921, B99945, BAC505, BB0555, BB077B, BB0B5B, BB0BB5, BB500A, BB53BC, BB53C5, BB5505, BB55BC, BB5BBA, BB5C0A, BB7BB4, BBB00A, BBB74B, BBBB54, BBBBAB, BC5054, BC5504, C00094, C00694, C009C4, C00C05, C03035, C050AA, C05309, C05404, C0544C, C05AC4, C05C39, C06092, C06694, C09035, C094CC, C09992, C09994, C09C4C, C09C95, C0CC3A, C0CC92, C33539, C35009, C4C555, C50309, C50AAA, C53009, C550A5, C555CA, C55A5A, C55CA5, C5AC55, C60094, C60694, C93335, C95405, C99094, CA05CA, CA0AC5, CA555C, CAC5CA, CC05A4, CC0AA5, CC0C05, CC3509, CC4555, CC5039, CC5554, CC555A, CC6092, CCC0C5, CCC353, CCC959, CCC9C2, 1000271, 1000802, 1000871, 1001771, 1001801, 1007078, 1008002, 1008107, 1008701, 1010117, 1027001, 1070771, 1077107, 1077701, 1080107, 1101077, 1110008, 1111078, 1115003, 1117777, 1170008, 1170101, 1700078, 1700777, 1800017, 1877017, 18B7772, 18BBB0B, 1999391, 1999931, 1BBBB3B, 2011001, 2107001, 2110001, 2700017, 2700707, 300000A, 3000019, 3000A33, 3003335, 3003395, 3009335, 300A05B, 3010009, 30A3333, 3335C09, 3339359, 3353777, 336A333, 3393959, 33AC333, 3537007, 3577777, 3636337, 3757777, 395C903, 3AC3333, 40003B5, 400B0B3, 400BBC3, 403B005, 405050B, 40B5555, 40BB555, 40CC555, 4436606, 4444306, 45C5555, 4BC5555, 4C55555, 4CC5004, 4CCC0C3, 500001B, 50003A5, 50005BA, 500B55B, 501000B, 505004B, 505B05B, 50B50B5, 50B550B, 50BB004, 5300009, 5400B0B, 54B000B, 5500BBB, 550B05B, 553000A, 5537777, 555054B, 55505BA, 5550B74, 5555054, 5555BAC, 5555C05, 555B005, 555C00A, 555CA55, 55AC005, 55AC555, 55B005B, 55CA0A5, 5A00004, 5AA5C05, 5B05B05, 5B50B05, 5B5C004, 5BBBBB5, 5BBBBCA, 5C00093, 5C003A5, 5C00A0A, 5C0A055, 5C505AA, 5C5555A, 6000692, 600A333, 606A333, 6363337, 6720002, 6906664, 7000112, 7000712, 7001201, 7001777, 7005553, 70088B7, 7009555, 7010771, 7070881, 7088107, 709800B, 70B9992, 7100021, 7100081, 7100087, 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C0608F, C0668F, C0844D, C0A0FF, C0AFF9, C0C3AF, C0C68F, C0CAAF, C0CDED, C0D0ED, C0E80D, C0EC2D, C0EC8D, C0FA0F, C0FAAF, C2CC63, C30CAF, C333AF, C3CAAF, C3CCAF, C4048D, C40D4D, C4404D, C4408D, C4440D, C44DDD, C4ACCB, C4DCCB, C4DD4D, C6068F, C66AAF, C68AAF, C6AA8F, C8044D, C8440D, C8666F, CA00FF, CA0FFF, CAAAAF, CAAFFF, CAFF0F, CBE0CD, CC008F, CC0C8F, CC3CAF, CC4ACB, CC608F, CC66AF, CCBECD, CCC4AB, CCCA0F, CCCC8F, CCCE8D, CE0C8D, CF0F23, CF0FAF, CFAFFF, CFCAAF, CFFAFF, D0005D, D00BA9, D05EDD, D077D7, D10CCB, D22207, D4000B, D4040D, D4044D, D40CCB, D70077, D7D00D, D90009, D900BB, DB00BB, DB4441, DD400D, DDD109, DDD1A9, DDD919, DDD941, DED00D, E00D4D, E00EEB, E0AAE1, E0AE41, E0AEA1, E0B44D, E0BCCD, E0BEBB, E0D0DD, E0E441, E4048D, E4448D, E800CD, E8200D, EA0E41, EAA0E1, EBB00B, ECCCAB, EDDDDD, EEBE0B, F00263, F0056F, F00A45, F02C63, F03F23, F05405, F060AF, F08585, F0A4A5, F0F2C3, F0F323, F2CCC3, F33203, F33C23, F5F66F, F5FF6F, F68CCF, F6AA8F, F888AF, FA0F45, FAA045, FAA545, FAFC69, FC0AAF, FC66AF, FCCCAF, FCFFAF, FF0323, FF056F, FF3203, FF7903, FFA045, FFA4A5, FFAA45, FFC0AF, FFF4A5, FFF575, FFFA45, FFFCAF, 10A009B, 20000D1, 2CCC663, 30A00FF, 30CCCAF, 30FA00F, 30FCCAF, 3333C23, 333C2C3, 33C3AAF, 33FCAAF, 33FFFAF, 3A0A00F, 3AAAA0F, 3AF000F, 3AFAAAF, 3C0CA0F, 3CCC3AF, 3CFF323, 3F33F23, 3FAA00F, 3FF3323, 4004441, 400DDD1, 400E00D, 400ED0D, 404404D, 404448D, 404E4DD, 440EDDD, 4440EDD, 44444ED, 4444E4D, 44DDDDD, 4A000A5, 4CCCCAB, 4D0CCCB, 4E4404D, 4E4444D, 4E4DDDD, 5000021, 5004221, 5006AAF, 500FF6F, 5042201, 508CCCD, 5400005, 5400AA5, 5555405, 5808007, 5AA4005, 5C0008D, 5CCC8CD, 5D4444D, 5EEEEEB, 5F40005, 5F554A5, 5F6AAAF, 60000AF, 60006A9, 600866F, 6008AAF, 600AA8F, 600F6A9, 606608F, 606686F, 608666F, 60AA08F, 60AAA8F, 66000AF, 66666A9, 6666AF9, 6866A8F, 6AAAAAF, 70070D7, 70077DD, 700DDDD, 707077D, 707D007, 70D00DD, 770077D, 770400D, 770740D, 7777775, 77777B7, 77777DD, 7777ACB, 77B88E7, 77DD00D, 77DDDDD, 7D0D00D, 7DD0D07, 7DDD00D, 800002D, 8000CED, 80C0E0D, 80CECCD, 840400D, 844000D, 844E00D, 868688F, 880444D, 884404D, 887D007, 8888801, 8888881, 8888E07, 8888F77, 8888FE7, 88A8AFF, 88AAAFF, 88FAFFF, 8A8AAAF, 8A8AAFF, 8AAA8FF, 8C00ECD, 8C8444D, 8E4400D, 8FCCCCF, 900BBAB, 90CC4AB, 9908AA1, 99E0E01, 9B00801, 9B6CCC9, A000FF9, A006069, A00A8FF, A01CCCB, A05F545, A0BEEEB, A0E4AA1, AA0008F, AA08FFF, AA40AA5, AA8FFFF, AAAA405, AE04AA1, AE44441, AE4AAA1, AECCCCB, AF40005, AFA5A45, AFFFC69, B000BAB, B000EBB, B0D0007, B222227, B6CCCC9, B8880A1, BA000EB, BA0BEEB, BAEEEEB, BB000CD, BB00C0D, BB0B00D, BC6CC69, BC6CCC9, BCCCC69, BCCCCED, C0000A9, C00068F, C000CFD, C000E2D, C000FAF, C004D4D, C00E20D, C00E8CD, C00F68F, C033A0F, C0802CD, C086AAF, C0A00AF, C0AFFFF, C0C086F, C0C0F8F, C0CA00F, C0CC08F, C0D044D, C0F0AFF, C0FF023, C0FFFAF, C33FA0F, C33FAAF, C3CA00F, C3FFCAF, C8002CD, C8200CD, CCC668F, CCCAA8F, CCCC0A9, CCCC3AF, CCCCCA9, CCCDC4B, CE0008D, CE2000D, CE8CCCD, CF000AF, CFF0AAF, CFFF0AF, D0000EB, D0005EB, D000775, D000EDD, D007077, D00DDD9, D00ED0D, D0AAA45, D0AAAA5, D0EDDDD, D19000B, D4404ED, D4440ED, D5BBBBB, DCCCC4B, DD00DD9, DD07077, DD0DD09, DD0DDD9, DD99999, DDD0D09, DDDD0D9, DDDD9E1, DDDDD09, DDDDD99, DE0DDDD, DEEEEEB, E00001B, E0004A1, E000CAB, E00A041, E00BB0B, E00BBBB, E00C80D, E00CCCB, E044DDD, E0AA4A1, E0AAA41, E0BBB0B, E0D444D, E40444D, E4DDD4D, E88CCCD, E8C000D, E8CCCCD, EA04441, EA0A4A1, EBB000D, EBCCCCD, ED0D00D, EEAAA01, EEBBBBB, EEE000B, F0002C3, F002CC3, F003323, F005545, F00F4A5, F033323, F0400A5, F0A5545, F333323, F333F23, F6660AF, F733333, FA00009, FA004A5, FAAAA45, FC6668F, FCC668F, FD00AA5, FEE7777, FF0F263, FF26003, FF3F323, FF5F887, FFAFF45, FFFF263, FFFF379, 2CCCCC63, 30CCA00F, 33333319, 3333FCAF, 3333FFAF, 33FFA00F, 3C00CCAF, 3C00FCAF, 3CF3FF23, 40000441, 40000CAB, 4000DAA1, 400440DD, 400ACCCB, 400CCCAB, 400E44DD, 4040D00D, 404400DD, 40444EDD, 4044D00D, 40ACCCCB, 40DDDDDD, 440000D1, 44000DDD, 4400DD0D, 44E400DD, 4A00004B, 4A0AAAA5, 5000C08D, 52000CCD, 555400A5, 55540A05, 58800007, 58888087, 5A540005, 5C00020D, 5F5400A5, 5F888887, 60006AAF, 600093AF, 600AAAAF, 608CCCCF, 6600686F, 6606866F, 6688AAAF, 7000077D, 70000D5D, 7000707B, 7000707D, 7000740D, 70500D0D, 7070040D, 707007DD, 7070777B, 7077744D, 7077777B, 77007D0D, 7700B44D, 7707000B, 7707D00D, 7770700D, 7770777B, 7777740D, 7777770B, 7777777D, 77777CAB, 7777B887, 778888E7, 788888E7, 79333333, 7ACCCCCB, 7D0000DD, 7D00D0DD, 7DD00D0D, 7DDDDDA9, 80000081, 80000087, 8000E0CD, 80400E4D, 80A0AAA1, 80EC000D, 84000E4D, 8404444D, 84400E4D, 868AAAAF, 86AAAA8F, 8884044D, 88FFFE77, 8C44444D, 8CCCCAAF, 8E40004D, 900000BB, 90000B0B, 90100009, 90800AA1, 93333AAF, 94AAAAA1, 980000A1, 998AAAA1, A00000F9, A0000EEB, A0005A45, A0055545, A00AAA45, A0666669, A0AAA045, A0AAAA45, A0AAE4A1, A0B44441, A4A00005, A6066669, A8AAFFFF, AA055545, AA0AA045, AAA00A45, AAAAA045, B00000AB, B000EEEB, B00EEE0B, B0900081, B0BBBBAB, B7777787, B9000081, B9008001, B9800001, BA00000B, BBBB0ABB, BCCCCCC9, ...} [/CODE] |
Largest minimal primes in simple families for bases 13 to 16 (written in decimal):
x{y} and {x}y: [CODE] 13, {1}, 1: 30941 13, 1, {1}: 30941 13, {1}, 2: 2381 13, 1, {2}: 197 13, {1}, 3: 883708283 13, 1, {3}: 211 13, {1}, 4: 17 13, 1, {4}: 17 13, 1, {5}: 239 13, {1}, 6: 19 13, 1, {6}: 19 13, 1, {7}: 253217502498750291800692183145337720992638880271493569431738157631027569095215561 13, 1, {8}: 281 13, {1}, 9: 191 13, 1, {9}: 27130132404659193376721686434661 13, {1}, 10: 23 13, 1, {10}: 23 13, {1}, 11: 193 13, 1, {11}: 820195757799727198696695842441476208994963187388611974376331352215246160014077762227387500472080168786083449145277773838858800795991958180632280974279961571375401216690067428169654353503039823288371270608465421195831630752643523277310127185899536482776399744897654945387104284638368882957170938237278749541598968947323011480936838588602269948473325034154837529102013688233548651627077006312693219100274314806851388327828512220445130387062692274032398838117351850000439212156970074507805886649022038706852334408911410638474707605620484783396663735375992701354072765197190488304749330337810593696686818871540035682031309739770552183078238960961 13, {1}, 12: 36898271981403391525359432679065451 13, 1, {12}: 337 13, {2}, 1: 4759 13, 2, {1}: 106637277112689077 13, {2}, 3: 29 13, 2, {3}: 29 13, {2}, 5: 31 13, 2, {5}: 31 13, {2}, 7: 804473 13, 2, {7}: 959173 13, {2}, 9: 373 13, {2}, 11: 37 13, 2, {11}: 37 13, {3}, 1: 547 13, 3, {1}: 521 13, {3}, 2: 41 13, 3, {2}: 41 13, {3}, 4: 43 13, 3, {4}: 43 13, {3}, 5: 6220138738168647434831423806501836269264860388724755065149565951442236175936796719067011700334477855412396475124991691344796424350196661821800919998163958711515136934394634097179800502201458093822871 13, 3, {5}: 577 13, 3, {7}: 2923035083 13, {3}, 8: 47 13, 3, {8}: 47 13, 3, {10}: 647 13, {3}, 11: 557 13, 3, {11}: 661 13, {4}, 1: 53 13, 4, {1}: 53 13, {4}, 3: 3534833123 13, 4, {3}: 9337 13, {4}, 5: 733 13, 4, {5}: 13799574804865291194219202692403868309075434457702185936718792737 13, {4}, 7: 59 13, 4, {7}: 59 13, {4}, 9: 61 13, 4, {9}: 61 13, {4}, 11: 739 13, {5}, 1: 911 13, 5, {1}: 859 13, {5}, 2: 67 13, 5, {2}: 67 13, {5}, 3: 4418541403 13, 5, {3}: 887 13, {5}, 4: 277256920492991599 13, 5, {4}: 11717 13, {5}, 6: 71 13, 5, {6}: 71 13, 5, {7}: 89921716241132417850870043495861784708189815671814466610282151602592361843582839405888101361717803180497433838124247307746249298923864405720133765156792108299126338169033221693415179426566058950783399084612596039386427021973466021161486411164221010228501523 13, {5}, 8: 73 13, 5, {8}: 73 13, {5}, 9: 919 13, 5, {9}: 971 13, {5}, 11: 3604339966408890811 13, 5, {11}: 815662743439 13, 5, {12}: 1013 13, {6}, 1: 79 13, 6, {1}: 79 13, {6}, 5: 83 13, 6, {5}: 83 13, {6}, 7: 14281 13, 6, {7}: 9624308578305020228623 13, {6}, 11: 89 13, 6, {11}: 89 13, {7}, 1: 178099219309623994113347865115830245881192672331136086644940932390281437773862542264058288467257753659801600961075130547842788841314191170557163588785806820352498775469685611756153126667905685532469070927112200362751893732722602014850726541568809237898479136618304580608401918281234930143064257920309170965969499129532757126377921069913907908471671716574775646406870993578835592914767961359736945068963155952491960373644449418976551172435322847621930721542144058540845006834135041368564051814099612330378729974668784588333951075548113502723356005589020090498660438660159816508080021058414799091444088419142837713138761057222122289170719363548408846549381700350017033339161308029642370902736513421829569951497819557491865511689513661142885392815700489495652470272195143512237819660601060635160718950091398065703223634277265560584842616957317643843802700518533058037207116004784352250055837483109124201522964355475650386943932211405298597905457035869654221665584889762003382262425915627119116388998518127939798007098237014217655662706251901683109346257926396141021838296111310372968225022079707942822795129744698997404408233889930805048836987175297692891091686536077480054995854699722419639970465336199946812908942288140516904083829095941075082576747429233198452931491612278651981884049653724609828879600848950258125404977404144280808141884961849927596265042459349498828074205132275149771178481317527349922743964367067517723323072236797216719438348669059261865669008875549670954883841127641347209610157306553559175511522826280497259599838188113180381333755455900349307408532955497746124837076056334168406604339828775363723268449080391071951364995984584279858031164184210011834081 13, {7}, 2: 2815633 13, 7, {2}: 425554642597531069476088349319265344520247822549860785329030452393239592223053102610453 13, {7}, 3: 1277 13, {7}, 4: 16657 13, 7, {4}: 16111 13, {7}, 5: 1279 13, {7}, 6: 97 13, 7, {6}: 97 13, {7}, 8: 16661 13, 7, {8}: 16843 13, {7}, 9: 1283 13, {7}, 10: 101 13, 7, {10}: 101 13, {7}, 11: 176677145512151 13, {7}, 12: 103 13, 7, {12}: 103 13, {8}, 1: 201916737728161 13, {8}, 3: 107 13, 8, {3}: 107 13, {8}, 5: 109 13, 8, {5}: 109 13, {8}, 7: 948090435833789898349839159224250448489822963231314463923791 13, 8, {7}: 91021952951 13, {8}, 9: 113 13, 8, {9}: 113 13, {8}, 11: 247531 13, 8, {11}: 107838136118779143544110144382426092596204365203552428434248370827269303781694390904131147239057991725836898042412261207750257957583868509005708271654322505648777710176276577432900281939636108415883809919452408705622432187985384594384558596142925993309255985009535307478523156581759235779167162869019703196846751228017411378948853640401871368029723240618590933942573810568367308325289 13, {9}, 1: 121215212228974701436931483878589891140303089832220953724825443452928225193998713337930372911067805791799869147809411855365238746332705480023760792811129961633638311977064834817213347105220337863208732908350674718836123175940417328579685338580206495293657407565505774954042639105332034212418635645417015477944123471063765451261909023510740139021 13, 9, {1}: 259429 13, 9, {2}: 1549 13, {9}, 4: 68799997644951462493799712001 13, 9, {4}: 36748846266526697 13, {9}, 5: 1513040167438129868806971467209988783047172742217267483154894329901717663980542315652349848778853894915555170807951322572378209558278635157172717513628242832984817118678991631720815452108000099210400379825153731084262244319493812890311789045794884092831356123242061758082732806604188005520228674939217665328388067373699901394102699076617987792004875548394303529748128202045356868637958421168851264492251414451671881645020719873271254395994874212298538928180953385418967331680881078358158544926445810098124494008083153487609262254200180462767265649895085196812223539172699394835278304012594889076972978898251549233575133572026086685197059747614441040152909798221820479642718133459387770777298554261474133030690753241864544153004060794064644652707512761735881885543957256884192628158805686561809594061924393932922840678322622784462425947093214038759553887929649916953155720669786551358215226363554462047120571340688059948977555505902980499655568680185582793237970604923030974220265795047325241015419809657491893019473031136829858500145730181190585716816328018616913990223894736777230358689553647254936178082288866348094694739653443883605310239766413123460690534793812785934073248428969751425870653389046775244015718236495145673518071665801891709920766181983586792103960637565600979310820525883807031028492116501924047338205921075081927116926896242518609770952106202341273961329572020549726342482699063714395183214771739428693004365223506346305628182421466017950778081664584660392184816346242270287808532367340974202505243 13, 9, {5}: 0 13, 9, {7}: 1619 13, {9}, 8: 21419 13, 9, {8}: 1332632087873 13, {9}, 10: 127 13, 9, {10}: 127 13, {9}, 11: 193272315039175487326233511816066059384236614936125602856112933108936878519237472904660314716980770567675038998906449425413153983914911198004481338508457524880312210951611198276223531290887108690292738026664067061295245572474703602610246125803179560118335836826468952746508374948156130103443176084512499609881640980577720176544209084145250756004293999763606271755552801884151163797064193306051490695131926580100826963694188422295198532753628423986864538180785129561486984798531474623258138617820724652921878726441978127950818375993469697424536456843865475449486163418975049457428208240361723655765920087962062440804142883445511641266703241468837936119086839611677856638014115541263266704520590558533279312304037176827720788243643056423797345536794811687465498057714283585292329995513313040502867333497441355894682583294242422882099383092926007095933296099723170595507590307152252179729407846066017402908790460912572771982724554085389116205103789017519828339606017138742770709364271683876516827264126734705389614550068455954555083870819276074206224473000405522699421186643568300931 13, 9, {11}: 188465890767567927768109 13, {10}, 1: 131 13, 10, {1}: 131 13, {10}, 3: 1823 13, {10}, 7: 137 13, 10, {7}: 137 13, {10}, 9: 139 13, 10, {9}: 139 13, {10}, 11: 1831 13, 10, {11}: 19564417634903 13, {11}, 1: 2003 13, 11, {1}: 1873 13, {11}, 2: 26171 13, 11, {2}: 24533 13, {11}, 3: 9720791083 13, 11, {3}: 1901 13, 11, {4}: 9244948171 13, {11}, 5: 16947116540528994433003374971133476766755585467759746042637787 13, 11, {5}: 55106069 13, {11}, 6: 149 13, 11, {6}: 149 13, 11, {7}: 275614952732329653238232466846964959764342234851112847239174811330605042147473869409955112827912038180459570611709057122713557738754852977301670028971129467515066229407333223877783672306282366369543871115968877415280318537960852589377566524955025517265527862915865085457853330890387002896522575722038083366786743993408179910457825505003824935898016650965949690327645173471015392053739386753124739151576511051033085127452855493526018723218218416151702223054562562122092583257452732969841804887384398911206575041813220337355043752278695720428137 13, {11}, 8: 151 13, 11, {8}: 151 13, {11}, 9: 2011 13, 11, {9}: 335591 13, {11}, 10: 52950113757237678592993805854491324369268684216704693554936717423836128996957000831399984295437037597224946915093310665683439 13, 11, {10}: 1999 13, {11}, 12: 16836900297891418080414469547118518955584357920776290786511507224819852347973193037600665289070901330976115445902783343792856149076064327963454445124840887022352433623214149015015943271257627167012185236811023315748308075343126054090560004563875124190448995227748073744916159908957819701603274854998000296763254125672206384758348891742961717040363229489213108521955314350073857925001010097317113705164622416602981584525394558649693204742511309000575073486313783914987497483013408328355077527202814535784777000148396721007194688339582681878366906510944731328876064735814127172451578146421749559114747412555063799277435883965467381 13, 11, {12}: 2027 13, {12}, 1: 157 13, 12, {1}: 157 13, {12}, 5: 1792160394029 13, {12}, 7: 163 13, 12, {7}: 163 13, {12}, 11: 167 13, 12, {11}: 167 14, {1}, 1: 211 14, 1, {1}: 211 14, {1}, 3: 17 14, 1, {3}: 17 14, {1}, 5: 19 14, 1, {5}: 19 14, {1}, 9: 23 14, 1, {9}: 23 14, {1}, 11: 41381 14, 1, {11}: 70921 14, {1}, 13: 223 14, 1, {13}: 76831 14, {2}, 1: 29 14, 2, {1}: 29 14, {2}, 3: 31 14, 2, {3}: 31 14, 2, {5}: 467 14, {2}, 9: 37 14, 2, {9}: 37 14, {2}, 11: 431 14, 2, {11}: 557 14, {2}, 13: 41 14, 2, {13}: 41 14, {3}, 1: 43 14, 3, {1}: 43 14, {3}, 5: 47 14, 3, {5}: 47 14, {3}, 11: 53 14, 3, {11}: 53 14, {3}, 13: 643 14, {4}, 1: 32434921 14, 4, {1}: 156619 14, {4}, 3: 59 14, 4, {3}: 59 14, {4}, 5: 61 14, 4, {5}: 61 14, 4, {9}: 919 14, {4}, 11: 67 14, 4, {11}: 67 14, {4}, 13: 853 14, 4, {13}: 0 14, {5}, 1: 71 14, 5, {1}: 71 14, {5}, 3: 73 14, 5, {3}: 73 14, {5}, 9: 79 14, 5, {9}: 79 14, {5}, 11: 1061 14, {5}, 13: 83 14, 5, {13}: 83 14, {6}, 1: 48652381 14, {6}, 5: 89 14, 6, {5}: 89 14, {6}, 11: 248231 14, {6}, 13: 97 14, 6, {13}: 97 14, {7}, 1: 1471 14, 7, {1}: 271867 14, {7}, 3: 101 14, 7, {3}: 101 14, {7}, 5: 103 14, 7, {5}: 103 14, {7}, 9: 107 14, 7, {9}: 107 14, {7}, 11: 109 14, 7, {11}: 109 14, {7}, 13: 1483 14, 7, {13}: 1567 14, {8}, 1: 113 14, 8, {1}: 113 14, 8, {3}: 1613 14, 8, {5}: 1527891201751406184274498849901197440218663 14, {8}, 9: 14893189141836674105809869073123801 14, 8, {9}: 333923 14, {8}, 11: 3178964943473909010162782853260586588933938914164889347525573201287387822569505763054988368562457051 14, 8, {11}: 1733 14, {8}, 13: 1693 14, 8, {13}: 0 14, {9}, 1: 127 14, 9, {1}: 127 14, {9}, 5: 131 14, 9, {5}: 131 14, {9}, 11: 137 14, 9, {11}: 137 14, {9}, 13: 139 14, 9, {13}: 139 14, {10}, 1: 81087301 14, {10}, 3: 450545636966997425132717095409966070465562893779498582938637002140743 14, 10, {3}: 5502353 14, {10}, 9: 149 14, 10, {9}: 149 14, {10}, 11: 151 14, 10, {11}: 151 14, {10}, 13: 2113 14, {11}, 1: 2311 14, 11, {1}: 0 14, {11}, 3: 157 14, 11, {3}: 157 14, 11, {5}: 437351 14, {11}, 9: 163 14, 11, {9}: 163 14, {11}, 13: 167 14, 11, {13}: 167 14, {12}, 1: 2521 14, 12, {1}: 1273070779 14, {12}, 5: 173 14, 12, {5}: 173 14, {12}, 11: 179 14, 12, {11}: 179 14, {12}, 13: 181 14, 12, {13}: 181 14, {13}, 1: 2731 14, 13, {1}: 19298779963 14, 13, {3}: 2593 14, {13}, 5: 0 14, 13, {5}: 276540164647 14, {13}, 9: 191 14, 13, {9}: 191 14, {13}, 11: 193 14, 13, {11}: 193 15, {1}, 1: 241 15, 1, {1}: 241 15, {1}, 2: 17 15, 1, {2}: 17 15, {1}, 4: 19 15, 1, {4}: 19 15, {1}, 7: 303629285816214089001622765197550636782710041318620954247 15, 1, {7}: 337 15, {1}, 8: 23 15, 1, {8}: 23 15, {1}, 11: 251 15, 1, {11}: 401 15, {1}, 13: 12204253 15, 1, {13}: 433 15, {1}, 14: 29 15, 1, {14}: 29 15, {2}, 1: 31 15, 2, {1}: 31 15, {2}, 7: 37 15, 2, {7}: 37 15, {2}, 11: 41 15, 2, {11}: 41 15, {2}, 13: 43 15, 2, {13}: 43 15, {3}, 1: 417041800362721 15, 3, {1}: 691 15, {3}, 2: 47 15, 3, {2}: 47 15, {3}, 4: 2440849 15, 3, {4}: 739 15, {3}, 7: 727 15, 3, {7}: 787 15, {3}, 8: 53 15, 3, {8}: 53 15, {3}, 11: 162731 15, 3, {11}: 491182564871651 15, {3}, 13: 733 15, 3, {13}: 883 15, {3}, 14: 59 15, 3, {14}: 59 15, {4}, 1: 61 15, 4, {1}: 61 15, {4}, 7: 67 15, 4, {7}: 67 15, {4}, 11: 71 15, 4, {11}: 71 15, {4}, 13: 73 15, 4, {13}: 73 15, {5}, 1: 1201 15, 5, {1}: 2924441266741 15, {5}, 2: 18077 15, 5, {2}: 3905357 15, {5}, 4: 79 15, 5, {4}: 79 15, {5}, 7: 13729771207 15, 5, {7}: 1237 15, {5}, 8: 83 15, 5, {8}: 83 15, {5}, 11: 271211 15, 5, {11}: 1301 15, {5}, 13: 1213 15, 5, {13}: 15194280133 15, {5}, 14: 89 15, 5, {14}: 89 15, {6}, 1: 4881691 15, 6, {1}: 1037360491 15, {6}, 7: 97 15, 6, {7}: 97 15, {6}, 11: 101 15, 6, {11}: 101 15, {6}, 13: 103 15, 6, {13}: 103 15, {7}, 1: 379681 15, 7, {1}: 80547991 15, {7}, 2: 107 15, 7, {2}: 107 15, {7}, 4: 109 15, 7, {4}: 109 15, {7}, 8: 113 15, 7, {8}: 113 15, {7}, 11: 218946945190429691 15, 7, {11}: 2588949882704871041434151 15, {7}, 13: 1693 15, 7, {13}: 1783 15, {8}, 1: 28921 15, 8, {1}: 27241 15, {8}, 7: 127 15, 8, {7}: 127 15, {8}, 11: 131 15, 8, {11}: 131 15, {8}, 13: 1933 15, 8, {13}: 30133 15, {9}, 1: 2161 15, 9, {1}: 103329241 15, {9}, 2: 137 15, 9, {2}: 137 15, {9}, 4: 139 15, 9, {4}: 139 15, {9}, 7: 24713588167 15, 9, {7}: 2137 15, {9}, 8: 1647572543 15, 9, {8}: 2153 15, {9}, 11: 488171 15, 9, {11}: 495401 15, {9}, 13: 5560557338173 15, 9, {13}: 502633 15, {9}, 14: 149 15, 9, {14}: 149 15, {10}, 1: 151 15, 10, {1}: 151 15, {10}, 7: 157 15, 10, {7}: 157 15, {10}, 11: 2411 15, 10, {11}: 8190401 15, {10}, 13: 163 15, 10, {13}: 163 15, {11}, 1: 881786829315764563424246641 15, 11, {1}: 560491 15, {11}, 2: 167 15, 11, {2}: 167 15, {11}, 4: 39769 15, 11, {4}: 2539 15, {11}, 7: 2647 15, 11, {7}: 130992187 15, {11}, 8: 173 15, 11, {8}: 173 15, {11}, 13: 596653 15, 11, {13}: 2683 15, {11}, 14: 179 15, 11, {14}: 179 15, {12}, 1: 181 15, 12, {1}: 181 15, {12}, 7: 2887 15, 12, {7}: 42187 15, {12}, 11: 191 15, 12, {11}: 191 15, {12}, 13: 193 15, 12, {13}: 193 15, {13}, 1: 3121 15, 13, {1}: 661741 15, {13}, 2: 197 15, 13, {2}: 197 15, {13}, 4: 199 15, 13, {4}: 199 15, {13}, 7: 705127 15, 13, {7}: 3037 15, {13}, 8: 10577003 15, 13, {8}: 2318805803 15, {13}, 11: 91488544954572405131 15, 13, {11}: 157027901 15, {13}, 14: 535461077009 15, 13, {14}: 408700964355468749 15, {14}, 1: 211 15, 14, {1}: 211 15, {14}, 11: 3371 15, 14, {11}: 42527645637007506364690405981881277901 15, {14}, 13: 223 15, 14, {13}: 223 16, {1}, 1: 17 16, 1, {1}: 17 16, {1}, 3: 19 16, 1, {3}: 19 16, {1}, 5: 277 16, 1, {5}: 0 16, {1}, 7: 23 16, 1, {7}: 23 16, {1}, 9: 281 16, 1, {9}: 409 16, {1}, 11: 283 16, 1, {11}: 443 16, {1}, 13: 29 16, 1, {13}: 29 16, {1}, 15: 31 16, 1, {15}: 31 16, {2}, 1: 8737 16, 2, {1}: 581714951868689 16, {2}, 3: 547 16, 2, {3}: 563 16, {2}, 5: 37 16, 2, {5}: 37 16, {2}, 7: 725935716098002055388532495854438851111 16, 2, {7}: 631 16, {2}, 9: 41 16, 2, {9}: 41 16, {2}, 11: 43 16, 2, {11}: 43 16, {2}, 13: 557 16, 2, {13}: 733 16, {2}, 15: 47 16, 2, {15}: 47 16, {3}, 1: 253530120045645880299340641073 16, 3, {1}: 13171233041 16, {3}, 5: 53 16, 3, {5}: 53 16, {3}, 7: 823 16, 3, {7}: 887 16, {3}, 11: 59 16, 3, {11}: 59 16, {3}, 13: 61 16, 3, {13}: 61 16, {4}, 1: 0 16, 4, {1}: 16657 16, {4}, 3: 67 16, 4, {3}: 67 16, {4}, 5: 1093 16, 4, {5}: 1109 16, {4}, 7: 71 16, 4, {7}: 71 16, {4}, 9: 73 16, 4, {9}: 73 16, {4}, 11: 17483 16, 4, {11}: 19387 16, 4, {13}: 444540081354816304286954136617869418478679481821 16, {4}, 15: 79 16, 4, {15}: 79 16, {5}, 1: 1361 16, 5, {1}: 1297 16, {5}, 3: 83 16, 5, {3}: 83 16, {5}, 7: 1367 16, 5, {7}: 1399 16, {5}, 9: 89 16, 5, {9}: 89 16, {5}, 11: 21851 16, 5, {11}: 1613789866474427 16, {5}, 13: 1373 16, 5, {13}: 24029 16, {6}, 1: 97 16, 6, {1}: 97 16, {6}, 5: 101 16, 6, {5}: 101 16, {6}, 7: 103 16, 6, {7}: 103 16, {6}, 11: 107 16, 6, {11}: 107 16, {6}, 13: 109 16, 6, {13}: 109 16, {7}, 1: 113 16, 7, {1}: 113 16, {7}, 3: 1907 16, 7, {3}: 0 16, {7}, 5: 125269877 16, 7, {5}: 1877 16, {7}, 9: 1913 16, 7, {9}: 498073 16, {7}, 11: 32069089147 16, 7, {11}: 1979 16, {7}, 13: 2004318077 16, 7, {13}: 9972184721795404625107398548957 16, {7}, 15: 127 16, 7, {15}: 127 16, {8}, 1: 143165569 16, 8, {1}: 0 16, {8}, 3: 131 16, 8, {3}: 131 16, {8}, 5: 34949 16, 8, {5}: 0 16, {8}, 7: 56166555556563832905556281431290897236744050880292859335632521351 16, 8, {7}: 34679 16, {8}, 9: 137 16, 8, {9}: 137 16, {8}, 11: 139 16, 8, {11}: 139 16, {8}, 13: 8947853 16, 8, {13}: 2269 16, 8, {15}: 0 16, {9}, 1: 39313 16, 9, {1}: 594193 16, {9}, 5: 149 16, 9, {5}: 149 16, {9}, 7: 151 16, 9, {7}: 151 16, {9}, 11: 2459 16, 9, {11}: 637883 16, {9}, 13: 157 16, 9, {13}: 157 16, {10}, 1: 733007751841 16, 10, {1}: 41233 16, {10}, 3: 163 16, 10, {3}: 163 16, {10}, 7: 167 16, 10, {7}: 167 16, {10}, 9: 2729 16, 10, {9}: 2713 16, {10}, 11: 2731 16, 10, {11}: 43963 16, {10}, 13: 173 16, 10, {13}: 173 16, {11}, 1: 48049 16, 11, {1}: 2833 16, {11}, 3: 179 16, 11, {3}: 179 16, {11}, 5: 181 16, 11, {5}: 181 16, {11}, 7: 2999 16, 11, {7}: 49248958327 16, {11}, 9: 3001 16, 11, {9}: 2969 16, {11}, 13: 12303293 16, 11, {13}: 3037 16, {11}, 15: 191 16, 11, {15}: 191 16, {12}, 1: 193 16, 12, {1}: 193 16, {12}, 5: 197 16, 12, {5}: 197 16, {12}, 7: 199 16, 12, {7}: 199 16, {12}, 11: 0 16, 12, {11}: 3259 16, {12}, 13: 0 16, 12, {13}: 843229 16, {13}, 1: 3722304977 16, 13, {1}: 0 16, {13}, 3: 211 16, 13, {3}: 211 16, {13}, 5: 3541 16, 13, {5}: 3413 16, {13}, 7: 908759 16, 13, {7}: 882551 16, {13}, 9: 999198637325934041 16, 13, {9}: 73749768669482915691491069321318626688914012237296060805206525525363591813836272035774910527919776180923677912554968891935394987986240496179621997893655532569315694783635803112700208508303413378891202384198997794213422176304573334413545606463639516918296257466344350258969657713796137622531892391975484473872559979575727003547753581022912486703477573912049826765132053211177341761946288632815391744689614375401028459100583268869723038988660204614984245471691470023113466364417874725337512714447532250846778586077760659205293618044147237229216306717637040861705334444181470200752974579322509544738704990857820454867202261704090678797538558326245584064671252468247095559023662993480878895077936090038599163027885118597295012047000583187251486272120581780591332114804425847265975339536610666721934463637123044596968540088909735294287236975077640291184075261370631154339959438296960077041200837369288569872632621900878776171600339056308989379011831334774153700978537796879728062464221403872190754080933322090903482454554035388405527848042146114638163297543714301665763439499883062039897590145382317298449300948509392928786633391596392577283746404680586571591203059252060135897745743650628381225179713605144028836751431506430002554673660083008864161778283325595456594614681 16, {13}, 11: 3547 16, 13, {11}: 0 16, {13}, 15: 223 16, 13, {15}: 223 16, {14}, 1: 61153 16, {14}, 3: 227 16, 14, {3}: 227 16, {14}, 5: 229 16, 14, {5}: 229 16, {14}, 9: 233 16, 14, {9}: 233 16, {14}, 11: 1300876803247619683256250232571154421182187 16, 14, {11}: 58304019973926508829195794288364830930948296694792337729075131089632305865113112154282586276837027138881157744159537913596428450753911485229682880296019616116874769754520009659 16, {14}, 13: 3821 16, 14, {13}: 15588829 16, {14}, 15: 239 16, 14, {15}: 239 16, {15}, 1: 241 16, 15, {1}: 241 16, {15}, 7: 0 16, 15, {7}: 66428827511 16, {15}, 11: 251 16, 15, {11}: 251 16, {15}, 13: 4093 16, 15, {13}: 1039837 [/CODE] x{0}y: [CODE] 13, 1, 4: 17 13, 1, 6: 19 13, 1, 10: 23 13, 1, 12: 181 13, 2, 3: 29 13, 2, 5: 31 13, 2, 9: 347 13, 2, 11: 37 13, 3, 2: 41 13, 3, 4: 43 13, 3, 8: 47 13, 3, 10: 14480437 13, 4, 1: 53 13, 4, 3: 54020737582614507942458917440610823901767221634062888289700852810815079724323258965596943883469013562594127941397398999016481603427622205011908900266071076096475248216593088654359952926621741299090558666311191216978954704152791 13, 4, 7: 59 13, 4, 9: 61 13, 5, 2: 67 13, 5, 6: 71 13, 5, 8: 73 13, 5, 12: 857 13, 6, 1: 79 13, 6, 5: 83 13, 6, 7: 1021 13, 6, 11: 89 13, 7, 4: 1187 13, 7, 6: 97 13, 7, 10: 101 13, 7, 12: 103 13, 8, 3: 107 13, 8, 5: 109 13, 8, 9: 113 13, 8, 11: 38614483 13, 9, 2: 1523 13, 9, 4: 19777 13, 9, 8: 43441289 13, 9, 10: 127 13, 10, 1: 131 13, 10, 3: 1693 13, 10, 7: 137 13, 10, 9: 139 13, 11, 2: 1861 13, 11, 6: 149 13, 11, 8: 151 13, 11, 12: 1871 13, 12, 1: 157 13, 12, 5: 2729251996728070131798006327033140931418231688924554627711654775633915597843291014832282995312758790354592964227251689466987151209150175544113 13, 12, 7: 163 13, 12, 11: 167 14, 1, 1: 197 14, 1, 3: 17 14, 1, 5: 19 14, 1, 9: 23 14, 1, 11: 0 14, 1, 13: 2177953337809371149 14, 2, 1: 29 14, 2, 3: 31 14, 2, 5: 397 14, 2, 9: 37 14, 2, 13: 41 14, 3, 1: 43 14, 3, 5: 47 14, 3, 11: 53 14, 3, 13: 601 14, 4, 1: 0 14, 4, 3: 59 14, 4, 5: 61 14, 4, 11: 67 14, 4, 13: 797 14, 5, 1: 71 14, 5, 3: 73 14, 5, 9: 79 14, 5, 11: 991 14, 5, 13: 83 14, 6, 1: 45177217 14, 6, 5: 89 14, 6, 11: 1187 14, 6, 13: 97 14, 7, 1: 1373 14, 7, 3: 101 14, 7, 5: 103 14, 7, 9: 107 14, 7, 11: 109 14, 7, 13: 0 14, 8, 1: 113 14, 8, 3: 1571 14, 8, 9: 21961 14, 8, 11: 1579 14, 8, 13: 0 14, 9, 1: 127 14, 9, 5: 131 14, 9, 11: 137 14, 9, 13: 139 14, 10, 1: 75295361 14, 10, 9: 149 14, 10, 11: 151 14, 10, 13: 1973 14, 11, 1: 0 14, 11, 3: 157 14, 11, 5: 2161 14, 11, 9: 163 14, 11, 13: 167 14, 12, 5: 173 14, 12, 11: 179 14, 12, 13: 181 14, 13, 1: 2549 14, 13, 3: 2551 14, 13, 5: 35677 14, 13, 9: 191 14, 13, 11: 193 15, 1, 2: 17 15, 1, 4: 19 15, 1, 8: 23 15, 1, 14: 29 15, 2, 1: 31 15, 2, 7: 37 15, 2, 11: 41 15, 2, 13: 43 15, 3, 2: 47 15, 3, 8: 53 15, 3, 14: 59 15, 4, 1: 61 15, 4, 7: 67 15, 4, 11: 71 15, 4, 13: 73 15, 5, 4: 79 15, 5, 8: 83 15, 5, 14: 89 15, 6, 7: 97 15, 6, 11: 101 15, 6, 13: 103 15, 7, 2: 107 15, 7, 4: 109 15, 7, 8: 113 15, 8, 1: 1801 15, 8, 7: 127 15, 8, 11: 131 15, 9, 2: 137 15, 9, 4: 139 15, 9, 8: 1537734383 15, 9, 14: 149 15, 10, 1: 151 15, 10, 7: 157 15, 10, 13: 163 15, 11, 2: 167 15, 11, 4: 8353129 15, 11, 8: 173 15, 11, 14: 179 15, 12, 1: 181 15, 12, 7: 2707 15, 12, 11: 191 15, 12, 13: 193 15, 13, 2: 197 15, 13, 4: 199 15, 13, 14: 2939 15, 14, 1: 211 15, 14, 11: 10631261 15, 14, 13: 223 16, 1, 1: 17 16, 1, 3: 19 16, 1, 7: 23 16, 1, 13: 29 16, 1, 15: 31 16, 2, 5: 37 16, 2, 9: 41 16, 2, 11: 43 16, 2, 15: 47 16, 3, 1: 769 16, 3, 5: 53 16, 3, 11: 59 16, 3, 13: 61 16, 4, 3: 67 16, 4, 7: 71 16, 4, 9: 73 16, 4, 13: 274877906957 16, 4, 15: 79 16, 5, 3: 83 16, 5, 9: 89 16, 5, 11: 1291 16, 6, 1: 97 16, 6, 5: 101 16, 6, 7: 103 16, 6, 11: 107 16, 6, 13: 109 16, 7, 1: 113 16, 7, 9: 1801 16, 7, 15: 127 16, 8, 3: 131 16, 8, 5: 2053 16, 8, 9: 137 16, 8, 11: 139 16, 8, 15: 2063 16, 9, 5: 149 16, 9, 7: 151 16, 9, 13: 157 16, 10, 1: 40961 16, 10, 3: 163 16, 10, 7: 167 16, 10, 9: 11529215046068469769 16, 10, 13: 173 16, 11, 3: 179 16, 11, 5: 181 16, 11, 15: 191 16, 12, 1: 193 16, 12, 5: 197 16, 12, 7: 199 16, 12, 11: 3083 16, 13, 1: 3329 16, 13, 3: 211 16, 13, 9: 55834574857 16, 13, 15: 223 16, 14, 3: 227 16, 14, 5: 229 16, 14, 9: 233 16, 14, 15: 239 16, 15, 1: 241 16, 15, 7: 3847 16, 15, 11: 251 16, 15, 13: 3853 [/CODE] |
[QUOTE=sweety439;568171]Largest minimal primes in simple families for bases 13 to 16 (written in decimal):
x{y} and {x}y: [CODE] 13, {1}, 1: 30941 13, 1, {1}: 30941 13, {1}, 2: 2381 13, 1, {2}: 197 13, {1}, 3: 883708283 13, 1, {3}: 211 13, {1}, 4: 17 13, 1, {4}: 17 13, 1, {5}: 239 13, {1}, 6: 19 13, 1, {6}: 19 13, 1, {7}: 253217502498750291800692183145337720992638880271493569431738157631027569095215561 13, 1, {8}: 281 13, {1}, 9: 191 13, 1, {9}: 27130132404659193376721686434661 13, {1}, 10: 23 13, 1, {10}: 23 13, {1}, 11: 193 13, 1, {11}: 820195757799727198696695842441476208994963187388611974376331352215246160014077762227387500472080168786083449145277773838858800795991958180632280974279961571375401216690067428169654353503039823288371270608465421195831630752643523277310127185899536482776399744897654945387104284638368882957170938237278749541598968947323011480936838588602269948473325034154837529102013688233548651627077006312693219100274314806851388327828512220445130387062692274032398838117351850000439212156970074507805886649022038706852334408911410638474707605620484783396663735375992701354072765197190488304749330337810593696686818871540035682031309739770552183078238960961 13, {1}, 12: 36898271981403391525359432679065451 13, 1, {12}: 337 13, {2}, 1: 4759 13, 2, {1}: 106637277112689077 13, {2}, 3: 29 13, 2, {3}: 29 13, {2}, 5: 31 13, 2, {5}: 31 13, {2}, 7: 804473 13, 2, {7}: 959173 13, {2}, 9: 373 13, {2}, 11: 37 13, 2, {11}: 37 13, {3}, 1: 547 13, 3, {1}: 521 13, {3}, 2: 41 13, 3, {2}: 41 13, {3}, 4: 43 13, 3, {4}: 43 13, 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15, 9, {14}: 149 15, {10}, 1: 151 15, 10, {1}: 151 15, {10}, 7: 157 15, 10, {7}: 157 15, {10}, 11: 2411 15, 10, {11}: 8190401 15, {10}, 13: 163 15, 10, {13}: 163 15, {11}, 1: 881786829315764563424246641 15, 11, {1}: 560491 15, {11}, 2: 167 15, 11, {2}: 167 15, {11}, 4: 39769 15, 11, {4}: 2539 15, {11}, 7: 2647 15, 11, {7}: 130992187 15, {11}, 8: 173 15, 11, {8}: 173 15, {11}, 13: 596653 15, 11, {13}: 2683 15, {11}, 14: 179 15, 11, {14}: 179 15, {12}, 1: 181 15, 12, {1}: 181 15, {12}, 7: 2887 15, 12, {7}: 42187 15, {12}, 11: 191 15, 12, {11}: 191 15, {12}, 13: 193 15, 12, {13}: 193 15, {13}, 1: 3121 15, 13, {1}: 661741 15, {13}, 2: 197 15, 13, {2}: 197 15, {13}, 4: 199 15, 13, {4}: 199 15, {13}, 7: 705127 15, 13, {7}: 3037 15, {13}, 8: 10577003 15, 13, {8}: 2318805803 15, {13}, 11: 91488544954572405131 15, 13, {11}: 157027901 15, {13}, 14: 535461077009 15, 13, {14}: 408700964355468749 15, {14}, 1: 211 15, 14, {1}: 211 15, {14}, 11: 3371 15, 14, {11}: 42527645637007506364690405981881277901 15, {14}, 13: 223 15, 14, {13}: 223 16, {1}, 1: 17 16, 1, {1}: 17 16, {1}, 3: 19 16, 1, {3}: 19 16, {1}, 5: 277 16, 1, {5}: 0 16, {1}, 7: 23 16, 1, {7}: 23 16, {1}, 9: 281 16, 1, {9}: 409 16, {1}, 11: 283 16, 1, {11}: 443 16, {1}, 13: 29 16, 1, {13}: 29 16, {1}, 15: 31 16, 1, {15}: 31 16, {2}, 1: 8737 16, 2, {1}: 581714951868689 16, {2}, 3: 547 16, 2, {3}: 563 16, {2}, 5: 37 16, 2, {5}: 37 16, {2}, 7: 725935716098002055388532495854438851111 16, 2, {7}: 631 16, {2}, 9: 41 16, 2, {9}: 41 16, {2}, 11: 43 16, 2, {11}: 43 16, {2}, 13: 557 16, 2, {13}: 733 16, {2}, 15: 47 16, 2, {15}: 47 16, {3}, 1: 253530120045645880299340641073 16, 3, {1}: 13171233041 16, {3}, 5: 53 16, 3, {5}: 53 16, {3}, 7: 823 16, 3, {7}: 887 16, {3}, 11: 59 16, 3, {11}: 59 16, {3}, 13: 61 16, 3, {13}: 61 16, {4}, 1: 0 16, 4, {1}: 16657 16, {4}, 3: 67 16, 4, {3}: 67 16, {4}, 5: 1093 16, 4, {5}: 1109 16, {4}, 7: 71 16, 4, {7}: 71 16, {4}, 9: 73 16, 4, {9}: 73 16, {4}, 11: 17483 16, 4, {11}: 19387 16, 4, {13}: 444540081354816304286954136617869418478679481821 16, {4}, 15: 79 16, 4, {15}: 79 16, {5}, 1: 1361 16, 5, {1}: 1297 16, {5}, 3: 83 16, 5, {3}: 83 16, {5}, 7: 1367 16, 5, {7}: 1399 16, {5}, 9: 89 16, 5, {9}: 89 16, {5}, 11: 21851 16, 5, {11}: 1613789866474427 16, {5}, 13: 1373 16, 5, {13}: 24029 16, {6}, 1: 97 16, 6, {1}: 97 16, {6}, 5: 101 16, 6, {5}: 101 16, {6}, 7: 103 16, 6, {7}: 103 16, {6}, 11: 107 16, 6, {11}: 107 16, {6}, 13: 109 16, 6, {13}: 109 16, {7}, 1: 113 16, 7, {1}: 113 16, {7}, 3: 1907 16, 7, {3}: 0 16, {7}, 5: 125269877 16, 7, {5}: 1877 16, {7}, 9: 1913 16, 7, {9}: 498073 16, {7}, 11: 32069089147 16, 7, {11}: 1979 16, {7}, 13: 2004318077 16, 7, {13}: 9972184721795404625107398548957 16, {7}, 15: 127 16, 7, {15}: 127 16, {8}, 1: 143165569 16, 8, {1}: 0 16, {8}, 3: 131 16, 8, {3}: 131 16, {8}, 5: 34949 16, 8, {5}: 0 16, {8}, 7: 56166555556563832905556281431290897236744050880292859335632521351 16, 8, {7}: 34679 16, {8}, 9: 137 16, 8, {9}: 137 16, {8}, 11: 139 16, 8, {11}: 139 16, {8}, 13: 8947853 16, 8, {13}: 2269 16, 8, {15}: 0 16, {9}, 1: 39313 16, 9, {1}: 594193 16, {9}, 5: 149 16, 9, {5}: 149 16, {9}, 7: 151 16, 9, {7}: 151 16, {9}, 11: 2459 16, 9, {11}: 637883 16, {9}, 13: 157 16, 9, {13}: 157 16, {10}, 1: 733007751841 16, 10, {1}: 41233 16, {10}, 3: 163 16, 10, {3}: 163 16, {10}, 7: 167 16, 10, {7}: 167 16, {10}, 9: 2729 16, 10, {9}: 2713 16, {10}, 11: 2731 16, 10, {11}: 43963 16, {10}, 13: 173 16, 10, {13}: 173 16, {11}, 1: 48049 16, 11, {1}: 2833 16, {11}, 3: 179 16, 11, {3}: 179 16, {11}, 5: 181 16, 11, {5}: 181 16, {11}, 7: 2999 16, 11, {7}: 49248958327 16, {11}, 9: 3001 16, 11, {9}: 2969 16, {11}, 13: 12303293 16, 11, {13}: 3037 16, {11}, 15: 191 16, 11, {15}: 191 16, {12}, 1: 193 16, 12, {1}: 193 16, {12}, 5: 197 16, 12, {5}: 197 16, {12}, 7: 199 16, 12, {7}: 199 16, {12}, 11: 0 16, 12, {11}: 3259 16, {12}, 13: 0 16, 12, {13}: 843229 16, {13}, 1: 3722304977 16, 13, {1}: 0 16, {13}, 3: 211 16, 13, {3}: 211 16, {13}, 5: 3541 16, 13, {5}: 3413 16, {13}, 7: 908759 16, 13, {7}: 882551 16, {13}, 9: 999198637325934041 16, 13, {9}: 73749768669482915691491069321318626688914012237296060805206525525363591813836272035774910527919776180923677912554968891935394987986240496179621997893655532569315694783635803112700208508303413378891202384198997794213422176304573334413545606463639516918296257466344350258969657713796137622531892391975484473872559979575727003547753581022912486703477573912049826765132053211177341761946288632815391744689614375401028459100583268869723038988660204614984245471691470023113466364417874725337512714447532250846778586077760659205293618044147237229216306717637040861705334444181470200752974579322509544738704990857820454867202261704090678797538558326245584064671252468247095559023662993480878895077936090038599163027885118597295012047000583187251486272120581780591332114804425847265975339536610666721934463637123044596968540088909735294287236975077640291184075261370631154339959438296960077041200837369288569872632621900878776171600339056308989379011831334774153700978537796879728062464221403872190754080933322090903482454554035388405527848042146114638163297543714301665763439499883062039897590145382317298449300948509392928786633391596392577283746404680586571591203059252060135897745743650628381225179713605144028836751431506430002554673660083008864161778283325595456594614681 16, {13}, 11: 3547 16, 13, {11}: 0 16, {13}, 15: 223 16, 13, {15}: 223 16, {14}, 1: 61153 16, {14}, 3: 227 16, 14, {3}: 227 16, {14}, 5: 229 16, 14, {5}: 229 16, {14}, 9: 233 16, 14, {9}: 233 16, {14}, 11: 1300876803247619683256250232571154421182187 16, 14, {11}: 58304019973926508829195794288364830930948296694792337729075131089632305865113112154282586276837027138881157744159537913596428450753911485229682880296019616116874769754520009659 16, {14}, 13: 3821 16, 14, {13}: 15588829 16, {14}, 15: 239 16, 14, {15}: 239 16, {15}, 1: 241 16, 15, {1}: 241 16, {15}, 7: 0 16, 15, {7}: 66428827511 16, {15}, 11: 251 16, 15, {11}: 251 16, {15}, 13: 4093 16, 15, {13}: 1039837 [/CODE] x{0}y: [CODE] 13, 1, 4: 17 13, 1, 6: 19 13, 1, 10: 23 13, 1, 12: 181 13, 2, 3: 29 13, 2, 5: 31 13, 2, 9: 347 13, 2, 11: 37 13, 3, 2: 41 13, 3, 4: 43 13, 3, 8: 47 13, 3, 10: 14480437 13, 4, 1: 53 13, 4, 3: 54020737582614507942458917440610823901767221634062888289700852810815079724323258965596943883469013562594127941397398999016481603427622205011908900266071076096475248216593088654359952926621741299090558666311191216978954704152791 13, 4, 7: 59 13, 4, 9: 61 13, 5, 2: 67 13, 5, 6: 71 13, 5, 8: 73 13, 5, 12: 857 13, 6, 1: 79 13, 6, 5: 83 13, 6, 7: 1021 13, 6, 11: 89 13, 7, 4: 1187 13, 7, 6: 97 13, 7, 10: 101 13, 7, 12: 103 13, 8, 3: 107 13, 8, 5: 109 13, 8, 9: 113 13, 8, 11: 38614483 13, 9, 2: 1523 13, 9, 4: 19777 13, 9, 8: 43441289 13, 9, 10: 127 13, 10, 1: 131 13, 10, 3: 1693 13, 10, 7: 137 13, 10, 9: 139 13, 11, 2: 1861 13, 11, 6: 149 13, 11, 8: 151 13, 11, 12: 1871 13, 12, 1: 157 13, 12, 5: 2729251996728070131798006327033140931418231688924554627711654775633915597843291014832282995312758790354592964227251689466987151209150175544113 13, 12, 7: 163 13, 12, 11: 167 14, 1, 1: 197 14, 1, 3: 17 14, 1, 5: 19 14, 1, 9: 23 14, 1, 11: 0 14, 1, 13: 2177953337809371149 14, 2, 1: 29 14, 2, 3: 31 14, 2, 5: 397 14, 2, 9: 37 14, 2, 13: 41 14, 3, 1: 43 14, 3, 5: 47 14, 3, 11: 53 14, 3, 13: 601 14, 4, 1: 0 14, 4, 3: 59 14, 4, 5: 61 14, 4, 11: 67 14, 4, 13: 797 14, 5, 1: 71 14, 5, 3: 73 14, 5, 9: 79 14, 5, 11: 991 14, 5, 13: 83 14, 6, 1: 45177217 14, 6, 5: 89 14, 6, 11: 1187 14, 6, 13: 97 14, 7, 1: 1373 14, 7, 3: 101 14, 7, 5: 103 14, 7, 9: 107 14, 7, 11: 109 14, 7, 13: 0 14, 8, 1: 113 14, 8, 3: 1571 14, 8, 9: 21961 14, 8, 11: 1579 14, 8, 13: 0 14, 9, 1: 127 14, 9, 5: 131 14, 9, 11: 137 14, 9, 13: 139 14, 10, 1: 75295361 14, 10, 9: 149 14, 10, 11: 151 14, 10, 13: 1973 14, 11, 1: 0 14, 11, 3: 157 14, 11, 5: 2161 14, 11, 9: 163 14, 11, 13: 167 14, 12, 5: 173 14, 12, 11: 179 14, 12, 13: 181 14, 13, 1: 2549 14, 13, 3: 2551 14, 13, 5: 35677 14, 13, 9: 191 14, 13, 11: 193 15, 1, 2: 17 15, 1, 4: 19 15, 1, 8: 23 15, 1, 14: 29 15, 2, 1: 31 15, 2, 7: 37 15, 2, 11: 41 15, 2, 13: 43 15, 3, 2: 47 15, 3, 8: 53 15, 3, 14: 59 15, 4, 1: 61 15, 4, 7: 67 15, 4, 11: 71 15, 4, 13: 73 15, 5, 4: 79 15, 5, 8: 83 15, 5, 14: 89 15, 6, 7: 97 15, 6, 11: 101 15, 6, 13: 103 15, 7, 2: 107 15, 7, 4: 109 15, 7, 8: 113 15, 8, 1: 1801 15, 8, 7: 127 15, 8, 11: 131 15, 9, 2: 137 15, 9, 4: 139 15, 9, 8: 1537734383 15, 9, 14: 149 15, 10, 1: 151 15, 10, 7: 157 15, 10, 13: 163 15, 11, 2: 167 15, 11, 4: 8353129 15, 11, 8: 173 15, 11, 14: 179 15, 12, 1: 181 15, 12, 7: 2707 15, 12, 11: 191 15, 12, 13: 193 15, 13, 2: 197 15, 13, 4: 199 15, 13, 14: 2939 15, 14, 1: 211 15, 14, 11: 10631261 15, 14, 13: 223 16, 1, 1: 17 16, 1, 3: 19 16, 1, 7: 23 16, 1, 13: 29 16, 1, 15: 31 16, 2, 5: 37 16, 2, 9: 41 16, 2, 11: 43 16, 2, 15: 47 16, 3, 1: 769 16, 3, 5: 53 16, 3, 11: 59 16, 3, 13: 61 16, 4, 3: 67 16, 4, 7: 71 16, 4, 9: 73 16, 4, 13: 274877906957 16, 4, 15: 79 16, 5, 3: 83 16, 5, 9: 89 16, 5, 11: 1291 16, 6, 1: 97 16, 6, 5: 101 16, 6, 7: 103 16, 6, 11: 107 16, 6, 13: 109 16, 7, 1: 113 16, 7, 9: 1801 16, 7, 15: 127 16, 8, 3: 131 16, 8, 5: 2053 16, 8, 9: 137 16, 8, 11: 139 16, 8, 15: 2063 16, 9, 5: 149 16, 9, 7: 151 16, 9, 13: 157 16, 10, 1: 40961 16, 10, 3: 163 16, 10, 7: 167 16, 10, 9: 11529215046068469769 16, 10, 13: 173 16, 11, 3: 179 16, 11, 5: 181 16, 11, 15: 191 16, 12, 1: 193 16, 12, 5: 197 16, 12, 7: 199 16, 12, 11: 3083 16, 13, 1: 3329 16, 13, 3: 211 16, 13, 9: 55834574857 16, 13, 15: 223 16, 14, 3: 227 16, 14, 5: 229 16, 14, 9: 233 16, 14, 15: 239 16, 15, 1: 241 16, 15, 7: 3847 16, 15, 11: 251 16, 15, 13: 3853 [/CODE][/QUOTE] Forms are not listed here means they have a NUMERICAL covering set (including trivial 1-cover) (only for families x{y} and {x}y, for families x{0}y, only families with trivial 1-cover (the trivial factor must divides either b or b-1) are not listed) Forms with 0 means either they make a full covering set with (all or partial) ALGEBRAIC factors or they are not ruled out as only contain composites but have no primes or PRPs with length <= 5000 |
All of the primes are minimal primes (start with b+1) except the case which the repeating digit (i.e. x in {x}y, or y in x{y}) is 1 (the x{0}y case is always minimal primes (start with b+1), but if x = 1 and the base is prime, then the corresponding primes are not minimal primes (start with b))
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[QUOTE=sweety439;568170]start searching bases 13 to 16
the minimal primes (start with b+1) are[/QUOTE] newest data: [CODE] 13: {14, 16, 1A, 23, 25, 2B, 32, 34, 38, 41, 47, 49, 52, 56, 58, 61, 65, 6B, 76, 7A, 7C, 83, 85, 89, 9A, A1, A7, A9, B6, B8, C1, C7, CB, 10C, 119, 11B, 122, 133, 155, 157, 173, 179, 17B, 188, 197, 1B1, 1B5, 1CC, 209, 212, 218, 229, 272, 274, 281, 287, 292, 296, 298, 29C, 2C9, 311, 313, 331, 33B, 355, 371, 373, 379, 397, 3A6, 3AA, 3B3, 3B9, 3BB, 3CA, 43C, 445, 44B, 45A, 463, 4A3, 4A5, 4B2, 4B4, 4BA, 50C, 511, 515, 533, 54A, 551, 559, 571, 575, 57B, 595, 599, 5B3, 5B9, 5CC, 607, 629, 63A, 643, 674, 704, 715, 724, 728, 731, 737, 739, 742, 751, 75B, 773, 775, 779, 782, 784, 791, 793, 797, 7B1, 812, 818, 874, 878, 8AB, 8B4, 902, 919, 922, 926, 92C, 937, 93B, 946, 95B, 962, 968, 971, 977, 979, 982, 98C, 9B3, 9B5, A03, A3C, A45, A4B, A54, AA3, AAB, B02, B0C, B11, B15, B17, B24, B33, B39, B42, B57, B59, B71, B93, B9B, BA4, BAA, BB1, BB9, BC2, BCC, C29, C43, C98, CA3, 1013, 1031, 1037, 105B, 1075, 10B7, 10BB, 1105, 1112, 1121, 1127, 113C, 1172, 1187, 1208, 1211, 1277, 12C8, 1307, 1309, 131C, 139C, 151C, 1721, 1727, 1787, 1901, 1909, 1912, 1918, 193C, 1981, 198B, 199C, 19B2, 19C3, 1B29, 1BB2, 1BBC, 1C28, 1C39, 2021, 2078, 2117, 2201, 2221, 2267, 2278, 2627, 2678, 2711, 2771, 2788, 3037, 3053, 306A, 3077, 3091, 309B, 30AC, 3305, 353C, 35AB, 35BA, 35BC, 3677, 3905, 390B, 39C5, 3A0C, 3AB5, 3B5C, 3C35, 3C59, 3C95, 403A, 40AB, 4333, 435B, 4403, 44C3, 4535, 4544, 454C, 45B5, 45BB, 480B, 4B35, 4B5B, 4C36, 5057, 5077, 509B, 50A4, 5107, 5305, 530B, 539C, 53AB, 53C9, 5444, 5455, 54C4, 5503, 5545, 55AB, 5774, 5794, 590B, 594B, 5974, 59B4, 5A4C, 5A53, 5AA4, 5AB5, 5ABB, 5ACA, 5B4B, 5B5A, 5BA5, 5CA4, 6227, 6278, 6667, 6698, 6733, 6872, 6928, 6944, 694C, 6973, 6986, 6997, 69C8, 6AC3, 6C92, 6C94, 7019, 7057, 70B5, 7103, 710B, 7118, 7127, 7129, 7172, 7178, 7192, 7211, 7217, 7219, 7271, 7303, 7408, 7433, 7444, 7505, 7507, 7574, 770B, 7774, 7778, 7787, 7871, 7877, 7888, 794B, 7994, 79B4, 7B43, 7B74, 7B94, 7BB2, 8027, 8072, 8081, 80BA, 8171, 8207, 821C, 848B, 8687, 8711, 8722, 87BB, 8867, 88B2, 88BA, 8B22, 8B2A, 8BAC, 9004, 9017, 9031, 9053, 9055, 9073, 9091, 90BB, 90C8, 9107, 9118, 913C, 9181, 91C3, 9284, 935C, 93C5, 9424, 9428, 9448, 9509, 959C, 96C4, 9703, 9743, 9745, 974B, 97B2, 9811, 981B, 987B, 98B1, 991C, 9967, 9998, 9B12, 9B74, 9B92, 9BBC, 9C55, 9C86, 9CC4, A0BA, A306, A436, A535, A5B5, A636, A6C3, A80B, AB04, AB22, AB35, AB3B, AB4C, AB55, ABAC, ABB5, AC36, ACA5, B044, B04A, B0B7, B129, B1B2, B219, B222, B291, B299, B2CA, B35A, B3A5, B404, B44C, B45B, B4B3, B501, B51C, B55A, B5A5, B5AB, B5C3, B707, B792, B794, B905, B912, B9C5, BA5B, BAB3, BB03, BB45, BB72, BBA5, BBB2, BC44, BC53, BC95, BC99, C30A, C36A, C395, C454, C535, C553, C593, C944, C953, C964, CC94, 10015, 10051, 10099, 10118, 10291, 10712, 10772, 10811, 10877, 10921, 10B92, 11111, 11135, 11171, 111C8, 11531, 11C03, 13001, 13177, 13777, 13915, 13951, 13991, 159BB, 17018, 17102, 17111, 17117, 17171, 17177, 17708, 17711, 17801, 18071, 18101, 18271, 18B27, 19003, 19153, 19315, 19351, 19591, 19913, 19951, 1C099, 20171, 20177, 20207, 20227, 20777, 21011, 21077, 2111C, 21707, 22207, 30017, 300B5, 301C9, 3033A, 303A3, 303C5, 3050B, 305C9, 3095C, 30B05, 31007, 3159B, 31999, 31C09, 3330A, 33353, 33593, 33757, 33C5C, 33CC5, 35003, 3591B, 39353, 39539, 39935, 39995, 3ACCC, 3C5C3, 3CC53, 40043, 40306, 405C4, 408BC, 40BBB, 40C54, 43066, 4366A, 4443A, 45055, 45505, 45554, 4555C, 455BC, 455C3, 45C04, 488BC, 4B03B, 4B0B5, 4B55C, 4BB0B, 4C003, 4C054, 4C5C4, 50053, 500B1, 5035A, 504B5, 5053A, 50554, 505B4, 50A35, 50B07, 50BBA, 5139B, 519BB, 51BB7, 535AC, 53A5C, 53AC5, 53BAC, 54004, 54035, 5403B, 545C3, 54B05, 54B5C, 54BBC, 54C53, 55357, 5535B, 553AC, 554BC, 55537, 55544, 5554C, 55577, 555A4, 555BB, 55A5C, 55B04, 55B55, 55B77, 55BB5, 55BC4, 55C54, 55C5A, 57403, 591BB, 59443, 59BB7, 5A044, 5AC04, 5AC35, 5B001, 5B007, 5B0AB, 5B0B4, 5B4C5, 5B544, 5B555, 5B5BB, 5B744, 5B777, 5BA0B, 5BB44, 5BB55, 5BBC4, 5BC54, 5C039, 5C35A, 5C53A, 60098, 60964, 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C0000FDD, C00033AF, C0003CAF, C000448D, C000AFFF, C000CF8F, C004444D, C00663AF, C00F00AF, C00FCCAF, C0FFCCAF, C844444D, CC3A000F, CCCCCBED, CCCCCE2D, CCCCD999, CCDCCC4B, CD44444D, CFAF000F, CFFFF023, D00400ED, D004404D, D00777A5, D00E00DD, D0444E0D, D40000ED, D444E00D, D7DDDDDD, DD00D007, DD0D0077, DD0D0707, DDD0040D, DDDDDD19, DDDDDDD1, E0000CCB, E0044441, E00A4AA1, E888820D, E8888CCD, E888C80D, E8AAAAA1, EB00C0CD, EBBC00CD, ECCCCCCB, F00006AF, F00040A5, F00066AF, F06666AF, F0F004A5, F33FFF23, F60006AF, F6AAA0AF, F88888F7, FE777777, FF33F2C3, FF3FFF23, FF588887, FFFF02C3, FFFF5F6F, FFFFF887, FFFFFF79, 10CCCCCAB, 266666603, ...} [/CODE] |
Please stop posting all these endless lists of numbers that nobody read nor needs. If you are really interested in studying these minimal sets, make a web page, put them in zip files, provide links, etc. The way you do it, "text mode", and with additional quote to quote to quote to quote, which makes the posts double size, you are only cluttering the forum, and some people believe that you are doing it intentionally (i.e. trolling), which means you are coming very close to a ban (subject already in discussion for a while on moderator threads!).
At least, wtf man? if you like your text mode so much, can't you at least use "code" tags? Scrolling down 20 pages of numbers (even more for mobile browsers) is a pain in the butt. Really. |
[QUOTE=LaurV;568284]Please stop posting all these endless lists of numbers that nobody read nor needs. If you are really interested in studying these minimal sets, make a web page, put them in zip files, provide links, etc. The way you do it, "text mode", and with additional quote to quote to quote to quote, which makes the posts double size, you are only cluttering the forum, and some people believe that you are doing it intentionally (i.e. trolling), which means you are coming very close to a ban (subject already in discussion for a while on moderator threads!).
At least, wtf man? if you like your text mode so much, can't you at least use "code" tags? Scrolling down 20 pages of numbers (even more for mobile browsers) is a pain in the butt. Really.[/QUOTE] OK, I stopped and going to make a web page about these primes. Should I delete these posts? |
[QUOTE=sweety439;568324]Should I delete these posts?[/QUOTE]
Putting the long text in "code tags" is enough, don't need to delete. You can however delete the quotes to former posts in each (which just duplicate the text). Anyhow, the issue is not the disk space (just some text), but the difficulty in reading through such long strings of numbers. |
[QUOTE=LaurV;568325]Putting the long text in "code tags" is enough, don't need to delete. You can however delete the quotes to former posts in each (which just duplicate the text). Anyhow, the issue is not the disk space (just some text), but the difficulty in reading through such long strings of numbers.[/QUOTE]
Is this OK? I put all my data for bases 13~16 with codes, must I also put data for bases <=12? (base 13~16 data is much longer) |
Newest status for bases 13~16:
[CODE] 13: {14, 16, 1A, 23, 25, 2B, 32, 34, 38, 41, 47, 49, 52, 56, 58, 61, 65, 6B, 76, 7A, 7C, 83, 85, 89, 9A, A1, A7, A9, B6, B8, C1, C7, CB, 10C, 119, 11B, 122, 133, 155, 157, 173, 179, 17B, 188, 197, 1B1, 1B5, 1CC, 209, 212, 218, 229, 272, 274, 281, 287, 292, 296, 298, 29C, 2C9, 311, 313, 331, 33B, 355, 371, 373, 379, 397, 3A6, 3AA, 3B3, 3B9, 3BB, 3CA, 43C, 445, 44B, 45A, 463, 4A3, 4A5, 4B2, 4B4, 4BA, 50C, 511, 515, 533, 54A, 551, 559, 571, 575, 57B, 595, 599, 5B3, 5B9, 5CC, 607, 629, 63A, 643, 674, 704, 715, 724, 728, 731, 737, 739, 742, 751, 75B, 773, 775, 779, 782, 784, 791, 793, 797, 7B1, 812, 818, 874, 878, 8AB, 8B4, 902, 919, 922, 926, 92C, 937, 93B, 946, 95B, 962, 968, 971, 977, 979, 982, 98C, 9B3, 9B5, A03, A3C, A45, A4B, A54, AA3, AAB, B02, B0C, B11, B15, B17, B24, B33, B39, B42, B57, B59, B71, B93, B9B, BA4, BAA, BB1, BB9, BC2, BCC, C29, C43, C98, CA3, 1013, 1031, 1037, 105B, 1075, 10B7, 10BB, 1105, 1112, 1121, 1127, 113C, 1172, 1187, 1208, 1211, 1277, 12C8, 1307, 1309, 131C, 139C, 151C, 1721, 1727, 1787, 1901, 1909, 1912, 1918, 193C, 1981, 198B, 199C, 19B2, 19C3, 1B29, 1BB2, 1BBC, 1C28, 1C39, 2021, 2078, 2117, 2201, 2221, 2267, 2278, 2627, 2678, 2711, 2771, 2788, 3037, 3053, 306A, 3077, 3091, 309B, 30AC, 3305, 353C, 35AB, 35BA, 35BC, 3677, 3905, 390B, 39C5, 3A0C, 3AB5, 3B5C, 3C35, 3C59, 3C95, 403A, 40AB, 4333, 435B, 4403, 44C3, 4535, 4544, 454C, 45B5, 45BB, 480B, 4B35, 4B5B, 4C36, 5057, 5077, 509B, 50A4, 5107, 5305, 530B, 539C, 53AB, 53C9, 5444, 5455, 54C4, 5503, 5545, 55AB, 5774, 5794, 590B, 594B, 5974, 59B4, 5A4C, 5A53, 5AA4, 5AB5, 5ABB, 5ACA, 5B4B, 5B5A, 5BA5, 5CA4, 6227, 6278, 6667, 6698, 6733, 6872, 6928, 6944, 694C, 6973, 6986, 6997, 69C8, 6AC3, 6C92, 6C94, 7019, 7057, 70B5, 7103, 710B, 7118, 7127, 7129, 7172, 7178, 7192, 7211, 7217, 7219, 7271, 7303, 7408, 7433, 7444, 7505, 7507, 7574, 770B, 7774, 7778, 7787, 7871, 7877, 7888, 794B, 7994, 79B4, 7B43, 7B74, 7B94, 7BB2, 8027, 8072, 8081, 80BA, 8171, 8207, 821C, 848B, 8687, 8711, 8722, 87BB, 8867, 88B2, 88BA, 8B22, 8B2A, 8BAC, 9004, 9017, 9031, 9053, 9055, 9073, 9091, 90BB, 90C8, 9107, 9118, 913C, 9181, 91C3, 9284, 935C, 93C5, 9424, 9428, 9448, 9509, 959C, 96C4, 9703, 9743, 9745, 974B, 97B2, 9811, 981B, 987B, 98B1, 991C, 9967, 9998, 9B12, 9B74, 9B92, 9BBC, 9C55, 9C86, 9CC4, A0BA, A306, A436, A535, A5B5, A636, A6C3, A80B, AB04, AB22, AB35, AB3B, AB4C, AB55, ABAC, ABB5, AC36, ACA5, B044, B04A, B0B7, B129, B1B2, B219, B222, B291, B299, B2CA, B35A, B3A5, B404, B44C, B45B, B4B3, B501, B51C, B55A, B5A5, B5AB, B5C3, B707, B792, B794, B905, B912, B9C5, BA5B, BAB3, BB03, BB45, BB72, BBA5, BBB2, BC44, BC53, BC95, BC99, C30A, C36A, C395, C454, C535, C553, C593, C944, C953, C964, CC94, 10015, 10051, 10099, 10118, 10291, 10712, 10772, 10811, 10877, 10921, 10B92, 11111, 11135, 11171, 111C8, 11531, 11C03, 13001, 13177, 13777, 13915, 13951, 13991, 159BB, 17018, 17102, 17111, 17117, 17171, 17177, 17708, 17711, 17801, 18071, 18101, 18271, 18B27, 19003, 19153, 19315, 19351, 19591, 19913, 19951, 1C099, 20171, 20177, 20207, 20227, 20777, 21011, 21077, 2111C, 21707, 22207, 30017, 300B5, 301C9, 3033A, 303A3, 303C5, 3050B, 305C9, 3095C, 30B05, 31007, 3159B, 31999, 31C09, 3330A, 33353, 33593, 33757, 33C5C, 33CC5, 35003, 3591B, 39353, 39539, 39935, 39995, 3ACCC, 3C5C3, 3CC53, 40043, 40306, 405C4, 408BC, 40BBB, 40C54, 43066, 4366A, 4443A, 45055, 45505, 45554, 4555C, 455BC, 455C3, 45C04, 488BC, 4B03B, 4B0B5, 4B55C, 4BB0B, 4C003, 4C054, 4C5C4, 50053, 500B1, 5035A, 504B5, 5053A, 50554, 505B4, 50A35, 50B07, 50BBA, 5139B, 519BB, 51BB7, 535AC, 53A5C, 53AC5, 53BAC, 54004, 54035, 5403B, 545C3, 54B05, 54B5C, 54BBC, 54C53, 55357, 5535B, 553AC, 554BC, 55537, 55544, 5554C, 55577, 555A4, 555BB, 55A5C, 55B04, 55B55, 55B77, 55BB5, 55BC4, 55C54, 55C5A, 57403, 591BB, 59443, 59BB7, 5A044, 5AC04, 5AC35, 5B001, 5B007, 5B0AB, 5B0B4, 5B4C5, 5B544, 5B555, 5B5BB, 5B744, 5B777, 5BA0B, 5BB44, 5BB55, 5BBC4, 5BC54, 5C039, 5C35A, 5C53A, 60098, 60964, 60988, 60A63, 66094, 66377, 66692, 66694, 669C2, 669C4, 66A36, 67022, 67099, 67222, 67277, 67772, 68627, 69088, 690C4, 69808, 69994, 6A663, 7007B, 70181, 70222, 70277, 70772, 70808, 70B0B, 70B29, 71113, 71711, 71908, 71999, 7199B, 71BB7, 71BBB, 74035, 74305, 7430B, 74503, 75443, 75454, 75535, 77072, 77108, 77177, 77717, 77BBB, 78011, 79BBB, 7B007, 7B7B7, 7B7BB, 7BBB3, 7BBB7, 80117, 80221, 80771, 80777, 80807, 8084B, 80B7B, 80BBB, 81107, 8400B, 86267, 87107, 87277, 87727, 87B27, 88111, 88201, 88702, 88771, 8888B, 88B77, 88BBB, 8B1BC, 8B727, 90035, 90059, 90088, 90095, 9009B, 90101, 90103, 90305, 90488, 904CC, 90574, 90644, 9064C, 90806, 908B7, 9090B, 90994, 90B09, 90C35, 90C59, 90C64, 91111, 91135, 91315, 9180B, 92008, 92408, 92488, 93359, 93395, 944C2, 944CC, 94505, 9455C, 94804, 94888, 94C0C, 94C33, 94C4C, 95045, 95504, 95573, 955C4, 95C54, 96044, 97BBB, 98066, 98408, 98444, 98804, 98848, 99001, 99005, 9900B, 99074, 990BC, 99113, 99175, 99278, 99335, 99454, 994C3, 99517, 99593, 9984B, 99881, 99904, 99917, 99935, 99955, 99973, 999BB, 999C2, 999C4, 99B99, 9B00B, 9B04B, 9B0B4, 9B1BB, 9BB04, 9C059, 9C244, 9C404, 9C44C, 9C488, 9C503, 9C5C9, 9C644, 9C664, 9CC88, 9CCC2, A00B4, A05BB, A08B2, A08BC, A0BC4, A3336, A3633, A443A, A4443, A50BB, A55C5, A5AAC, A5BBA, A5C53, A5C55, AACC5, AB05B, AB0BB, AB40A, ABBBC, ABC4A, ACC5A, ACCC3, B0053, B0075, B010B, B0455, B0743, B0774, B0909, B0BB4, B2277, B2A2C, B3005, B351B, B37B5, B3A0B, B3ABC, B3B0A, B400A, B4035, B403B, B4053, B4305, B4BC5, B4C0A, B504B, B50BA, B530A, B5454, B54BC, B54C5, B5544, B55B5, B5B44, B5B4C, B5BB5, B7403, B7535, B77BB, B7955, B7B7B, B9207, B9504, B9999, BA055, BA305, BABC5, BAC35, BB054, BB05A, BB207, BB3B5, BB4C3, BB504, BB544, BB54C, BB5B5, BB753, BB7B7, BBABC, BBB04, BBB4C, BBB55, BBBAC, BC035, BC455, C0353, C0359, C03AC, C0904, C0959, C0A5A, C0CC5, C3059, C335C, C5A0A, C5A44, C5AAC, C6692, C69C2, C904C, C9305, C9905, C995C, C99C5, C9C04, C9C59, C9CC2, CA50A, CA5AC, CAA05, CAA5A, CC335, CC544, CC5AA, CC935, CC955, 100039, 100178, 100718, 100903, 101177, 101708, 101711, 101777, 102017, 102071, 103999, 107081, 107777, 108217, 109111, 109151, 110078, 110108, 110717, 111017, 111103, 1111C3, 111301, 111707, 113501, 115103, 117017, 117107, 117181, 117701, 120701, 13C999, 159103, 170717, 177002, 177707, 180002, 187001, 18C002, 19111C, 199903, 1B0007, 1BB077, 1BBB07, 1C0903, 1C8002, 1C9993, 200027, 207107, 217777, 219991, 220027, 222227, 270008, 271007, 277777, 290444, 300059, 300509, 303359, 303995, 309959, 30B50A, 3336AC, 333707, 33395C, 335707, 3360A3, 350009, 36660A, 3666AC, 370007, 377B07, 39001C, 399503, 3BC005, 400366, 400555, 400B3B, 400B53, 400BB5, 400CC3, 4030B5, 40B053, 40B30B, 40B505, 43600A, 450004, 4A088B, 4B0503, 4B5C05, 4BBBB5, 4BC505, 500039, 50045B, 50405B, 504B0B, 50555B, 5055B5, 505B0A, 509003, 50A50B, 50B045, 50B054, 539B01, 550054, 5500BA, 55040B, 553BC5, 5553C5, 55550B, 5555C3, 555C04, 55B00A, 55BB0B, 570007, 5A500B, 5A555B, 5AC505, 5B055B, 5B0B5B, 5B5B5C, 5B5BC5, 5BB05B, 5BBB0B, 5BBB54, 5BBBB4, 5BBC0A, 5BC405, 5C5A5A, 5CA5A5, 600694, 6060A3, 609992, 637777, 6606A3, 6660A3, 667727, 667808, 668777, 669664, 670088, 679988, 696064, 69C064, 6A6333, 700727, 700811, 700909, 70098B, 700B92, 701117, 701171, 701717, 707027, 707111, 707171, 707201, 707801, 70788B, 7080BB, 708101, 70881B, 70887B, 70B227, 710012, 710177, 711002, 711017, 711071, 717707, 718001, 718111, 720077, 722002, 727777, 74BB3B, 74BB53, 770102, 770171, 770801, 777112, 777202, 777727, 777772, 778801, 77B772, 780008, 78087B, 781001, 788B07, 79088B, 794555, 7B000B, 7B0535, 7B077B, 7B2777, 7B4BBB, 7BB4BB, 800021, 800717, 801077, 80BB07, 811117, 870077, 8777B7, 877B77, 880177, 88071B, 88077B, 8808BC, 887017, 88707B, 888227, 88877B, 8887B7, 888821, 888827, 888BB7, 8B001B, 8B00BB, 8BBB77, 8BBBB7, 900097, 900BC9, 901115, 903935, 904033, 90440C, 908008, 908866, 909359, 909C05, 90B944, 90C95C, 90CC95, 91008B, 91115C, 911503, 920888, 930335, 933503, 935903, 940033, 94040C, 940808, 94CCCC, 950005, 950744, 95555C, 9555C5, 95C003, 95C005, 96400C, 96440C, 96664C, 966664, 966994, 969094, 969964, 97008B, 97080B, 975554, 97800B, 97880B, 980006, 980864, 980B07, 984884, 986006, 986606, 986644, 988006, 988088, 988664, 988817, 988886, 988B0B, 98B007, 990115, 990151, 990694, 990B44, 990C5C, 991501, 993059, 99408B, 994555, 995404, 995435, 996694, 9978BB, 998087, 999097, 999103, 99944C, 999503, 9995C3, 999754, 999901, 99990B, 999B09, 99B4C4, 99C0C5, 99C539, 99CC05, 9B9444, 9B9909, 9C0484, 9C0808, 9C2888, 9C400C, 9C4CCC, 9C6994, 9C90C5, 9C9C5C, 9CC008, 9CC5C3, 9CC905, 9CCC08, A0055B, A005AC, A0088B, A00B2C, A00BBB, A0555C, A05CAA, A0A5AC, A0A5CA, A0AC05, A0AC5A, A0B50B, A0BB0B, A0BBB4, A0C5AC, A3660A, A5050B, A555AC, A5B00B, AA0C05, AAA05C, AAA0C5, AAC05C, AB4444, ABB00B, AC050A, AC333A, B0001B, B00099, B0030B, B004B5, B00A35, B00B54, B030BA, B05043, B0555B, B05B0A, B05B5B, B07B53, B09074, B09755, B09975, B09995, B0AB0B, B0B04B, B0B535, B0BB53, B4C055, B50003, B5003A, B500A3, B50504, B50B04, B53BC5, B54BBB, B550BB, B555BC, B55C55, B5B004, B5B0BB, B5B50B, B5B554, B5B55C, B5B5B4, B5BBB4, B5BBBC, B5BC0A, B5C045, B5C054, B70995, B70B3B, B74555, B74B55, B99921, B99945, BAC505, BB0555, BB077B, BB0B5B, BB0BB5, BB500A, BB53BC, BB53C5, BB5505, BB55BC, BB5BBA, BB5C0A, BB7BB4, BBB00A, BBB74B, BBBB54, BBBBAB, BC5054, BC5504, C00094, C00694, C009C4, C00C05, C03035, C050AA, C05309, C05404, C0544C, C05AC4, C05C39, C06092, C06694, C09035, C094CC, C09992, C09994, C09C4C, C09C95, C0CC3A, C0CC92, C33539, C35009, C4C555, C50309, C50AAA, C53009, C550A5, C555CA, C55A5A, C55CA5, C5AC55, C60094, C60694, C93335, C95405, C99094, CA05CA, CA0AC5, CA555C, CAC5CA, CC05A4, CC0AA5, CC0C05, CC3509, CC4555, CC5039, CC5554, CC555A, CC6092, CCC0C5, CCC353, CCC959, CCC9C2, 1000271, 1000802, 1000871, 1001771, 1001801, 1007078, 1008002, 1008107, 1008701, 1010117, 1027001, 1070771, 1077107, 1077701, 1080107, 1101077, 1110008, 1111078, 1115003, 1117777, 1170008, 1170101, 1700078, 1700777, 1800017, 1877017, 18B7772, 18BBB0B, 1999391, 1999931, 1BBBB3B, 2011001, 2107001, 2110001, 2700017, 2700707, 300000A, 3000019, 3000A33, 3003335, 3003395, 3009335, 300A05B, 3010009, 30A3333, 3335C09, 3339359, 3353777, 336A333, 3393959, 33AC333, 3537007, 3577777, 3636337, 3757777, 395C903, 3AC3333, 40003B5, 400B0B3, 400BBC3, 403B005, 405050B, 40B5555, 40BB555, 40CC555, 4436606, 4444306, 45C5555, 4BC5555, 4C55555, 4CC5004, 4CCC0C3, 500001B, 50003A5, 50005BA, 500B55B, 501000B, 505004B, 505B05B, 50B50B5, 50B550B, 50BB004, 5300009, 5400B0B, 54B000B, 5500BBB, 550B05B, 553000A, 5537777, 555054B, 55505BA, 5550B74, 5555054, 5555BAC, 5555C05, 555B005, 555C00A, 555CA55, 55AC005, 55AC555, 55B005B, 55CA0A5, 5A00004, 5AA5C05, 5B05B05, 5B50B05, 5B5C004, 5BBBBB5, 5BBBBCA, 5C00093, 5C003A5, 5C00A0A, 5C0A055, 5C505AA, 5C5555A, 6000692, 600A333, 606A333, 6363337, 6720002, 6906664, 7000112, 7000712, 7001201, 7001777, 7005553, 70088B7, 7009555, 7010771, 7070881, 7088107, 709800B, 70B9992, 7100021, 7100081, 7100087, 7101107, 7110101, 7120001, 7170077, 7200202, 7270007, 74BBB05, 7700027, 7700201, 7700221, 7700881, 7701017, 7701101, 7707101, 7707701, 7711001, 7770101, 7771201, 7777001, 7777021, 7777102, 77777B7, 777B207, 777B777, 7780001, 77881BB, 788001B, 798000B, 7B00955, 7B00995, 7B55553, 7B55555, 7B77722, 7BB777B, 7BBB40B, 800000B, 8000BB7, 8001B0B, 8010011, 8010101, 8020111, 80B100B, 81B000B, 8677777, 8770001, 8777071, 8801B07, 88040BC, 8822177, 8880007, 8882777, 8887772, 8888087, 8888801, 888B07B, 888B10B, 8B0B00B, 8B777B2, 8BB000B, 9000008, 9000013, 9001151, 9086666, 9088864, 9094003, 9097808, 9099905, 90B99C9, 9151003, 9170008, 91BBBB7, 9244444, 9290111, 940C444, 9430003, 944404C, 94444C4, 944C044, 944C444, 9555005, 9555557, 9644404, 964444C, 96640CC, 9800008, 98800B7, 98884BB, 9888844, 9888884, 98BBB0B, 990888B, 9909C95, 990C94C, 9939953, 9944443, 9955555, 9988807, 998BB07, 99905C9, 9990C95, 9991115, 9994033, 9996644, 9997B44, 999B201, 999CC95, 99CCC5C, 9B20001, 9BBBB44, 9C03335, 9C04444, 9C08888, 9C640CC, 9C80008, 9C99994, 9CC9959, A00AA5C, A00AAC5, A00C50A, A00C555, A00C5AA, A05C00A, A0C005A, A0C0555, A0C555A, A30000A, A33500A, A55553A, A55555C, A5C00AA, A5CAAAA, A8BBB0A, AA00AC5, AA00C5A, AA05C0A, AA5CAAA, AAAC5AA, AAC0555, AC005AA, AC0555A, AC5000A, AC5505A, AC5550A, AC66663, ACC0555, B00007B, B0003AB, B000435, B0004BB, B000A3B, B000B5A, B000BA3, B003777, B005054, B005504, B0055BB, B00777B, B007B3B, B00A0BB, B00AB05, B00B0BA, B00B555, B00B55B, B00BB5B, B00BBB3, B040B0B, B04B00B, B050054, B0500B4, B0554BB, B05B055, B070005, B073B05, B0B00AB, B0B0A0B, B0B50BB, B0B550B, B0B554B, B0BABBB, B0BB305, B1BBB3B, B30000B, B377B77, B400B0B, B4C5005, B5000B4, B5003B5, B505505, B550004, B550055, B555555, B555C05, B5B005B, B5C5505, B70000B, B7B300B, B7BB777, B7BBBBB, B920001, B99545C, B99954C, B999744, BA000BB, BABBB0B, BB000AB, BB0055B, BB05B0B, BB074BB, BB0BABB, BB4000B, BB4430A, BB500BB, BB540BB, BB5555B, BB5BBBB, BB74B0B, BB77B44, BB7B40B, BBB005B, BBB0077, BBB00B5, BBB3007, BBB4444, BBB4B0B, BBB500B, BBB7B3B, BBB7BB5, BBBAB0B, BBBB375, BBBB3B7, 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FF26003, FF3F323, FF5F887, FFAFF45, FFFF263, FFFF379, 2CCCCC63, 30CCA00F, 33333319, 3333FCAF, 3333FFAF, 33FFA00F, 3C00CCAF, 3C00FCAF, 3CF3FF23, 40000441, 40000CAB, 4000DAA1, 400440DD, 400ACCCB, 400CCCAB, 400E44DD, 4040D00D, 404400DD, 40444EDD, 4044D00D, 40ACCCCB, 40DDDDDD, 440000D1, 44000DDD, 4400DD0D, 44E400DD, 4A00004B, 4A0AAAA5, 5000C08D, 52000CCD, 555400A5, 55540A05, 58800007, 58888087, 5A540005, 5C00020D, 5F5400A5, 5F888887, 60006AAF, 600093AF, 600AAAAF, 608CCCCF, 6600686F, 6606866F, 6688AAAF, 7000077D, 70000D5D, 7000707B, 7000707D, 7000740D, 70500D0D, 7070040D, 707007DD, 7070777B, 7077744D, 7077777B, 77007D0D, 7700B44D, 7707000B, 7707D00D, 7770700D, 7770777B, 7777740D, 7777770B, 7777777D, 77777CAB, 7777B887, 778888E7, 788888E7, 79333333, 7ACCCCCB, 7D0000DD, 7D00D0DD, 7DD00D0D, 7DDDDDA9, 80000081, 80000087, 8000E0CD, 80400E4D, 80A0AAA1, 80EC000D, 84000E4D, 8404444D, 84400E4D, 868AAAAF, 86AAAA8F, 8884044D, 88FFFE77, 8C44444D, 8CCCCAAF, 8E40004D, 900000BB, 90000B0B, 90100009, 90800AA1, 93333AAF, 94AAAAA1, 980000A1, 998AAAA1, A00000F9, A0000EEB, A0005A45, A0055545, A00AAA45, A0666669, A0AAA045, A0AAAA45, A0AAE4A1, A0B44441, A4A00005, A6066669, A8AAFFFF, AA055545, AA0AA045, AAA00A45, AAAAA045, B00000AB, B000EEEB, B00EEE0B, B0900081, B0BBBBAB, B7777787, B9000081, B9008001, B9800001, BA00000B, BBBB0ABB, BCCCCCC9, C000004D, C000086F, C0000AFF, C0000E8D, C0000FDD, C00033AF, C0003CAF, C000448D, C000AFFF, C000CF8F, C004444D, C00663AF, C00F00AF, C00FCCAF, C0FFCCAF, C844444D, CC3A000F, CCCCCBED, CCCCCE2D, CCCCD999, CCDCCC4B, CD44444D, CFAF000F, CFFFF023, D00400ED, D004404D, D00777A5, D00E00DD, D0444E0D, D40000ED, D444E00D, D7DDDDDD, DD00D007, DD0D0077, DD0D0707, DDD0040D, DDDDDD19, DDDDDDD1, E0000CCB, E0044441, E00A4AA1, E888820D, E8888CCD, E888C80D, E8AAAAA1, EB00C0CD, EBBC00CD, ECCCCCCB, F00006AF, F00040A5, F00066AF, F06666AF, F0F004A5, F33FFF23, F60006AF, F6AAA0AF, F88888F7, FE777777, FF33F2C3, FF3FFF23, FF588887, FFFF02C3, FFFF5F6F, FFFFF887, FFFFFF79, 10CCCCCAB, 266666603, 333333AAF, 333333F23, 3333FF2C3, 333CCCCAF, 333FFCCAF, 3A000000F, 3FA00000F, 40000048D, 4000004DD, 4000040D1, 40000ACCB, 4000400D1, 4040000DD, 404D0000D, 40A000005, 40E00444D, 40ED0000D, 444E000DD, 444ED000D, ...} [/CODE] |
[QUOTE=sweety439;568171]Largest minimal primes in simple families for bases 13 to 16 (written in decimal):
[/QUOTE] For the families with "0": x{y} and {x}y: 13, 9, {5}: unsolved family (113*13^n-5)/12 14, 4, {13}: first prime is 4D[SUB]19698[/SUB], found by CRUS 14, 8, {13}: even D's: difference of squares, odd D's: factor of 5 14, 11, {1}: even 1's: difference of squares, odd 1's: factor of 5 14, {13}, 5: even D's: difference of squares, odd D's: factor of 5 16, 1, {5}: difference of squares 16, {4}, 1: difference of squares 16, 7, {3}: difference of squares 16, 8, {1}: difference of squares 16, 8, {5}: difference of squares 16, 8, {15}: difference of squares 16, {12}, 11: difference of squares 16, {12}, 13: x^4+4*y^4 16, 13, {1}: difference of squares 16, 13, {11}: unsolved family (206*16^n-11)/15 16, {15}, 7: difference of squares x{0}y: 14, 1, 11: covering set {3, 5} 14, 4, 1: covering set {3, 5} 14, 7, 13: covering set {3, 5} 14, 8, 13: covering set {3, 5} 14, 11, 1: covering set {3, 5} |
Still no (probable) prime found for base 11 5{7} family, tested to around n=25K
the formula of this family is (57*11^n-7)/10 |
Some known large minimal primes (start with base+1) and unsolved families in bases 2~16: (base 13~16 families are all found for x{y}, {x}y, x{0}y, unless other referenced listed)
For more such primes and more unsolved families containing neither prime digits nor (digit 1 before digit 0), see [URL="https://github.com/curtisbright/mepn-data/tree/master/data"]https://github.com/curtisbright/mepn-data/tree/master/data[/URL] (base 2~30, "minimal n" for primes, "unsolved n" for unsolved families) and [URL="https://github.com/RaymondDevillers/primes"]https://github.com/RaymondDevillers/primes[/URL] (base 28~50, "kernel n" for primes, "left n" for unsolved families) Base 5: 10[SUB]93[/SUB]13 Base 7: 3[SUB]16[/SUB]1 Base 8: 7[SUB]12[/SUB]1 5[SUB]13[/SUB]25 4[SUB]220[/SUB]7 Base 9: 54[SUB]11[/SUB] 20[SUB]11[/SUB]7 561[SUB]36[/SUB] 76[SUB]329[/SUB]2 Base 10: 5[SUB]11[/SUB]1 50[SUB]28[/SUB]27 Base 11: 1[SUB]17[/SUB] 14[SUB]12[/SUB]1111 4[SUB]14[/SUB]111 A14[SUB]15[/SUB] A9[SUB]15[/SUB]6 (already minimal when primes <=b are not excluded) 8[SUB]17[/SUB]3 14[SUB]18[/SUB] (already minimal when primes <=b are not excluded) 7[SUB]18[/SUB]1 40[SUB]15[/SUB]A041 (already minimal when primes <=b are not excluded) A9[SUB]21[/SUB] (already minimal when primes <=b are not excluded) A4[SUB]25[/SUB]1 (already minimal when primes <=b are not excluded) 150[SUB]25[/SUB]7 40[SUB]26[/SUB]41 (already minimal when primes <=b are not excluded) 440[SUB]27[/SUB]1 (already minimal when primes <=b are not excluded) 9[SUB]32[/SUB]1 (already minimal when primes <=b are not excluded) A47[SUB]41[/SUB] 4[SUB]44[/SUB]1 (already minimal when primes <=b are not excluded) 5{7} (unsolved family) Base 12: B0[SUB]27[/SUB]9B Base 13: 3[SUB]178[/SUB]5 40[SUB]202[/SUB]3 57[SUB]230[/SUB] 9[SUB]308[/SUB]1 (already minimal when primes <=b are not excluded) 8B[SUB]343[/SUB] B7[SUB]486[/SUB] B[SUB]563[/SUB]C 1B[SUB]576[/SUB] 9[SUB]968[/SUB]B 9[SUB]1362[/SUB]5 7[SUB]1504[/SUB]1 930[SUB]1551[/SUB]1 (found by CRUS) 390[SUB]6266[/SUB]1 (found by CRUS) 80[SUB]32017[/SUB]111 (already minimal when primes <=b are not excluded) 9{5} (unsolved family) Base 14: 10[SUB]15[/SUB]D 8[SUB]29[/SUB]9 (already minimal when primes <=b are not excluded) 85[SUB]36[/SUB] 9[SUB]36[/SUB]89 A[SUB]59[/SUB]3 4[SUB]63[/SUB]09 (already minimal when primes <=b are not excluded) 40[SUB]83[/SUB]49 (already minimal when primes <=b are not excluded) 8[SUB]86[/SUB]B 4D[SUB]19698[/SUB] (found by CRUS) Base 15: 7[SUB]14[/SUB]B DE[SUB]14[/SUB] D[SUB]16[/SUB]B 7B[SUB]20[/SUB] B[SUB]22[/SUB]1 EB[SUB]31[/SUB] 1[SUB]48[/SUB]7 (*** not minimal) 96[SUB]104[/SUB]08 (already minimal when primes <=b are not excluded) Base 16: D[SUB]14[/SUB]9 3[SUB]24[/SUB]1 7D[SUB]25[/SUB] 2[SUB]32[/SUB]7 E[SUB]34[/SUB]B 4D[SUB]39[/SUB] 8[SUB]53[/SUB]7 EB[SUB]145[/SUB] 8888F[SUB]201[/SUB] (already minimal when primes <=b are not excluded) 88F[SUB]545[/SUB] (already minimal when primes <=b are not excluded) D9[SUB]1052[/SUB] F8[SUB]1517[/SUB]F (already minimal when primes <=b are not excluded) 90[SUB]3542[/SUB]91 (already minimal when primes <=b are not excluded) D{B} (unsolved family) |
Newest status:
[CODE] 13: {14, 16, 1A, 23, 25, 2B, 32, 34, 38, 41, 47, 49, 52, 56, 58, 61, 65, 6B, 76, 7A, 7C, 83, 85, 89, 9A, A1, A7, A9, B6, B8, C1, C7, CB, 10C, 119, 11B, 122, 133, 155, 157, 173, 179, 17B, 188, 197, 1B1, 1B5, 1CC, 209, 212, 218, 229, 272, 274, 281, 287, 292, 296, 298, 29C, 2C9, 311, 313, 331, 33B, 355, 371, 373, 379, 397, 3A6, 3AA, 3B3, 3B9, 3BB, 3CA, 43C, 445, 44B, 45A, 463, 4A3, 4A5, 4B2, 4B4, 4BA, 50C, 511, 515, 533, 54A, 551, 559, 571, 575, 57B, 595, 599, 5B3, 5B9, 5CC, 607, 629, 63A, 643, 674, 704, 715, 724, 728, 731, 737, 739, 742, 751, 75B, 773, 775, 779, 782, 784, 791, 793, 797, 7B1, 812, 818, 874, 878, 8AB, 8B4, 902, 919, 922, 926, 92C, 937, 93B, 946, 95B, 962, 968, 971, 977, 979, 982, 98C, 9B3, 9B5, A03, A3C, A45, A4B, A54, AA3, AAB, B02, B0C, B11, B15, B17, B24, B33, B39, B42, B57, B59, B71, B93, B9B, BA4, BAA, BB1, BB9, BC2, BCC, C29, C43, C98, CA3, 1013, 1031, 1037, 105B, 1075, 10B7, 10BB, 1105, 1112, 1121, 1127, 113C, 1172, 1187, 1208, 1211, 1277, 12C8, 1307, 1309, 131C, 139C, 151C, 1721, 1727, 1787, 1901, 1909, 1912, 1918, 193C, 1981, 198B, 199C, 19B2, 19C3, 1B29, 1BB2, 1BBC, 1C28, 1C39, 2021, 2078, 2117, 2201, 2221, 2267, 2278, 2627, 2678, 2711, 2771, 2788, 3037, 3053, 306A, 3077, 3091, 309B, 30AC, 3305, 353C, 35AB, 35BA, 35BC, 3677, 3905, 390B, 39C5, 3A0C, 3AB5, 3B5C, 3C35, 3C59, 3C95, 403A, 40AB, 4333, 435B, 4403, 44C3, 4535, 4544, 454C, 45B5, 45BB, 480B, 4B35, 4B5B, 4C36, 5057, 5077, 509B, 50A4, 5107, 5305, 530B, 539C, 53AB, 53C9, 5444, 5455, 54C4, 5503, 5545, 55AB, 5774, 5794, 590B, 594B, 5974, 59B4, 5A4C, 5A53, 5AA4, 5AB5, 5ABB, 5ACA, 5B4B, 5B5A, 5BA5, 5CA4, 6227, 6278, 6667, 6698, 6733, 6872, 6928, 6944, 694C, 6973, 6986, 6997, 69C8, 6AC3, 6C92, 6C94, 7019, 7057, 70B5, 7103, 710B, 7118, 7127, 7129, 7172, 7178, 7192, 7211, 7217, 7219, 7271, 7303, 7408, 7433, 7444, 7505, 7507, 7574, 770B, 7774, 7778, 7787, 7871, 7877, 7888, 794B, 7994, 79B4, 7B43, 7B74, 7B94, 7BB2, 8027, 8072, 8081, 80BA, 8171, 8207, 821C, 848B, 8687, 8711, 8722, 87BB, 8867, 88B2, 88BA, 8B22, 8B2A, 8BAC, 9004, 9017, 9031, 9053, 9055, 9073, 9091, 90BB, 90C8, 9107, 9118, 913C, 9181, 91C3, 9284, 935C, 93C5, 9424, 9428, 9448, 9509, 959C, 96C4, 9703, 9743, 9745, 974B, 97B2, 9811, 981B, 987B, 98B1, 991C, 9967, 9998, 9B12, 9B74, 9B92, 9BBC, 9C55, 9C86, 9CC4, A0BA, A306, A436, A535, A5B5, A636, A6C3, A80B, AB04, AB22, AB35, AB3B, AB4C, AB55, ABAC, ABB5, AC36, ACA5, B044, B04A, B0B7, B129, B1B2, B219, B222, B291, B299, B2CA, B35A, B3A5, B404, B44C, B45B, B4B3, B501, B51C, B55A, B5A5, B5AB, B5C3, B707, B792, B794, B905, B912, B9C5, BA5B, BAB3, BB03, BB45, BB72, BBA5, BBB2, BC44, BC53, BC95, BC99, C30A, C36A, C395, C454, C535, C553, C593, C944, C953, C964, CC94, 10015, 10051, 10099, 10118, 10291, 10712, 10772, 10811, 10877, 10921, 10B92, 11111, 11135, 11171, 111C8, 11531, 11C03, 13001, 13177, 13777, 13915, 13951, 13991, 159BB, 17018, 17102, 17111, 17117, 17171, 17177, 17708, 17711, 17801, 18071, 18101, 18271, 18B27, 19003, 19153, 19315, 19351, 19591, 19913, 19951, 1C099, 20171, 20177, 20207, 20227, 20777, 21011, 21077, 2111C, 21707, 22207, 30017, 300B5, 301C9, 3033A, 303A3, 303C5, 3050B, 305C9, 3095C, 30B05, 31007, 3159B, 31999, 31C09, 3330A, 33353, 33593, 33757, 33C5C, 33CC5, 35003, 3591B, 39353, 39539, 39935, 39995, 3ACCC, 3C5C3, 3CC53, 40043, 40306, 405C4, 408BC, 40BBB, 40C54, 43066, 4366A, 4443A, 45055, 45505, 45554, 4555C, 455BC, 455C3, 45C04, 488BC, 4B03B, 4B0B5, 4B55C, 4BB0B, 4C003, 4C054, 4C5C4, 50053, 500B1, 5035A, 504B5, 5053A, 50554, 505B4, 50A35, 50B07, 50BBA, 5139B, 519BB, 51BB7, 535AC, 53A5C, 53AC5, 53BAC, 54004, 54035, 5403B, 545C3, 54B05, 54B5C, 54BBC, 54C53, 55357, 5535B, 553AC, 554BC, 55537, 55544, 5554C, 55577, 555A4, 555BB, 55A5C, 55B04, 55B55, 55B77, 55BB5, 55BC4, 55C54, 55C5A, 57403, 591BB, 59443, 59BB7, 5A044, 5AC04, 5AC35, 5B001, 5B007, 5B0AB, 5B0B4, 5B4C5, 5B544, 5B555, 5B5BB, 5B744, 5B777, 5BA0B, 5BB44, 5BB55, 5BBC4, 5BC54, 5C039, 5C35A, 5C53A, 60098, 60964, 60988, 60A63, 66094, 66377, 66692, 66694, 669C2, 669C4, 66A36, 67022, 67099, 67222, 67277, 67772, 68627, 69088, 690C4, 69808, 69994, 6A663, 7007B, 70181, 70222, 70277, 70772, 70808, 70B0B, 70B29, 71113, 71711, 71908, 71999, 7199B, 71BB7, 71BBB, 74035, 74305, 7430B, 74503, 75443, 75454, 75535, 77072, 77108, 77177, 77717, 77BBB, 78011, 79BBB, 7B007, 7B7B7, 7B7BB, 7BBB3, 7BBB7, 80117, 80221, 80771, 80777, 80807, 8084B, 80B7B, 80BBB, 81107, 8400B, 86267, 87107, 87277, 87727, 87B27, 88111, 88201, 88702, 88771, 8888B, 88B77, 88BBB, 8B1BC, 8B727, 90035, 90059, 90088, 90095, 9009B, 90101, 90103, 90305, 90488, 904CC, 90574, 90644, 9064C, 90806, 908B7, 9090B, 90994, 90B09, 90C35, 90C59, 90C64, 91111, 91135, 91315, 9180B, 92008, 92408, 92488, 93359, 93395, 944C2, 944CC, 94505, 9455C, 94804, 94888, 94C0C, 94C33, 94C4C, 95045, 95504, 95573, 955C4, 95C54, 96044, 97BBB, 98066, 98408, 98444, 98804, 98848, 99001, 99005, 9900B, 99074, 990BC, 99113, 99175, 99278, 99335, 99454, 994C3, 99517, 99593, 9984B, 99881, 99904, 99917, 99935, 99955, 99973, 999BB, 999C2, 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4F, 53, 59, 61, 65, 67, 6B, 6D, 71, 7F, 83, 89, 8B, 95, 97, 9D, A3, A7, AD, B3, B5, BF, C1, C5, C7, D3, DF, E3, E5, E9, EF, F1, FB, 14B, 15B, 185, 199, 1A5, 1BB, 1C9, 1EB, 223, 22D, 233, 241, 277, 281, 287, 28D, 2A1, 2D7, 2DD, 2E7, 301, 337, 373, 377, 38F, 3A1, 3A9, 41B, 42D, 445, 455, 45D, 481, 4B1, 4BD, 4CD, 4D5, 4E1, 4EB, 50B, 515, 51B, 527, 551, 557, 55D, 577, 581, 58F, 5AB, 5CB, 5CF, 5D1, 5D5, 5DB, 5E7, 623, 709, 727, 737, 745, 74B, 755, 757, 773, 779, 78D, 7BB, 7C3, 7C9, 7CD, 7DB, 7EB, 7ED, 805, 80F, 815, 821, 827, 841, 851, 85D, 85F, 8A5, 8DD, 8E1, 8F5, 923, 98F, 99B, 9A9, 9EB, A21, A6F, A81, A85, A99, A9F, AA9, AAB, ACF, B1B, B2D, B7B, B8D, B99, B9B, BB7, BB9, BCB, BDD, BE1, C0B, CB9, CBB, CEB, D01, D21, D2D, D55, D69, D79, D81, D85, D87, D8D, DAB, DB7, DBD, DC9, DCD, DD5, DDB, DE7, E21, E27, E4B, E7D, E87, EB1, EB7, ED1, EDB, EED, F07, F0D, F4D, FD9, FFD, 1069, 1505, 1609, 1669, 16A9, 19AB, 1A69, 1AB9, 2027, 204D, 2063, 207D, 20C3, 20ED, 2221, 22E1, 2327, 244D, 26C3, 274D, 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B04B, B069, B07D, B087, B0B1, B0ED, B1A9, B201, B40B, B40D, B609, B70D, B7A9, B807, B9A1, BA41, BAA1, BB4B, BBB1, BBDB, BBED, BD19, BD41, BDBB, BDEB, BE07, BEE7, C0D9, C203, C24D, C6A9, C88D, C88F, C8CF, C8ED, C9AF, C9CB, CA09, CA4B, CA69, CAC9, CC0D, CC23, CC4D, CC9B, CD09, CDD9, CE4D, CEDD, CFA9, CFCD, D04B, D099, D405, D415, D44B, D4A5, D4DD, D50D, D70B, D74D, D77B, D7CB, D91B, D991, DA05, DA09, DA15, DA51, DB91, DBEB, DD7D, DDA1, DDED, DE0B, DE41, DE4D, DEA1, E02D, E07B, E0D7, E1CB, E2CD, E401, E801, EABB, EACB, EAEB, EBAB, EC4D, ECDD, ED07, EDD7, EE7B, EE81, EEAB, EEE1, F08F, F0A9, F227, F2ED, F3AF, F485, F58D, F72D, F763, F769, F787, F7A5, F7E7, F82D, F86F, F877, F88D, F8D7, F8E7, F8FF, FCCD, FED7, FF85, FF8F, FFA9, 100AB, 10BA9, 1A0CB, 1BA09, 200E1, 2C603, 2CC03, 30227, 303AF, 30AAF, 32003, 32207, 32CC3, 330AF, 33169, 33221, 33391, 33881, 33AFF, 38807, 38887, 3AFFF, 3F203, 3F887, 3FAFF, 400BB, 4084D, 40A4B, 42001, 44221, 44401, 444D1, 4480D, 4488D, 44CCB, 44D4D, 44E8D, 4804D, 4840D, 4A0CB, 4A54B, 4CACB, 4D0DD, 4D40D, 4D44D, 5004D, 50075, 502CD, 5044D, 50887, 50EE1, 5448D, 548ED, 55A45, 55F45, 5844D, 5A4A5, 5AE41, 5B0CD, 5B44D, 5BBCD, 5D4ED, 5E0E1, 5EB4D, 5EC8D, 5ECCD, 5EE41, 5F06F, 5F7DD, 5F885, 5F8CD, 5FC8D, 5FF75, 6088F, 60AFF, 630AF, 633AF, 660A9, 668CF, 669AF, 66A09, 66A0F, 66FA9, 6886F, 6A00F, 6A0FF, 6A8AF, 6AFFF, 7002D, 7024D, 70B0D, 70B7D, 7200D, 73363, 73999, 7444D, 770B7, 777D7, 77B07, 77D7D, 77DD7, 79003, 79999, 7B00D, 7D05D, 7D7DD, 8007D, 800D1, 8074D, 82CCD, 82E4D, 8448D, 8484D, 8704D, 8724D, 87887, 88001, 8800D, 880CD, 88507, 88555, 8866F, 8872D, 8877D, 888D1, 888D7, 88AA1, 88C2D, 88D57, 88D75, 88D77, 8AFAF, 8C2CD, 8C40D, 8C8CD, 8CCED, 8CE2D, 8CFED, 8E007, 8E20D, 8E24D, 8F6FF, 8FAAF, 900CB, 901AB, 90901, 909A1, 90AB1, 90AE1, 90EE1, 910AB, 93331, 940AB, 963AF, 966AF, 99019, 99109, 99A01, 9AAE1, 9B00B, 9B0AB, 9B441, 9BABB, 9BBBB, 9E441, A00BB, A0405, A044B, A08AF, A0A51, A0B91, A0C4B, A1B09, A54A5, A5B41, A6609, A904B, A94A1, A9C4B, A9E01, A9E41, AA0A1, AA441, AA501, AA8AF, AAEE1, AAF45, AAF8F, ABBA1, ACC69, AE0BB, AE0EB, AEAE1, AEE0B, AEEA1, AEECB, AF045, AF4A5, AFA8F, B00A1, B00D7, B044D, B0777, B0A0B, B0A91, B0BBD, B0BCD, B0C09, B0DA9, B0EAB, B2207, B4001, B6669, B7707, B7D07, B8081, B9021, BA091, BA109, BA4BB, BB001, BB0EB, BB8A1, BBBEB, BBE0B, BBEBB, BC009, BCECD, BD0A9, BE44D, BEB0D, BEBBB, BEEBB, C0263, C02C3, C02ED, C040D, C0CA9, C0CCD, C2663, C2CED, C32C3, C3323, C400D, C40ED, C44CB, C44ED, C480D, C484D, C4CAB, C60AF, C686F, C6A0F, C86FF, C8C2D, CAA0F, CAFAF, CBCED, CC0AF, CC44B, CC82D, CC8FF, CCAF9, CCAFF, CCCFD, CCFAF, CD00D, CD4CB, CD4ED, CDDDD, CF2C3, CFC8F, CFE8D, D0045, D07DD, D09BB, D0D4D, D0DD7, D0EBB, D0EEB, D1009, D1045, D10B9, D1BA9, D54BB, D54ED, D5AE1, D5D07, D5EE1, D70DD, D7707, D7777, D77DD, D7DD7, D9441, D9AE1, D9B0B, DA9A1, DA9E1, DAA41, DAAA1, DBB0B, DBBA1, DC4CB, DD227, DD44D, DDDD7, E0081, E00E1, E010B, E088D, E08CD, E0B0D, E0BBD, E100B, E4D0D, E777B, E77AB, E7CCB, E844D, E848D, E884D, E88A1, EB0BB, EBB4D, EBBEB, EBEEB, EC8CD, ECBCD, ECC8D, ED04D, EE001, EE0EB, EE4A1, EEEBB, F0085, F09AF, F0C23, F0CAF, F2663, F2C03, F3799, F3887, F4A05, F4AA5, F506F, F5845, F5885, F5C2D, F5ECD, F5F45, F66A9, F688F, F6AFF, F7399, F777D, F8545, F8555, F8AAF, F8F87, F9AAF, FA0F9, FA405, FA669, FAFF9, FC263, FCA0F, FCAFF, FCE8D, FCF23, FD777, FDDDD, FDEDD, FEC2D, FEC8D, FF545, FF6AF, FF739, FF775, FF9AF, FFC23, 100055, 100555, 10A9CB, 1A090B, 1A900B, 1CACCB, 1CCACB, 20DEE1, 266003, 3000AF, 300A0F, 300AFF, 308087, 308E07, 3323E1, 333A0F, 339331, 33CA0F, 33CF23, 33CFAF, 33F323, 380087, 3A00AF, 3A0F0F, 3AA0FF, 3AAF0F, 3C33AF, 3C3A0F, 3C3FAF, 3CCAAF, 3F0FAF, 3F32C3, 3FF0AF, 3FFAAF, 4004CB, 400A05, 4048ED, 404DDD, 40AA05, 40D04D, 40DD4D, 40E0DD, 40E48D, 440041, 44008D, 44044D, 4404DD, 44440D, 4448ED, 4484ED, 448E4D, 44E44D, 48888D, 4AA005, 4DD00D, 4DD04D, 4DDD0D, 4E048D, 4E448D, 4E880D, 5000DD, 500201, 50066F, 5008CD, 500C2D, 500D7D, 50C20D, 520C0D, 544EDD, 54AA05, 54AAA5, 54ED4D, 566AAF, 57D00D, 580087, 5A5545, 5C20CD, 5C8CCD, 5CC2CD, 5D000D, 5D070D, 5F666F, 5FAA45, 5FFF45, 60008F, 600A0F, 603AAF, 6060AF, 6066AF, 60A0AF, 63AA0F, 6663AF, 66668F, 666AAF, 668A8F, 66AFF9, 68888F, 693AAF, 7007B7, 70404D, 70770B, 70770D, 707BE7, 70DD0D, 733339, 733699, 74004D, 74040D, 77007B, 770CCB, 777B4D, 777BE7, 777CCB, 77ACCB, 77B74D, 77D0DD, 7A0CCB, 7B744D, 7CACCB, 7DDD99, 80044D, 800807, 80200D, 8044ED, 80C04D, 80CC2D, 80E44D, 8404ED, 84888D, 84E04D, 84E40D, 86686F, 8668AF, 8686AF, 86F66F, 86FFFF, 87000D, 87744D, 880807, 886AFF, 88824D, 88870D, 888787, 88884D, 88886F, 88887D, 88888D, 888C4D, 888FAF, 88AA8F, 88CC8D, 88F6AF, 88F8AF, 88FA8F, 88FF6F, 88FF87, 88FFAF, 8A8FFF, 8C0C2D, 8C802D, 8CCFFF, 8CE00D, 8CE0CD, 8CFCCF, 8E00CD, 8E044D, 8E0CCD, 8EC0CD, 8F68AF, 8F88F7, 8FCFCF, 8FF887, 8FFCCF, 8FFF6F, 9002E1, 9004AB, 9008A1, 900919, 900ABB, 900B21, 90B801, 90CCCB, 9332E1, 944441, 94ACCB, 990001, 9900A1, 9A4441, 9A4AA1, 9AA4A1, 9AAA41, 9AAAAF, 9B66C9, 9BBA0B, 9BC0C9, 9BC669, 9BC6C9, 9C4ACB, A0094B, A00ECB, A09441, A0A08F, A0E0CB, A0ECCB, A0F669, A40A05, A4AAA5, A50E41, A5AA45, A60069, A8FAFF, A9AA41, AA5E41, AAA4A5, AAA545, AC6669, ACCC4B, ACCCC9, AEAA41, AFF405, AFF669, AFFA45, AFFFF9, B00921, B00BEB, B00CC9, B00D91, B08801, B0D077, B70077, B70E77, B77E77, B88877, B88881, B94421, BAE00B, BB00AB, BB0DA1, BB444D, BB44D1, BB8881, BBBBBD, BBBC4D, BBCCCD, BC0CC9, BC66C9, BCC669, BCC6C9, BCCC09, BE000D, BE00BD, BE0B4D, BE0CCD, BEA00B, BECCCD, C0084D, C00A0F, C0608F, C0668F, C0844D, C0A0FF, C0AFF9, C0C3AF, C0C68F, C0CAAF, C0CDED, C0D0ED, C0E80D, C0EC2D, C0EC8D, C0FA0F, C0FAAF, C2CC63, C30CAF, C333AF, C3CAAF, C3CCAF, C4048D, C40D4D, C4404D, C4408D, C4440D, C44DDD, C4ACCB, C4DCCB, C4DD4D, C6068F, C66AAF, C68AAF, C6AA8F, C8044D, C8440D, C8666F, CA00FF, CA0FFF, CAAAAF, CAAFFF, CAFF0F, CBE0CD, CC008F, CC0C8F, CC3CAF, CC4ACB, CC608F, CC66AF, CCBECD, CCC4AB, CCCA0F, CCCC8F, CCCE8D, CE0C8D, CF0F23, CF0FAF, CFAFFF, CFCAAF, CFFAFF, D0005D, D00BA9, D05EDD, D077D7, D10CCB, D22207, D4000B, D4040D, D4044D, D40CCB, D70077, D7D00D, D90009, D900BB, DB00BB, DB4441, DD400D, DDD109, DDD1A9, DDD919, DDD941, DED00D, E00D4D, E00EEB, E0AAE1, E0AE41, E0AEA1, E0B44D, E0BCCD, E0BEBB, E0D0DD, E0E441, E4048D, E4448D, E800CD, E8200D, EA0E41, EAA0E1, EBB00B, ECCCAB, EDDDDD, EEBE0B, F00263, F0056F, F00A45, F02C63, F03F23, F05405, F060AF, F08585, F0A4A5, F0F2C3, F0F323, F2CCC3, F33203, F33C23, F5F66F, F5FF6F, F68CCF, F6AA8F, F888AF, FA0F45, FAA045, FAA545, FAFC69, FC0AAF, FC66AF, FCCCAF, FCFFAF, FF0323, FF056F, FF3203, FF7903, FFA045, FFA4A5, FFAA45, FFC0AF, FFF4A5, FFF575, FFFA45, FFFCAF, 10A009B, 20000D1, 2CCC663, 30A00FF, 30CCCAF, 30FA00F, 30FCCAF, 3333C23, 333C2C3, 33C3AAF, 33FCAAF, 33FFFAF, 3A0A00F, 3AAAA0F, 3AF000F, 3AFAAAF, 3C0CA0F, 3CCC3AF, 3CFF323, 3F33F23, 3FAA00F, 3FF3323, 4004441, 400DDD1, 400E00D, 400ED0D, 404404D, 404448D, 404E4DD, 440EDDD, 4440EDD, 44444ED, 4444E4D, 44DDDDD, 4A000A5, 4CCCCAB, 4D0CCCB, 4E4404D, 4E4444D, 4E4DDDD, 5000021, 5004221, 5006AAF, 500FF6F, 5042201, 508CCCD, 5400005, 5400AA5, 5555405, 5808007, 5AA4005, 5C0008D, 5CCC8CD, 5D4444D, 5EEEEEB, 5F40005, 5F554A5, 5F6AAAF, 60000AF, 60006A9, 600866F, 6008AAF, 600AA8F, 600F6A9, 606608F, 606686F, 608666F, 60AA08F, 60AAA8F, 66000AF, 66666A9, 6666AF9, 6866A8F, 6AAAAAF, 70070D7, 70077DD, 700DDDD, 707077D, 707D007, 70D00DD, 770077D, 770400D, 770740D, 7777775, 77777B7, 77777DD, 7777ACB, 77B88E7, 77DD00D, 77DDDDD, 7D0D00D, 7DD0D07, 7DDD00D, 800002D, 8000CED, 80C0E0D, 80CECCD, 840400D, 844000D, 844E00D, 868688F, 880444D, 884404D, 887D007, 8888801, 8888881, 8888E07, 8888F77, 8888FE7, 88A8AFF, 88AAAFF, 88FAFFF, 8A8AAAF, 8A8AAFF, 8AAA8FF, 8C00ECD, 8C8444D, 8E4400D, 8FCCCCF, 900BBAB, 90CC4AB, 9908AA1, 99E0E01, 9B00801, 9B6CCC9, A000FF9, A006069, A00A8FF, A01CCCB, A05F545, A0BEEEB, A0E4AA1, AA0008F, AA08FFF, AA40AA5, AA8FFFF, AAAA405, AE04AA1, AE44441, AE4AAA1, AECCCCB, AF40005, AFA5A45, AFFFC69, B000BAB, B000EBB, B0D0007, B222227, B6CCCC9, B8880A1, BA000EB, BA0BEEB, BAEEEEB, BB000CD, BB00C0D, BB0B00D, BC6CC69, BC6CCC9, BCCCC69, BCCCCED, C0000A9, C00068F, C000CFD, C000E2D, C000FAF, C004D4D, C00E20D, C00E8CD, C00F68F, C033A0F, C0802CD, C086AAF, C0A00AF, C0AFFFF, C0C086F, C0C0F8F, C0CA00F, C0CC08F, C0D044D, C0F0AFF, C0FF023, C0FFFAF, C33FA0F, C33FAAF, C3CA00F, C3FFCAF, C8002CD, C8200CD, CCC668F, CCCAA8F, CCCC0A9, CCCC3AF, CCCCCA9, CCCDC4B, CE0008D, CE2000D, CE8CCCD, CF000AF, CFF0AAF, CFFF0AF, D0000EB, D0005EB, D000775, D000EDD, D007077, D00DDD9, D00ED0D, D0AAA45, D0AAAA5, D0EDDDD, D19000B, D4404ED, D4440ED, D5BBBBB, DCCCC4B, DD00DD9, DD07077, DD0DD09, DD0DDD9, DD99999, DDD0D09, DDDD0D9, DDDD9E1, DDDDD09, DDDDD99, DE0DDDD, DEEEEEB, E00001B, E0004A1, E000CAB, E00A041, E00BB0B, E00BBBB, E00C80D, E00CCCB, E044DDD, E0AA4A1, E0AAA41, E0BBB0B, E0D444D, E40444D, E4DDD4D, E88CCCD, E8C000D, E8CCCCD, EA04441, EA0A4A1, EBB000D, EBCCCCD, ED0D00D, EEAAA01, EEBBBBB, EEE000B, F0002C3, F002CC3, F003323, F005545, F00F4A5, F033323, F0400A5, F0A5545, F333323, F333F23, F6660AF, F733333, FA00009, FA004A5, FAAAA45, FC6668F, FCC668F, FD00AA5, FEE7777, FF0F263, FF26003, FF3F323, FF5F887, FFAFF45, FFFF263, FFFF379, 2CCCCC63, 30CCA00F, 33333319, 3333FCAF, 3333FFAF, 33FFA00F, 3C00CCAF, 3C00FCAF, 3CF3FF23, 40000441, 40000CAB, 4000DAA1, 400440DD, 400ACCCB, 400CCCAB, 400E44DD, 4040D00D, 404400DD, 40444EDD, 4044D00D, 40ACCCCB, 40DDDDDD, 440000D1, 44000DDD, 4400DD0D, 44E400DD, 4A00004B, 4A0AAAA5, 5000C08D, 52000CCD, 555400A5, 55540A05, 58800007, 58888087, 5A540005, 5C00020D, 5F5400A5, 5F888887, 60006AAF, 600093AF, 600AAAAF, 608CCCCF, 6600686F, 6606866F, 6688AAAF, 7000077D, 70000D5D, 7000707B, 7000707D, 7000740D, 70500D0D, 7070040D, 707007DD, 7070777B, 7077744D, 7077777B, 77007D0D, 7700B44D, 7707000B, 7707D00D, 7770700D, 7770777B, 7777740D, 7777770B, 7777777D, 77777CAB, 7777B887, 778888E7, 788888E7, 79333333, 7ACCCCCB, 7D0000DD, 7D00D0DD, 7DD00D0D, 7DDDDDA9, 80000081, 80000087, 8000E0CD, 80400E4D, 80A0AAA1, 80EC000D, 84000E4D, 8404444D, 84400E4D, 868AAAAF, 86AAAA8F, 8884044D, 88FFFE77, 8C44444D, 8CCCCAAF, 8E40004D, 900000BB, 90000B0B, 90100009, 90800AA1, 93333AAF, 94AAAAA1, 980000A1, 998AAAA1, A00000F9, A0000EEB, A0005A45, A0055545, A00AAA45, A0666669, A0AAA045, A0AAAA45, A0AAE4A1, A0B44441, A4A00005, A6066669, A8AAFFFF, AA055545, AA0AA045, AAA00A45, AAAAA045, B00000AB, B000EEEB, B00EEE0B, B0900081, B0BBBBAB, B7777787, B9000081, B9008001, B9800001, BA00000B, BBBB0ABB, BCCCCCC9, C000004D, C000086F, C0000AFF, C0000E8D, C0000FDD, C00033AF, C0003CAF, C000448D, C000AFFF, C000CF8F, C004444D, C00663AF, C00F00AF, C00FCCAF, C0FFCCAF, C844444D, CC3A000F, CCCCCBED, CCCCCE2D, CCCCD999, CCDCCC4B, CD44444D, CFAF000F, CFFFF023, D00400ED, D004404D, D00777A5, D00E00DD, D0444E0D, D40000ED, D444E00D, D7DDDDDD, DD00D007, DD0D0077, DD0D0707, DDD0040D, DDDDDD19, DDDDDDD1, E0000CCB, E0044441, E00A4AA1, E888820D, E8888CCD, E888C80D, E8AAAAA1, EB00C0CD, EBBC00CD, ECCCCCCB, F00006AF, F00040A5, F00066AF, F06666AF, F0F004A5, F33FFF23, F60006AF, F6AAA0AF, F88888F7, FE777777, FF33F2C3, FF3FFF23, FF588887, FFFF02C3, FFFF5F6F, FFFFF887, FFFFFF79, 10CCCCCAB, 266666603, 333333AAF, 333333F23, 3333FF2C3, 333CCCCAF, 333FFCCAF, 3A000000F, 3FA00000F, 40000048D, 4000004DD, 4000040D1, 40000ACCB, 4000400D1, 4040000DD, 404D0000D, 40A000005, 40E00444D, 40ED0000D, 444E000DD, 444ED000D, 48444444D, 4A0000005, 4AAAAAAA5, 500000C8D, 500000F8D, 50CCCCC8D, 50FFFFF6F, 5AAAAAA45, 5C020000D, 5E444444D, 666666AFF, 70000044D, 70000440D, 700007CCB, 700007D07, 70044000D, 70070007D, 77070007D, 77700040D, 77700070D, 77707044D, 77770000D, 77777777B, 777888887, ...} [/CODE] |
Compare with bases 2 to 12 (with first minimal primes and some large minimal primes):
[CODE] 2: {11} 3: {12, 21, 111} 4: {11, 13, 23, 31, 221} 5: {12, 21, 23, 32, 34, 43, 104, 111, 131, 133, 313, 401, 414, 3101, 10103, 14444, 30301, 33001, 33331, 44441, 300031, 100000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000013} 6: {11, 15, 21, 25, 31, 35, 45, 51, 4401, 4441, 40041} 7: {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, 1022, 1051, 1112, 1202, 1211, 1222, 2111, 3031, 3055, 3334, 3503, 3505, 3545, 4504, 4555, 5011, 5455, 5545, 5554, 6034, 6634, 11111, 11201, 30011, 30101, 31001, 31111, 33001, 33311, 35555, 40054, 100121, 150001, 300053, 351101, 531101, 1100021, 33333301, 5100000001, ..., 33333333333333331, ...} 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, 77774444441, 7777777777771, 555555555555525, 44444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444447} 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, 5161111111, 8333333335, ..., 300000000035, ..., 544444444444, ..., 2000000000007, ..., 56111111111111111111111111111111111111, ..., 7666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666662, ...} 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, ..., 555555555551, ..., 5000000000000000000000000000027, ...} 11: {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, 10A, 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, 1101, 11A9, 1305, 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, 5105, 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, 9101, 9107, 9305, 9505, 9703, A052, A119, A151, A175, A515, A517, A575, A577, A5A8, A719, A779, A911, AAA9, 10011, 10075, 10091, 10109, 10411, 10444, 10705, 10709, 10774, 10901, 11104, 11131, 11144, 11191, 1141A, 114A1, 13757, 1411A, 14477, 144A4, 14A04, 14A11, 17045, 17704, 1774A, 17777, 177A4, 17A47, 1A091, 1A109, 1A114, 1A404, 1A411, 1A709, 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, 70103, 70471, 70583, 70714, 71474, 717A4, 71A09, 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, A1404, A4177, A4401, A4717, A5228, A52AA, A5558, A580A, A5822, A58AA, A5A59, A5AA2, A6096, A6966, A6999, A7051, A7778, A7808, A9055, A9091, A9699, A9969, AA52A, AA58A, 100019, 100079, 101113, 101119, 101911, 107003, 140004, 144011, 144404, 1A0019, 1A0141, 1A5001, 1A7005, 1A9001, 222223, 222823, 300107, 300202, 300323, 303203, 307577, 310007, 332003, 370777, 400555, 401A11, 404001, 404111, 405AAA, 41A011, 440A41, 441011, 451777, 455555, 470051, 470444, 474404, 4A0401, 4A4041, 500015, 500053, 500077, 500507, 505577, 522A2A, 525223, 528A2A, 531707, 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, 701074, 701777, 704408, 704417, 704457, 704484, 707041, 707441, 707708, 707744, 707784, 710777, 717044, 717077, 740008, 74484A, 770441, 770744, 770748, 770771, 777017, 777071, 777448, 777484, 777701, 7778A8, 777A19, 777A48, 778883, 78A808, 790003, 7A1009, 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, A01901, A05509, A0A058, A0A955, A10114, A555A2, A55999, A59991, A5A222, A5A22A, A60609, A66069, A66906, A69006, A79005, A87888, A90099, A90996, A96006, A96666, A97177, A97771, AA0A58, AA5A22, AAA522, 1000501, 1011141, 1030007, 1070047, 111114A, 1111A14, 1111A41, 1144441, 14A4444, 1700005, 1700474, 1A44444, 2555505, 2845055, 3030023, 3100003, 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, 7100447, 7174404, 717444A, 7400404, 7700717, 7701077, 7701707, 7707778, 7774004, 7777104, 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, A001091, A006906, A010044, A040041, A0AAA58, A141111, A5222A2, A600A69, A906606, A909009, A990009, A997701, AA55A52, AAA5552, AAAAA52, 10004747, 10005007, 17000744, 22888823, 28888223, 30010111, 30555777, 31011111, 33000023, 40A00041, 45000055, 47040004, 50377777, 50555553, 5282AAA2, 55505003, 555A5A52, 555AAA2A, 55A5A552, 5AAAAA2A, 60A99999, 70000057, 70070474, 70074704, 70174004, 70700078, 70700474, 70704704, 70710707, 70771007, 70777177, 71074004, 74470001, 77000177, 77070477, 77100077, 77470004, 77700404, 77710007, 77717707, 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, A1900001, A5555009, A5A55552, A9700001, A9909006, A9990006, A9990606, A9999917, A9999966, 100000507, 100035077, 100050777, 100057707, 101111114, 15A000001, 170000447, 300577777, 40000A401, 447771777, 44A444441, 474000004, 477700004, 477777774, 505000003, 55555AA2A, 5555A5A2A, 700000147, 700017004, 700044004, 700077774, 700170004, 701000047, 701700004, 704000044, 704040004, 707070774, 707077704, 707770704, 707777004, 717000004, 717700007, 770000078, 770004704, 770070747, 770070774, 770700008, 770700084, 770707074, 777000044, 777000774, 777717007, 777770477, 777770747, 7777777A4, 77A700008, 888888302, 900000091, 900090799, 970009099, 990990007, 997000077, 999999997, A0000AA58, A00990001, A05555559, A44444111, A44444777, A44477777, A66666669, A90000606, A99999006, A99999099, 1000007447, 1005000007, 1500000001, ..., A000144444, ..., A900000066, ..., 33333333337, ..., 44444444447, ..., A0000000001, ..., A0014444444, ..., 40000000A0041, ..., A000000014444, ..., A044444444441, ..., A144444444411, ..., 40000000000401, ..., A0000044444441, ..., A00000000444441, ..., 11111111111111111, ..., 14444444444441111, ..., 44444444444444111, ..., A1444444444444444, ..., A9999999999999996, ..., 888888888888888883, ..., 1444444444444444444, ..., 7777777777777777771, ..., 4000000000000000A041, ..., A999999999999999999999, ..., A44444444444444444444444441, ..., 1500000000000000000000000007, ..., 40000000000000000000000000041, ..., 440000000000000000000000000001, ..., 999999999999999999999999999999991, ..., A477777777777777777777777777777777777777777, ..., 444444444444444444444444444444444444444444441, ...} 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, BBBBBB99B, ..., B0000000000000000000000000009B, ...} [/CODE] |
Now, I try to prove base 10 (may find some minimal primes not in my current list) like base 8:
In base 10, 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), (1,7), (1,9), (2,1), (2,3), (2,7), (2,9), (3,1), (3,3), (3,7), (3,9), (4,1), (4,3), (4,7), (4,9), (5,1), (5,3), (5,7), (5,9), (6,1), (6,3), (6,7), (6,9), (7,1), (7,3), (7,7), (7,9), (8,1), (8,3), (8,7), (8,9), (9,1), (9,3), (9,7), (9,9) * Case (1,1): ** [B]11[/B] is prime, and thus the only minimal prime in this family. * Case (1,3): ** [B]13[/B] is prime, and thus the only minimal prime in this family. * Case (1,7): ** [B]17[/B] is prime, and thus the only minimal prime in this family. * Case (1,9): ** [B]19[/B] is prime, and thus the only minimal prime in this family. * Case (2,1): ** Since 23, 29, 11, 31, 41, 61, 71, [B]251[/B], [B]281[/B] are primes, we only need to consider the family 2{0,2}1 (since any digits 1, 3, 4, 5, 6, 7, 8, 9 between them will produce smaller primes) *** Since [B]2221[/B] and [B]20201[/B] are primes, we only need to consider the families 2{0}1, 2{0}21, 22{0}1 (since any digits combo 22 or 020 between them will produce smaller primes) **** All numbers of the form 2{0}1 are divisible by 3, thus cannot be prime. **** The smallest prime of the form 2{0}21 is [B]20021[/B] **** The smallest prime of the form 22{0}1 is [B]22000001[/B] * Case (2,3): ** [B]23[/B] is prime, and thus the only minimal prime in this family. * Case (2,7): ** Since 23, 29, 17, 37, 47, 67, 97 [B]227[/B], [B]257[/B], [B]277[/B] are primes, we only need to consider the family 2{0,8}7 (since any digits 1, 2, 3, 4, 5, 6, 7, 9 between them will produce smaller primes) *** Since 887 and [B]2087[/B] are primes, we only need to consider the families 2{0}7 and 28{0}7 (since any digit combo 08 or 88 between them will produce smaller primes) **** All numbers of the form 2{0}7 are divisible by 3, thus cannot be prime. **** All numbers of the form 28{0}7 are divisible by 7, thus cannot be prime. * Case (2,9): ** [B]29[/B] is prime, and thus the only minimal prime in this family. |
* Case (3,1):
** [B]31[/B] is prime, and thus the only minimal prime in this family. * Case (3,3): ** Since 31, 37, 13, 23, 43, 53, 73, 83 are primes, we only need to consider the family 3{0,3,6,9}3 (since any digits 1, 2, 4, 5, 7, 8 between them will produce smaller primes) *** All numbers of the form 3{0,3,6,9}3 are divisible by 3, thus cannot be prime. * Case (3,7): ** [B]37[/B] is prime, and thus the only minimal prime in this family. * Case (3,9): ** Since 31, 37, 19, 29, 59, 79, 89, [B]349[/B] are primes, we only need to consider the family 3{0,3,6,9}9 (since any digits 1, 2, 4, 5, 7, 8 between them will produce smaller primes) *** All numbers of the form 3{0,3,6,9}9 are divisible by 3, thus cannot be prime. * Case (4,1): ** [B]41[/B] is prime, and thus the only minimal prime in this family. * Case (4,3): ** [B]43[/B] is prime, and thus the only minimal prime in this family. * Case (4,7): ** [B]47[/B] is prime, and thus the only minimal prime in this family. * Case (4,9): ** Since 41, 43, 47, 19, 29, 59, 79, 89, [B]409[/B], [B]449[/B], [B]499[/B] are primes, we only need to consider the family 4{6}9 (since any digits 0, 1, 2, 3, 4, 5, 7, 8, 9 between them will produce smaller primes) *** All numbers of the form 4{6}9 are divisible by 7, thus cannot be prime. |
* Case (5,1):
** Since 53, 59, 11, 31, 41, 61, 71, [B]521[/B] are primes, we only need to consider the family 5{0,5,8}1 (since any digits 1, 2, 3, 4, 6, 7, 9 between them will produce smaller primes) *** Since 881 is prime, we only need to consider the families 5{0,5}1 and 5{0,5}8{0,5}1 (since any digit combo 88 between them will produce smaller primes) **** For the 5{0,5}1 family, since [B]5051[/B] and [B]5501[/B] are primes, we only need to consider the families 5{0}1 and 5{5}1 (since any digit combo 05 or 50 between them will produce smaller primes) ***** All numbers of the form 5{0}1 are divisible by 3, thus cannot be prime. ***** The smallest prime of the form 5{5}1 is [B]555555555551[/B] **** For the 5{0,5}8{0,5}1 family, since [B]5081[/B], [B]5581[/B], [B]5801[/B], [B]5851[/B] are primes, we only need to consider the number 581 ***** 581 is not prime. * Case (5,3): ** [B]53[/B] is prime, and thus the only minimal prime in this family. * Case (5,7): ** Since 53, 59, 17, 37, 47, 67, 97, [B]557[/B], [B]577[/B], [B]587[/B] are primes, we only need to consider the family 5{0,2}7 (since any digits 1, 3, 4, 5, 6, 7, 8, 9 between them will produce smaller primes) *** Since 227 and [B]50207[/B] are primes, we only need to consider the families 5{0}7, 5{0}27, 52{0}7 (since any digits combo 22 or 020 between them will produce smaller primes) **** All numbers of the form 5{0}7 are divisible by 3, thus cannot be prime. **** The smallest prime of the form 5{0}27 is [B]5000000000000000000000000000027[/B] **** The smallest prime of the form 52{0}7 is [B]5200007[/B] * Case (5,9): ** [B]59[/B] is prime, and thus the only minimal prime in this family. |
* Case (6,1):
** [B]61[/B] is prime, and thus the only minimal prime in this family. * Case (6,3): ** Since 61, 67, 13, 23, 43, 53, 73, 83 are primes, we only need to consider the family 6{0,3,6,9}3 (since any digits 1, 2, 4, 5, 7, 8 between them will produce smaller primes) *** All numbers of the form 6{0,3,6,9}3 are divisible by 3, thus cannot be prime. * Case (6,7): ** [B]67[/B] is prime, and thus the only minimal prime in this family. * Case (6,9): ** Since 61, 67, 19, 29, 59, 79, 89 are primes, we only need to consider the family 6{0,3,4,6,9}9 (since any digits 1, 2, 5, 7, 8 between them will produce smaller primes) *** Since 449 is prime, we only need to consider the families 6{0,3,6,9}9 and 6{0,3,6,9}4{0,3,6,9}9 (since any digit combo 44 between them will produce smaller primes) **** All numbers of the form 6{0,3,6,9}9 are divisible by 3, thus cannot be prime. **** For the 6{0,3,6,9}4{0,3,6,9}9 family, since 409, 43, [B]6469[/B], 499 are primes, we only need to consider the family 6{0,3,6,9}49 ***** Since 349, [B]6949[/B] are primes, we only need to consider the family 6{0,6}49 ****** Since [B]60649[/B] is prime, we only need to consider the family 6{6}{0}49 (since any digits combo 06 between {6,49} will produce smaller primes) ******* The smallest prime of the form 6{6}49 is [B]666649[/B] ******** Since this prime has just 4 6's, we only need to consider the families with <=3 6's ********* The smallest prime of the form 6{0}49 is [B]60000049[/B] ********* The smallest prime of the form 66{0}49 is [B]66000049[/B] ********* The smallest prime of the form 666{0}49 is [B]66600049[/B] * Case (7,1): ** [B]71[/B] is prime, and thus the only minimal prime in this family. * Case (7,3): ** [B]73[/B] is prime, and thus the only minimal prime in this family. * Case (7,7): ** Since 71, 73, 79, 17, 37, 47, 67, 97, [B]727[/B], [B]757[/B], [B]787[/B] are primes, we only need to consider the family 7{0,7}7 (since any digits 1, 2, 3, 4, 5, 6, 8, 9 between them will produce smaller primes) *** All numbers of the form 7{0,7}7 are divisible by 7, thus cannot be prime. * Case (7,9): ** [B]79[/B] is prime, and thus the only minimal prime in this family. |
* Case (8,1):
** Since 83, 89, 11, 31, 41, 61, 71, [B]821[/B], [B]881[/B] are primes, we only need to consider the family 8{0,5}1 (since any digits 1, 2, 3, 4, 6, 7, 8, 9 between them will produce smaller primes) *** Since [B]8501[/B] is prime, we only need to consider the family 8{0}{5}1 (since any digits combo 50 between them will produce smaller primes) **** Since [B]80051[/B] is prime, we only need to consider the families 8{0}1, 8{5}1, 80{5}1 (since any digits combo 005 between them will produce smaller primes) ***** All numbers of the form 8{0}1 are divisible by 3, thus cannot be prime. ***** The smallest prime of the form 8{5}1 is 8555555555555555555551 (not minimal prime, since 555555555551 is prime) ***** The smallest prime of the form 80{5}1 is [B]80555551[/B] * Case (8,3): ** [B]83[/B] is prime, and thus the only minimal prime in this family. * Case (8,7): ** Since 83, 89, 17, 37, 47, 67, 97, [B]827[/B], [B]857[/B], [B]877[/B], [B]887[/B] are primes, we only need to consider the family 8{0}7 (since any digits 1, 2, 3, 4, 5, 6, 7, 8, 9 between them will produce smaller primes) *** All numbers of the form 8{0}7 are divisible by 3, thus cannot be prime. * Case (8,9): ** [B]89[/B] is prime, and thus the only minimal prime in this family. * Case (9,1): ** Since 97, 11, 31, 41, 61, 71, [B]991[/B] are primes, we only need to consider the family 9{0,2,5,8}1 (since any digits 1, 3, 4, 6, 7, 9 between them will produce smaller primes) *** Since 251, 281, 521, 821, 881, [B]9001[/B], [B]9221[/B], [B]9551[/B], [B]9851[/B] are primes, we only need to consider the families 9{2,5,8}0{2,5,8}1, 9{0}2{0}1, 9{0}5{0,8}1, 9{0,5}8{0}1 (since any digits combo 00, 22, 25, 28, 52, 55, 82, 85, 88 between them will produce smaller primes) **** For the 9{2,5,8}0{2,5,8}1 family, since any digits combo 22, 25, 28, 52, 55, 82, 85, 88 between (9,1) will produce smaller primes, we only need to consider the numbers 901, 9021, 9051, 9081, 9201, 9501, 9801, 90581, 95081, 95801 ***** 95801 is the only prime among 901, 9021, 9051, 9081, 9201, 9501, 9801, 90581, 95081, 95801, but it is not minimal prime since 5801 is prime. **** For the 9{0}2{0}1 family, since 9001 is prime, we only need to consider the numbers 921, 9201, 9021 ***** None of 921, 9201, 9021 are primes. **** For the 9{0}5{0,8}1 family, since 9001 and 881 are primes, we only need to consider the numbers 951, 9051, 9501, 9581, 90581, 95081, 95801 ***** 95801 is the only prime among 951, 9051, 9501, 9581, 90581, 95081, 95801, but it is not minimal prime since 5801 is prime. **** For the 9{0,5}8{0}1 family, since 9001 and 5581 are primes, we only need to consider the numbers 981, 9081, 9581, 9801, 90581, 95081, 95801 ***** 95801 is the only prime among 981, 9081, 9581, 9801, 90581, 95081, 95801, but it is not minimal prime since 5801 is prime. * Case (9,3): ** Since 97, 13, 23, 43, 53, 73, 83 are primes, we only need to consider the family 9{0,3,6,9}3 (since any digits 1, 2, 4, 5, 7, 8 between them will produce smaller primes) *** All numbers of the form 9{0,3,6,9}3 are divisible by 3, thus cannot be prime. * Case (9,7): ** [B]97[/B] is prime, and thus the only minimal prime in this family. * Case (9,9): ** Since 97, 19, 29, 59, 79, 89 are primes, we only need to consider the family 9{0,3,4,6,9}9 (since any digits 1, 2, 5, 7, 8 between them will produce smaller primes) *** Since 449 is prime, we only need to consider the families 9{0,3,6,9}9 and 9{0,3,6,9}4{0,3,6,9}9 (since any digit combo 44 between them will produce smaller primes) **** All numbers of the form 9{0,3,6,9}9 are divisible by 3, thus cannot be prime. **** For the 9{0,3,6,9}4{0,3,6,9}9 family, since [B]9049[/B], 349, [B]9649[/B], [B]9949[/B] are primes, we only need to consider the family 94{0,3,6,9}9 ***** Since 409, 43, 499 are primes, we only need to consider the family 94{6}9 (since any digits 0, 3, 9 between (94,9) will produce smaller primes) ****** The smallest prime of the form 94{6}9 is [B]946669[/B] |
Now, we proved the set of minimal primes (start with b+1, which is equivalent to start with b, if b is composite) of base b=10:
[CODE] 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 555555555551 5000000000000000000000000000027 [/CODE] |
There are totally 77 minimal primes (start with 2 digits) in base 10, there are 75 such primes in base 8
|
[QUOTE=sweety439;567723]Proof of base 4:[/QUOTE]
Proof of base 6: The possible (first digit,last digit) for an element with >=3 digits in the minimal set of the strings for primes with at least two digits are: (1,1), (1,5), (2,1), (2,5), (3,1), (3,5), (4,1), (4,5), (5,1), (5,5) * Case (1,1): ** [B]11[/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 (2,1): ** [B]21[/B] is prime, and thus the only minimal prime in this family. * Case (2,5): ** [B]25[/B] is prime, and thus the only minimal prime in this family. * Case (3,1): ** [B]31[/B] is prime, and thus the only minimal prime in this family. * Case (3,5): ** [B]35[/B] is prime, and thus the only minimal prime in this family. * Case (4,1): ** Since 45, 11, 21, 31, 51 are primes, we only need to consider the family 4{0,4}1 (since any digits 1, 2, 3, 5 between them will produce smaller primes) *** Since [B]4401[/B] and [B]4441[/B] are primes, we only need to consider the families 4{0}1 and 4{0}41 (since any digits combo 40 and 44 between them will produce smaller primes) **** All numbers of the form 4{0}1 are divisible by 5, thus cannot be prime. **** The smallest prime of the form 4{0}41 is [B]40041[/B] * Case (4,5): ** [B]45[/B] is prime, and thus the only minimal prime in this family. * Case (5,1): ** [B]51[/B] is prime, and thus the only minimal prime in this family. * Case (5,5): ** Since 51, 15, 25, 35, 45 are primes, we only need to consider the family 5{0,5}5 (since any digits 1, 2, 3, 4 between them will produce smaller primes) *** All numbers of the form 5{0,5}5 are divisible by 5, thus cannot be prime. |
Currently status for bases 13 to 16:
[CODE] 13: {14, 16, 1A, 23, 25, 2B, 32, 34, 38, 41, 47, 49, 52, 56, 58, 61, 65, 6B, 76, 7A, 7C, 83, 85, 89, 9A, A1, A7, A9, B6, B8, C1, C7, CB, 10C, 119, 11B, 122, 133, 155, 157, 173, 179, 17B, 188, 197, 1B1, 1B5, 1CC, 209, 212, 218, 229, 272, 274, 281, 287, 292, 296, 298, 29C, 2C9, 311, 313, 331, 33B, 355, 371, 373, 379, 397, 3A6, 3AA, 3B3, 3B9, 3BB, 3CA, 43C, 445, 44B, 45A, 463, 4A3, 4A5, 4B2, 4B4, 4BA, 50C, 511, 515, 533, 54A, 551, 559, 571, 575, 57B, 595, 599, 5B3, 5B9, 5CC, 607, 629, 63A, 643, 674, 704, 715, 724, 728, 731, 737, 739, 742, 751, 75B, 773, 775, 779, 782, 784, 791, 793, 797, 7B1, 812, 818, 874, 878, 8AB, 8B4, 902, 919, 922, 926, 92C, 937, 93B, 946, 95B, 962, 968, 971, 977, 979, 982, 98C, 9B3, 9B5, A03, A3C, A45, A4B, A54, AA3, AAB, B02, B0C, B11, B15, B17, B24, B33, B39, B42, B57, B59, B71, B93, B9B, BA4, BAA, BB1, BB9, BC2, BCC, C29, C43, C98, CA3, 1013, 1031, 1037, 105B, 1075, 10B7, 10BB, 1105, 1112, 1121, 1127, 113C, 1172, 1187, 1208, 1211, 1277, 12C8, 1307, 1309, 131C, 139C, 151C, 1721, 1727, 1787, 1901, 1909, 1912, 1918, 193C, 1981, 198B, 199C, 19B2, 19C3, 1B29, 1BB2, 1BBC, 1C28, 1C39, 2021, 2078, 2117, 2201, 2221, 2267, 2278, 2627, 2678, 2711, 2771, 2788, 3037, 3053, 306A, 3077, 3091, 309B, 30AC, 3305, 353C, 35AB, 35BA, 35BC, 3677, 3905, 390B, 39C5, 3A0C, 3AB5, 3B5C, 3C35, 3C59, 3C95, 403A, 40AB, 4333, 435B, 4403, 44C3, 4535, 4544, 454C, 45B5, 45BB, 480B, 4B35, 4B5B, 4C36, 5057, 5077, 509B, 50A4, 5107, 5305, 530B, 539C, 53AB, 53C9, 5444, 5455, 54C4, 5503, 5545, 55AB, 5774, 5794, 590B, 594B, 5974, 59B4, 5A4C, 5A53, 5AA4, 5AB5, 5ABB, 5ACA, 5B4B, 5B5A, 5BA5, 5CA4, 6227, 6278, 6667, 6698, 6733, 6872, 6928, 6944, 694C, 6973, 6986, 6997, 69C8, 6AC3, 6C92, 6C94, 7019, 7057, 70B5, 7103, 710B, 7118, 7127, 7129, 7172, 7178, 7192, 7211, 7217, 7219, 7271, 7303, 7408, 7433, 7444, 7505, 7507, 7574, 770B, 7774, 7778, 7787, 7871, 7877, 7888, 794B, 7994, 79B4, 7B43, 7B74, 7B94, 7BB2, 8027, 8072, 8081, 80BA, 8171, 8207, 821C, 848B, 8687, 8711, 8722, 87BB, 8867, 88B2, 88BA, 8B22, 8B2A, 8BAC, 9004, 9017, 9031, 9053, 9055, 9073, 9091, 90BB, 90C8, 9107, 9118, 913C, 9181, 91C3, 9284, 935C, 93C5, 9424, 9428, 9448, 9509, 959C, 96C4, 9703, 9743, 9745, 974B, 97B2, 9811, 981B, 987B, 98B1, 991C, 9967, 9998, 9B12, 9B74, 9B92, 9BBC, 9C55, 9C86, 9CC4, A0BA, A306, A436, A535, A5B5, A636, A6C3, A80B, AB04, AB22, AB35, AB3B, AB4C, AB55, ABAC, ABB5, AC36, ACA5, B044, B04A, B0B7, B129, B1B2, B219, B222, B291, B299, B2CA, B35A, B3A5, B404, B44C, B45B, B4B3, B501, B51C, B55A, B5A5, B5AB, B5C3, B707, B792, B794, B905, B912, B9C5, BA5B, BAB3, BB03, BB45, BB72, BBA5, BBB2, BC44, BC53, BC95, BC99, C30A, C36A, C395, C454, C535, C553, C593, C944, C953, C964, CC94, 10015, 10051, 10099, 10118, 10291, 10712, 10772, 10811, 10877, 10921, 10B92, 11111, 11135, 11171, 111C8, 11531, 11C03, 13001, 13177, 13777, 13915, 13951, 13991, 159BB, 17018, 17102, 17111, 17117, 17171, 17177, 17708, 17711, 17801, 18071, 18101, 18271, 18B27, 19003, 19153, 19315, 19351, 19591, 19913, 19951, 1C099, 20171, 20177, 20207, 20227, 20777, 21011, 21077, 2111C, 21707, 22207, 30017, 300B5, 301C9, 3033A, 303A3, 303C5, 3050B, 305C9, 3095C, 30B05, 31007, 3159B, 31999, 31C09, 3330A, 33353, 33593, 33757, 33C5C, 33CC5, 35003, 3591B, 39353, 39539, 39935, 39995, 3ACCC, 3C5C3, 3CC53, 40043, 40306, 405C4, 408BC, 40BBB, 40C54, 43066, 4366A, 4443A, 45055, 45505, 45554, 4555C, 455BC, 455C3, 45C04, 488BC, 4B03B, 4B0B5, 4B55C, 4BB0B, 4C003, 4C054, 4C5C4, 50053, 500B1, 5035A, 504B5, 5053A, 50554, 505B4, 50A35, 50B07, 50BBA, 5139B, 519BB, 51BB7, 535AC, 53A5C, 53AC5, 53BAC, 54004, 54035, 5403B, 545C3, 54B05, 54B5C, 54BBC, 54C53, 55357, 5535B, 553AC, 554BC, 55537, 55544, 5554C, 55577, 555A4, 555BB, 55A5C, 55B04, 55B55, 55B77, 55BB5, 55BC4, 55C54, 55C5A, 57403, 591BB, 59443, 59BB7, 5A044, 5AC04, 5AC35, 5B001, 5B007, 5B0AB, 5B0B4, 5B4C5, 5B544, 5B555, 5B5BB, 5B744, 5B777, 5BA0B, 5BB44, 5BB55, 5BBC4, 5BC54, 5C039, 5C35A, 5C53A, 60098, 60964, 60988, 60A63, 66094, 66377, 66692, 66694, 669C2, 669C4, 66A36, 67022, 67099, 67222, 67277, 67772, 68627, 69088, 690C4, 69808, 69994, 6A663, 7007B, 70181, 70222, 70277, 70772, 70808, 70B0B, 70B29, 71113, 71711, 71908, 71999, 7199B, 71BB7, 71BBB, 74035, 74305, 7430B, 74503, 75443, 75454, 75535, 77072, 77108, 77177, 77717, 77BBB, 78011, 79BBB, 7B007, 7B7B7, 7B7BB, 7BBB3, 7BBB7, 80117, 80221, 80771, 80777, 80807, 8084B, 80B7B, 80BBB, 81107, 8400B, 86267, 87107, 87277, 87727, 87B27, 88111, 88201, 88702, 88771, 8888B, 88B77, 88BBB, 8B1BC, 8B727, 90035, 90059, 90088, 90095, 9009B, 90101, 90103, 90305, 90488, 904CC, 90574, 90644, 9064C, 90806, 908B7, 9090B, 90994, 90B09, 90C35, 90C59, 90C64, 91111, 91135, 91315, 9180B, 92008, 92408, 92488, 93359, 93395, 944C2, 944CC, 94505, 9455C, 94804, 94888, 94C0C, 94C33, 94C4C, 95045, 95504, 95573, 955C4, 95C54, 96044, 97BBB, 98066, 98408, 98444, 98804, 98848, 99001, 99005, 9900B, 99074, 990BC, 99113, 99175, 99278, 99335, 99454, 994C3, 99517, 99593, 9984B, 99881, 99904, 99917, 99935, 99955, 99973, 999BB, 999C2, 999C4, 99B99, 9B00B, 9B04B, 9B0B4, 9B1BB, 9BB04, 9C059, 9C244, 9C404, 9C44C, 9C488, 9C503, 9C5C9, 9C644, 9C664, 9CC88, 9CCC2, A00B4, A05BB, A08B2, A08BC, A0BC4, A3336, A3633, A443A, A4443, A50BB, A55C5, A5AAC, A5BBA, A5C53, A5C55, AACC5, AB05B, AB0BB, AB40A, ABBBC, ABC4A, ACC5A, ACCC3, B0053, B0075, B010B, B0455, B0743, B0774, B0909, B0BB4, B2277, B2A2C, B3005, B351B, B37B5, B3A0B, B3ABC, B3B0A, B400A, B4035, B403B, B4053, B4305, B4BC5, B4C0A, B504B, B50BA, B530A, B5454, B54BC, B54C5, B5544, B55B5, B5B44, B5B4C, B5BB5, B7403, B7535, B77BB, B7955, B7B7B, B9207, B9504, B9999, BA055, BA305, BABC5, BAC35, BB054, BB05A, BB207, BB3B5, BB4C3, BB504, BB544, BB54C, BB5B5, BB753, BB7B7, BBABC, BBB04, BBB4C, BBB55, BBBAC, BC035, BC455, C0353, C0359, C03AC, C0904, C0959, C0A5A, C0CC5, C3059, C335C, C5A0A, C5A44, C5AAC, C6692, C69C2, C904C, C9305, C9905, C995C, C99C5, C9C04, C9C59, C9CC2, CA50A, CA5AC, CAA05, CAA5A, CC335, CC544, CC5AA, CC935, CC955, 100039, 100178, 100718, 100903, 101177, 101708, 101711, 101777, 102017, 102071, 103999, 107081, 107777, 108217, 109111, 109151, 110078, 110108, 110717, 111017, 111103, 1111C3, 111301, 111707, 113501, 115103, 117017, 117107, 117181, 117701, 120701, 13C999, 159103, 170717, 177002, 177707, 180002, 187001, 18C002, 19111C, 199903, 1B0007, 1BB077, 1BBB07, 1C0903, 1C8002, 1C9993, 200027, 207107, 217777, 219991, 220027, 222227, 270008, 271007, 277777, 290444, 300059, 300509, 303359, 303995, 309959, 30B50A, 3336AC, 333707, 33395C, 335707, 3360A3, 350009, 36660A, 3666AC, 370007, 377B07, 39001C, 399503, 3BC005, 400366, 400555, 400B3B, 400B53, 400BB5, 400CC3, 4030B5, 40B053, 40B30B, 40B505, 43600A, 450004, 4A088B, 4B0503, 4B5C05, 4BBBB5, 4BC505, 500039, 50045B, 50405B, 504B0B, 50555B, 5055B5, 505B0A, 509003, 50A50B, 50B045, 50B054, 539B01, 550054, 5500BA, 55040B, 553BC5, 5553C5, 55550B, 5555C3, 555C04, 55B00A, 55BB0B, 570007, 5A500B, 5A555B, 5AC505, 5B055B, 5B0B5B, 5B5B5C, 5B5BC5, 5BB05B, 5BBB0B, 5BBB54, 5BBBB4, 5BBC0A, 5BC405, 5C5A5A, 5CA5A5, 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EE7B, EE81, EEAB, EEE1, F08F, F0A9, F227, F2ED, F3AF, F485, F58D, F72D, F763, F769, F787, F7A5, F7E7, F82D, F86F, F877, F88D, F8D7, F8E7, F8FF, FCCD, FED7, FF85, FF8F, FFA9, 100AB, 10BA9, 1A0CB, 1BA09, 200E1, 2C603, 2CC03, 30227, 303AF, 30AAF, 32003, 32207, 32CC3, 330AF, 33169, 33221, 33391, 33881, 33AFF, 38807, 38887, 3AFFF, 3F203, 3F887, 3FAFF, 400BB, 4084D, 40A4B, 42001, 44221, 44401, 444D1, 4480D, 4488D, 44CCB, 44D4D, 44E8D, 4804D, 4840D, 4A0CB, 4A54B, 4CACB, 4D0DD, 4D40D, 4D44D, 5004D, 50075, 502CD, 5044D, 50887, 50EE1, 5448D, 548ED, 55A45, 55F45, 5844D, 5A4A5, 5AE41, 5B0CD, 5B44D, 5BBCD, 5D4ED, 5E0E1, 5EB4D, 5EC8D, 5ECCD, 5EE41, 5F06F, 5F7DD, 5F885, 5F8CD, 5FC8D, 5FF75, 6088F, 60AFF, 630AF, 633AF, 660A9, 668CF, 669AF, 66A09, 66A0F, 66FA9, 6886F, 6A00F, 6A0FF, 6A8AF, 6AFFF, 7002D, 7024D, 70B0D, 70B7D, 7200D, 73363, 73999, 7444D, 770B7, 777D7, 77B07, 77D7D, 77DD7, 79003, 79999, 7B00D, 7D05D, 7D7DD, 8007D, 800D1, 8074D, 82CCD, 82E4D, 8448D, 8484D, 8704D, 8724D, 87887, 88001, 8800D, 880CD, 88507, 88555, 8866F, 8872D, 8877D, 888D1, 888D7, 88AA1, 88C2D, 88D57, 88D75, 88D77, 8AFAF, 8C2CD, 8C40D, 8C8CD, 8CCED, 8CE2D, 8CFED, 8E007, 8E20D, 8E24D, 8F6FF, 8FAAF, 900CB, 901AB, 90901, 909A1, 90AB1, 90AE1, 90EE1, 910AB, 93331, 940AB, 963AF, 966AF, 99019, 99109, 99A01, 9AAE1, 9B00B, 9B0AB, 9B441, 9BABB, 9BBBB, 9E441, A00BB, A0405, A044B, A08AF, A0A51, A0B91, A0C4B, A1B09, A54A5, A5B41, A6609, A904B, A94A1, A9C4B, A9E01, A9E41, AA0A1, AA441, AA501, AA8AF, AAEE1, AAF45, AAF8F, ABBA1, ACC69, AE0BB, AE0EB, AEAE1, AEE0B, AEEA1, AEECB, AF045, AF4A5, AFA8F, B00A1, B00D7, B044D, B0777, B0A0B, B0A91, B0BBD, B0BCD, B0C09, B0DA9, B0EAB, B2207, B4001, B6669, B7707, B7D07, B8081, B9021, BA091, BA109, BA4BB, BB001, BB0EB, BB8A1, BBBEB, BBE0B, BBEBB, BC009, BCECD, BD0A9, BE44D, BEB0D, BEBBB, BEEBB, C0263, C02C3, C02ED, C040D, C0CA9, C0CCD, C2663, C2CED, C32C3, C3323, C400D, C40ED, C44CB, C44ED, C480D, C484D, C4CAB, C60AF, C686F, C6A0F, C86FF, C8C2D, CAA0F, CAFAF, CBCED, CC0AF, CC44B, CC82D, CC8FF, CCAF9, CCAFF, CCCFD, CCFAF, CD00D, CD4CB, CD4ED, CDDDD, CF2C3, CFC8F, CFE8D, D0045, D07DD, D09BB, D0D4D, D0DD7, D0EBB, D0EEB, D1009, D1045, D10B9, D1BA9, D54BB, D54ED, D5AE1, D5D07, D5EE1, D70DD, D7707, D7777, D77DD, D7DD7, D9441, D9AE1, D9B0B, DA9A1, DA9E1, DAA41, DAAA1, DBB0B, DBBA1, DC4CB, DD227, DD44D, DDDD7, E0081, E00E1, E010B, E088D, E08CD, E0B0D, E0BBD, E100B, E4D0D, E777B, E77AB, E7CCB, E844D, E848D, E884D, E88A1, EB0BB, EBB4D, EBBEB, EBEEB, EC8CD, ECBCD, ECC8D, ED04D, EE001, EE0EB, EE4A1, EEEBB, F0085, F09AF, F0C23, F0CAF, F2663, F2C03, F3799, F3887, F4A05, F4AA5, F506F, F5845, F5885, F5C2D, F5ECD, F5F45, F66A9, F688F, F6AFF, F7399, F777D, F8545, F8555, F8AAF, F8F87, F9AAF, FA0F9, FA405, FA669, FAFF9, FC263, FCA0F, FCAFF, FCE8D, FCF23, FD777, FDDDD, FDEDD, FEC2D, FEC8D, FF545, FF6AF, FF739, FF775, FF9AF, FFC23, 100055, 100555, 10A9CB, 1A090B, 1A900B, 1CACCB, 1CCACB, 20DEE1, 266003, 3000AF, 300A0F, 300AFF, 308087, 308E07, 3323E1, 333A0F, 339331, 33CA0F, 33CF23, 33CFAF, 33F323, 380087, 3A00AF, 3A0F0F, 3AA0FF, 3AAF0F, 3C33AF, 3C3A0F, 3C3FAF, 3CCAAF, 3F0FAF, 3F32C3, 3FF0AF, 3FFAAF, 4004CB, 400A05, 4048ED, 404DDD, 40AA05, 40D04D, 40DD4D, 40E0DD, 40E48D, 440041, 44008D, 44044D, 4404DD, 44440D, 4448ED, 4484ED, 448E4D, 44E44D, 48888D, 4AA005, 4DD00D, 4DD04D, 4DDD0D, 4E048D, 4E448D, 4E880D, 5000DD, 500201, 50066F, 5008CD, 500C2D, 500D7D, 50C20D, 520C0D, 544EDD, 54AA05, 54AAA5, 54ED4D, 566AAF, 57D00D, 580087, 5A5545, 5C20CD, 5C8CCD, 5CC2CD, 5D000D, 5D070D, 5F666F, 5FAA45, 5FFF45, 60008F, 600A0F, 603AAF, 6060AF, 6066AF, 60A0AF, 63AA0F, 6663AF, 66668F, 666AAF, 668A8F, 66AFF9, 68888F, 693AAF, 7007B7, 70404D, 70770B, 70770D, 707BE7, 70DD0D, 733339, 733699, 74004D, 74040D, 77007B, 770CCB, 777B4D, 777BE7, 777CCB, 77ACCB, 77B74D, 77D0DD, 7A0CCB, 7B744D, 7CACCB, 7DDD99, 80044D, 800807, 80200D, 8044ED, 80C04D, 80CC2D, 80E44D, 8404ED, 84888D, 84E04D, 84E40D, 86686F, 8668AF, 8686AF, 86F66F, 86FFFF, 87000D, 87744D, 880807, 886AFF, 88824D, 88870D, 888787, 88884D, 88886F, 88887D, 88888D, 888C4D, 888FAF, 88AA8F, 88CC8D, 88F6AF, 88F8AF, 88FA8F, 88FF6F, 88FF87, 88FFAF, 8A8FFF, 8C0C2D, 8C802D, 8CCFFF, 8CE00D, 8CE0CD, 8CFCCF, 8E00CD, 8E044D, 8E0CCD, 8EC0CD, 8F68AF, 8F88F7, 8FCFCF, 8FF887, 8FFCCF, 8FFF6F, 9002E1, 9004AB, 9008A1, 900919, 900ABB, 900B21, 90B801, 90CCCB, 9332E1, 944441, 94ACCB, 990001, 9900A1, 9A4441, 9A4AA1, 9AA4A1, 9AAA41, 9AAAAF, 9B66C9, 9BBA0B, 9BC0C9, 9BC669, 9BC6C9, 9C4ACB, A0094B, A00ECB, A09441, A0A08F, A0E0CB, A0ECCB, A0F669, A40A05, A4AAA5, A50E41, A5AA45, A60069, A8FAFF, A9AA41, AA5E41, AAA4A5, AAA545, AC6669, ACCC4B, ACCCC9, AEAA41, AFF405, AFF669, AFFA45, AFFFF9, B00921, B00BEB, B00CC9, B00D91, B08801, B0D077, B70077, B70E77, B77E77, B88877, B88881, B94421, BAE00B, BB00AB, BB0DA1, BB444D, BB44D1, BB8881, BBBBBD, BBBC4D, BBCCCD, BC0CC9, BC66C9, BCC669, BCC6C9, BCCC09, BE000D, BE00BD, BE0B4D, BE0CCD, BEA00B, BECCCD, C0084D, C00A0F, C0608F, C0668F, C0844D, C0A0FF, C0AFF9, C0C3AF, C0C68F, C0CAAF, C0CDED, C0D0ED, C0E80D, C0EC2D, C0EC8D, C0FA0F, C0FAAF, C2CC63, C30CAF, C333AF, C3CAAF, C3CCAF, C4048D, C40D4D, C4404D, C4408D, C4440D, C44DDD, C4ACCB, C4DCCB, C4DD4D, C6068F, C66AAF, C68AAF, C6AA8F, C8044D, C8440D, C8666F, CA00FF, CA0FFF, CAAAAF, CAAFFF, CAFF0F, CBE0CD, CC008F, CC0C8F, CC3CAF, CC4ACB, CC608F, CC66AF, CCBECD, CCC4AB, CCCA0F, CCCC8F, CCCE8D, CE0C8D, CF0F23, CF0FAF, CFAFFF, CFCAAF, CFFAFF, D0005D, D00BA9, D05EDD, D077D7, D10CCB, D22207, D4000B, D4040D, D4044D, D40CCB, D70077, D7D00D, D90009, D900BB, DB00BB, DB4441, DD400D, DDD109, DDD1A9, DDD919, DDD941, DED00D, E00D4D, E00EEB, E0AAE1, E0AE41, E0AEA1, E0B44D, E0BCCD, E0BEBB, E0D0DD, E0E441, E4048D, E4448D, E800CD, E8200D, EA0E41, EAA0E1, EBB00B, ECCCAB, EDDDDD, EEBE0B, F00263, F0056F, F00A45, F02C63, F03F23, F05405, F060AF, F08585, F0A4A5, F0F2C3, F0F323, F2CCC3, F33203, F33C23, F5F66F, F5FF6F, F68CCF, F6AA8F, F888AF, FA0F45, FAA045, FAA545, FAFC69, FC0AAF, FC66AF, FCCCAF, FCFFAF, FF0323, FF056F, FF3203, FF7903, FFA045, FFA4A5, FFAA45, FFC0AF, FFF4A5, FFF575, FFFA45, FFFCAF, 10A009B, 20000D1, 2CCC663, 30A00FF, 30CCCAF, 30FA00F, 30FCCAF, 3333C23, 333C2C3, 33C3AAF, 33FCAAF, 33FFFAF, 3A0A00F, 3AAAA0F, 3AF000F, 3AFAAAF, 3C0CA0F, 3CCC3AF, 3CFF323, 3F33F23, 3FAA00F, 3FF3323, 4004441, 400DDD1, 400E00D, 400ED0D, 404404D, 404448D, 404E4DD, 440EDDD, 4440EDD, 44444ED, 4444E4D, 44DDDDD, 4A000A5, 4CCCCAB, 4D0CCCB, 4E4404D, 4E4444D, 4E4DDDD, 5000021, 5004221, 5006AAF, 500FF6F, 5042201, 508CCCD, 5400005, 5400AA5, 5555405, 5808007, 5AA4005, 5C0008D, 5CCC8CD, 5D4444D, 5EEEEEB, 5F40005, 5F554A5, 5F6AAAF, 60000AF, 60006A9, 600866F, 6008AAF, 600AA8F, 600F6A9, 606608F, 606686F, 608666F, 60AA08F, 60AAA8F, 66000AF, 66666A9, 6666AF9, 6866A8F, 6AAAAAF, 70070D7, 70077DD, 700DDDD, 707077D, 707D007, 70D00DD, 770077D, 770400D, 770740D, 7777775, 77777B7, 77777DD, 7777ACB, 77B88E7, 77DD00D, 77DDDDD, 7D0D00D, 7DD0D07, 7DDD00D, 800002D, 8000CED, 80C0E0D, 80CECCD, 840400D, 844000D, 844E00D, 868688F, 880444D, 884404D, 887D007, 8888801, 8888881, 8888E07, 8888F77, 8888FE7, 88A8AFF, 88AAAFF, 88FAFFF, 8A8AAAF, 8A8AAFF, 8AAA8FF, 8C00ECD, 8C8444D, 8E4400D, 8FCCCCF, 900BBAB, 90CC4AB, 9908AA1, 99E0E01, 9B00801, 9B6CCC9, A000FF9, A006069, A00A8FF, A01CCCB, A05F545, A0BEEEB, A0E4AA1, AA0008F, AA08FFF, AA40AA5, AA8FFFF, AAAA405, AE04AA1, AE44441, AE4AAA1, AECCCCB, AF40005, AFA5A45, AFFFC69, B000BAB, B000EBB, B0D0007, B222227, B6CCCC9, B8880A1, BA000EB, BA0BEEB, BAEEEEB, BB000CD, BB00C0D, BB0B00D, BC6CC69, BC6CCC9, BCCCC69, BCCCCED, C0000A9, C00068F, C000CFD, C000E2D, C000FAF, C004D4D, C00E20D, C00E8CD, C00F68F, C033A0F, C0802CD, C086AAF, C0A00AF, C0AFFFF, C0C086F, C0C0F8F, C0CA00F, C0CC08F, C0D044D, C0F0AFF, C0FF023, C0FFFAF, C33FA0F, C33FAAF, C3CA00F, C3FFCAF, C8002CD, C8200CD, CCC668F, CCCAA8F, CCCC0A9, CCCC3AF, CCCCCA9, CCCDC4B, CE0008D, CE2000D, CE8CCCD, CF000AF, CFF0AAF, CFFF0AF, D0000EB, D0005EB, D000775, D000EDD, D007077, D00DDD9, D00ED0D, D0AAA45, D0AAAA5, D0EDDDD, D19000B, D4404ED, D4440ED, D5BBBBB, DCCCC4B, DD00DD9, DD07077, DD0DD09, DD0DDD9, DD99999, DDD0D09, DDDD0D9, DDDD9E1, DDDDD09, DDDDD99, DE0DDDD, DEEEEEB, E00001B, E0004A1, E000CAB, E00A041, E00BB0B, E00BBBB, E00C80D, E00CCCB, E044DDD, E0AA4A1, E0AAA41, E0BBB0B, E0D444D, E40444D, E4DDD4D, E88CCCD, E8C000D, E8CCCCD, EA04441, EA0A4A1, EBB000D, EBCCCCD, ED0D00D, EEAAA01, EEBBBBB, EEE000B, F0002C3, F002CC3, F003323, F005545, F00F4A5, F033323, F0400A5, F0A5545, F333323, F333F23, F6660AF, F733333, FA00009, FA004A5, FAAAA45, FC6668F, FCC668F, FD00AA5, FEE7777, FF0F263, FF26003, FF3F323, FF5F887, FFAFF45, FFFF263, FFFF379, 2CCCCC63, 30CCA00F, 33333319, 3333FCAF, 3333FFAF, 33FFA00F, 3C00CCAF, 3C00FCAF, 3CF3FF23, 40000441, 40000CAB, 4000DAA1, 400440DD, 400ACCCB, 400CCCAB, 400E44DD, 4040D00D, 404400DD, 40444EDD, 4044D00D, 40ACCCCB, 40DDDDDD, 440000D1, 44000DDD, 4400DD0D, 44E400DD, 4A00004B, 4A0AAAA5, 5000C08D, 52000CCD, 555400A5, 55540A05, 58800007, 58888087, 5A540005, 5C00020D, 5F5400A5, 5F888887, 60006AAF, 600093AF, 600AAAAF, 608CCCCF, 6600686F, 6606866F, 6688AAAF, 7000077D, 70000D5D, 7000707B, 7000707D, 7000740D, 70500D0D, 7070040D, 707007DD, 7070777B, 7077744D, 7077777B, 77007D0D, 7700B44D, 7707000B, 7707D00D, 7770700D, 7770777B, 7777740D, 7777770B, 7777777D, 77777CAB, 7777B887, 778888E7, 788888E7, 79333333, 7ACCCCCB, 7D0000DD, 7D00D0DD, 7DD00D0D, 7DDDDDA9, 80000081, 80000087, 8000E0CD, 80400E4D, 80A0AAA1, 80EC000D, 84000E4D, 8404444D, 84400E4D, 868AAAAF, 86AAAA8F, 8884044D, 88FFFE77, 8C44444D, 8CCCCAAF, 8E40004D, 900000BB, 90000B0B, 90100009, 90800AA1, 93333AAF, 94AAAAA1, 980000A1, 998AAAA1, A00000F9, A0000EEB, A0005A45, A0055545, A00AAA45, A0666669, A0AAA045, A0AAAA45, A0AAE4A1, A0B44441, A4A00005, A6066669, A8AAFFFF, AA055545, AA0AA045, AAA00A45, AAAAA045, B00000AB, B000EEEB, B00EEE0B, B0900081, B0BBBBAB, B7777787, B9000081, B9008001, B9800001, BA00000B, BBBB0ABB, BCCCCCC9, C000004D, C000086F, C0000AFF, C0000E8D, C0000FDD, C00033AF, C0003CAF, C000448D, C000AFFF, C000CF8F, C004444D, C00663AF, C00F00AF, C00FCCAF, C0FFCCAF, C844444D, CC3A000F, CCCCCBED, CCCCCE2D, CCCCD999, CCDCCC4B, CD44444D, CFAF000F, CFFFF023, D00400ED, D004404D, D00777A5, D00E00DD, D0444E0D, D40000ED, D444E00D, D7DDDDDD, DD00D007, DD0D0077, DD0D0707, DDD0040D, DDDDDD19, DDDDDDD1, E0000CCB, E0044441, E00A4AA1, E888820D, E8888CCD, E888C80D, E8AAAAA1, EB00C0CD, EBBC00CD, ECCCCCCB, F00006AF, F00040A5, F00066AF, F06666AF, F0F004A5, F33FFF23, F60006AF, F6AAA0AF, F88888F7, FE777777, FF33F2C3, FF3FFF23, FF588887, FFFF02C3, FFFF5F6F, FFFFF887, FFFFFF79, 10CCCCCAB, 266666603, 333333AAF, 333333F23, 3333FF2C3, 333CCCCAF, 333FFCCAF, 3A000000F, 3FA00000F, 40000048D, 4000004DD, 4000040D1, 40000ACCB, 4000400D1, 4040000DD, 404D0000D, 40A000005, 40E00444D, 40ED0000D, 444E000DD, 444ED000D, 48444444D, 4A0000005, 4AAAAAAA5, 500000C8D, 500000F8D, 50CCCCC8D, 50FFFFF6F, 5AAAAAA45, 5C020000D, 5E444444D, 666666AFF, 70000044D, 70000440D, 700007CCB, 700007D07, 70044000D, 70070007D, 77070007D, 77700040D, 77700070D, 77707044D, 77770000D, 77777777B, 777888887, 7D0DDDDDD, 7DD0000D7, 8008880A1, 800888A01, 800C000ED, 888800087, 88888AF8F, 888CCCCCD, 88CCCCCCD, 8AAAAAFFF, 8AAFFFFFF, 8CECCCCCD, 8CFFFFCFF, 8EC00000D, 900010009, 908A0AAA1, 9800AAAA1, 9B0CCCCC9, A00000669, A00005545, A0000A545, A000FFF45, A0AAAAA8F, A4000004B, A55540005, A5F554005, AA0A0AA45, AA0AAA8FF, AA4000005, AAA0AA8FF, AAAA0A8FF, AAAA0AA8F, ...} [/CODE] |
Proven minimal primes (start with b+1) set: (bases b = 2, 3, 4, 5, 6, 8, 10, 12) (base b = 7 is not proven, but I cannot find other such primes)
[CODE] 2: {11} 3: {12, 21, 111} 4: {11, 13, 23, 31, 221} 5: {12, 21, 23, 32, 34, 43, 104, 111, 131, 133, 313, 401, 414, 3101, 10103, 14444, 30301, 33001, 33331, 44441, 300031, 100000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000013} 6: {11, 15, 21, 25, 31, 35, 45, 51, 4401, 4441, 40041} 7: {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, 1022, 1051, 1112, 1202, 1211, 1222, 2111, 3031, 3055, 3334, 3503, 3505, 3545, 4504, 4555, 5011, 5455, 5545, 5554, 6034, 6634, 11111, 11201, 30011, 30101, 31001, 31111, 33001, 33311, 35555, 40054, 100121, 150001, 300053, 351101, 531101, 1100021, 33333301, 5100000001, 33333333333333331} 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, 77774444441, 7777777777771, 555555555555525, 44444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444447} 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, 555555555551, 5000000000000000000000000000027} 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, BBBBBB99B, B0000000000000000000000000009B, 400000000000000000000000000000000000000077} [/CODE] |
Let L(b) be the minimal set of the strings for the primes >b in base b
[CODE] b |L(b)| largest element in L(b) largest element in L(b) in base b written in decimal 2 1 11 3 3 3 111 13 4 5 221 41 5 22 10[SUB]93[/SUB]13 5^95+8 6 11 40041 5209 7 71 3[SUB]16[/SUB]1 (7^17-5)/2 8 75 4[SUB]220[/SUB]7 (4*8^221+17)/7 10 77 50[SUB]28[/SUB]27 5*10^30+27 12 106 40[SUB]39[/SUB]77 4*12^41+91 [/CODE] (base b = 7 is not proven, but I cannot find other such primes) |
[URL="http://factordb.com/index.php?id=3"]base 2 minimal primes (start with 2 digits)[/URL]
[URL="http://factordb.com/index.php?id=455"]base 3 minimal primes (start with 2 digits)[/URL] [URL="http://factordb.com/index.php?id=205205"]base 4 minimal primes (start with 2 digits)[/URL] [URL="http://factordb.com/index.php?id=1100000002457822814"]base 5 minimal primes (start with 2 digits)[/URL] [URL="http://factordb.com/index.php?id=1100000002457821560"]base 6 minimal primes (start with 2 digits)[/URL] [URL="http://factordb.com/index.php?id=1100000002457825324"]base 7 minimal primes (start with 2 digits) (not proven, but I cannot find other such primes)[/URL] [URL="http://factordb.com/index.php?id=1100000002371473795"]base 8 minimal primes (start with 2 digits)[/URL] [URL="http://factordb.com/index.php?id=1100000002370859491"]base 10 minimal primes (start with 2 digits)[/URL] [URL="http://factordb.com/index.php?id=1100000002457818232"]base 12 minimal primes (start with 2 digits)[/URL] |
Haha, quite ingenious putting them in fdb in a product. Hat off! :bow: You made me click on the links because I could not imagine how you can put a [U]list[/U] of numbers in fdb under a [U]single[/U] identifier. Pleasantly surprised.
(this encouragement post should not be taken as a license to restart spamming the forum, I really do appreciate the cleverness of the solution, but that's all :razz:) |
[QUOTE=LaurV;568619]Haha, quite ingenious putting them in fdb in a product. Hat off! :bow: You made me click on the links because I could not imagine how you can put a [U]list[/U] of numbers in fdb under a [U]single[/U] identifier. Pleasantly surprised.
(this encouragement post should not be taken as a license to restart spamming the forum, I really do appreciate the cleverness of the solution, but that's all :razz:)[/QUOTE] Now I computed the set for bases 2, 3, 4, 5, 6, 8, 10 and proved that my sets are all complete (by hand), see post [URL="https://mersenneforum.org/showpost.php?p=568564&postcount=93"]#93[/URL], in [URL="https://mersenneforum.org/showpost.php?p=531632&postcount=7"]this post[/URL], you said that my set in post [URL="https://mersenneforum.org/showpost.php?p=531436&postcount=1"]#1[/URL] for bases 7 and 8 are not complete, and then I tried to solved base 8, and found there are only 4 primes not in the list in post [URL="https://mersenneforum.org/showpost.php?p=531436&postcount=1"]#1[/URL]: [CODE] 77774444441 7777777777771 555555555555525 44444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444447 [/CODE] For the fully proof of base 8, see post [URL="https://mersenneforum.org/showpost.php?p=568017&postcount=62"]#62[/URL] For the fully proof of base 10, see posts #84~#88 in [URL="https://mersenneforum.org/showthread.php?t=24972&page=8"]page 8[/URL] Now I try to proof base 7, 9, 12 (other bases are too difficult to prove by hand, but can still prove by computer program, but I have no such program that can compute the simple families), base 9 may be harder, since this base has many families which can be ruled out to only contain composites (e.g. {1}, 3{1}, 5{1}, 6{1}, 3{8}, {8}5 and even the non-simple family {1}6{1}) |
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