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

 sweety439 2019-11-25 18:28

Minimal set of the strings for primes with at least two digits

[COLOR="Red"]Edit:

The GitHub page for this project: [URL="https://github.com/xayahrainie4793/non-single-digit-primes"]https://github.com/xayahrainie4793/non-single-digit-primes[/URL]

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----------------------------------------------------------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.

 LaurV 2019-11-26 05:46

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...

 uau 2019-11-26 06:05

[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]

[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.

 LaurV 2019-11-26 06:10

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)

 LaurV 2019-11-26 06:46

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...

 LaurV 2019-11-26 11:21

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.

 LaurV 2019-11-28 11:29

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.

 sweety439 2019-11-28 22:16

[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.

 sweety439 2019-11-28 22:40

[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)

 LaurV 2019-11-29 02:17

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.

 sweety439 2019-11-29 06:50

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 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;
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]);
char* temp;
char* last;
int inttemp;
int intlast;
for(int j=0; j<n; j++)
{ temp = strlist[j];
strlist[j] = last;
last = temp;
inttemp = listpos[j];
listpos[j] = intlast;
intlast = inttemp;
}
else if(strcmp(str,strlist[j])>0)
last = strlist[j];
strlist[j] = str;
intlast = listpos[j];
listpos[j] = n;
}
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)
last = str;
intlast = n;
}
}
{ 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;
}

{ 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));
}
}

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_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)
}
else if(side==0)
}
}
else
}
}

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))
#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)
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(&copyf);
copyfamily(&copyf, *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));
if(k==i)
{ newrepeats = malloc(copyf.numrepeats[k]*sizeof(char));
memcpy(newrepeats, copyf.repeats[k], copyf.numrepeats[k]*sizeof(char));
}
}

#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(&copyf);
clearfamily(&newf);

return 1;
}

tripleinstancestring(str, *f, i, j, i, j, i, j);

if(!nosubword(str))
{
family copyf;
familyinit(&copyf);
copyfamily(&copyf, *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));
if(k==i)
{ newrepeats = malloc(copyf.numrepeats[k]*sizeof(char));
memcpy(newrepeats, copyf.repeats[k], copyf.numrepeats[k]*sizeof(char));
}
}

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));
if(k==i)
{ newrepeats = malloc(copyf.numrepeats[k]*sizeof(char));
memcpy(newrepeats, copyf.repeats[k], copyf.numrepeats[k]*sizeof(char));
newrepeats = malloc(copyf.numrepeats[k]*sizeof(char));
memcpy(newrepeats, copyf.repeats[k], copyf.numrepeats[k]*sizeof(char));
}
}

#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(&copyf);
clearfamily(&newf);
clearfamily(&newf2);

return 1;
}

quadinstancestring(str, *f, i, j, i, j, i, j, i, j);

if(!nosubword(str))
{
family copyf;
familyinit(&copyf);
copyfamily(&copyf, *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));
if(k==i)
{ newrepeats = malloc(copyf.numrepeats[k]*sizeof(char));
memcpy(newrepeats, copyf.repeats[k], copyf.numrepeats[k]*sizeof(char));
}
}

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));
if(k==i)
{ newrepeats = malloc(copyf.numrepeats[k]*sizeof(char));
memcpy(newrepeats, copyf.repeats[k], copyf.numrepeats[k]*sizeof(char));
newrepeats = malloc(copyf.numrepeats[k]*sizeof(char));
memcpy(newrepeats, copyf.repeats[k], copyf.numrepeats[k]*sizeof(char));
}
}

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));
if(k==i)
{ newrepeats = malloc(copyf.numrepeats[k]*sizeof(char));
memcpy(newrepeats, copyf.repeats[k], copyf.numrepeats[k]*sizeof(char));
newrepeats = malloc(copyf.numrepeats[k]*sizeof(char));
memcpy(newrepeats, copyf.repeats[k], copyf.numrepeats[k]*sizeof(char));
newrepeats = malloc(copyf.numrepeats[k]*sizeof(char));
memcpy(newrepeats, copyf.repeats[k], copyf.numrepeats[k]*sizeof(char));
}
}

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(&copyf);
clearfamily(&newf);
clearfamily(&newf2);
clearfamily(&newf3);

return 1;
}

quintinstancestring(str, *f, i, j);

if(!nosubword(str))
{
family copyf;
familyinit(&copyf);
copyfamily(&copyf, *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;

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));
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));
}
}
}

clearfamily(&newf);

}

clearfamily(&copyf);

#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(&copyf);
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));
if(i==ii)
{ repeatscopy = malloc(f->numrepeats[ii]*sizeof(char));
memcpy(repeatscopy, f->repeats[ii], f->numrepeats[ii]*sizeof(char));
}
}
#ifdef PRINTSPLITEXT
familystring(str, *f);
printf("%s splits into ", str);
familystring(str, copyf);
printf("%s\n", str);
#endif
clearfamily(&copyf);
return 1;
}
}

}
}
return 0;
}

int split2(family* f, list* unsolved, char insplit)
{ if(insplit==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(&copyf);
copyfamily(&copyf, *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;

#ifdef PRINTSPLIT
char str[MAXSTRING];
familystring(str, *f);
printf("%s splits into ", str);
familystring(str, copyf);
printf("%s and ", str);
#endif

clearfamily(&copyf);

familyinit(&copyf);
copyfamily(&copyf, *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;

#ifdef PRINTSPLIT
familystring(str, copyf);
printf("%s [because of %s, %s]\n", str, str1, str2);
#endif

clearfamily(&copyf);

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));
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));

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;
}
}

#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));
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));

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;
}
}

#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));
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));

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));

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;
}
}

#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));
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));

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));

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;
}
}

#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));
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));

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));

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));

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;
}
}

#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));
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));

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));

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));

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;
}
}

#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(&copyf);
copyfamily(&copyf, *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;

#ifdef PRINTSPLIT
char str[MAXSTRING];
familystring(str, *f);
printf("%s splits into ", str);
familystring(str, copyf);
printf("%s and ", str);
#endif

clearfamily(&copyf);

familyinit(&copyf);
copyfamily(&copyf, *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;

#ifdef PRINTSPLIT
familystring(str, copyf);
printf("%s [because of %s]\n", str, str1);
#endif

clearfamily(&copyf);

return 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))

#ifdef PRINTEXPLORE
familystring(str, newf);
printf("%s, ", str);
#endif

clearfamily(&newf);
}

family copyf;
familyinit(&copyf);
copyfamily(&copyf, f);
copyf.repeats[i] = NULL;
copyf.numrepeats[i] = 0;
if(examine(&copyf))

#ifdef PRINTEXPLORE
familystring(str, copyf);
printf("%s\n", str);
#endif

clearfamily(&copyf);

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);
}
}

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);
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);
#ifdef PRINTRESUME
#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]!='{')
}
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]);
}
i = k+i+2-(line[i]=='{'?1:0)+1;
}
}
#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])))

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])))

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)

 sweety439 2019-11-29 09:08

[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.

 yae9911 2019-11-29 14:39

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.

 sweety439 2019-11-30 00:26

[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}

 sweety439 2019-11-30 00:59

[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".

 sweety439 2019-11-30 06:43

[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

 sweety439 2019-11-30 13:42

[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?

 sweety439 2019-12-03 03:58

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,

 sweety439 2020-11-24 03:50

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.

 sweety439 2020-11-25 04:52

* 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.

 sweety439 2020-11-25 04:57

* 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.

 sweety439 2020-12-12 10:39

* 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)

 sweety439 2020-12-12 10:56

* 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.

 sweety439 2020-12-13 05:06

* 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.

 sweety439 2020-12-13 05:38

1 Attachment(s)
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])

 sweety439 2020-12-13 07:53

Found a minimal prime (start with 2 digits) in base 13: 7[SUB]1504[/SUB]1, which equals (7*13^1505-79)/12

 sweety439 2020-12-13 09:46

1 Attachment(s)
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, {5}: 53
12, {4}, 7: 631
12, 4, {7}: 8011
12, {4}, 11: 59
12, 4, {11}: 59
12, {5}, 1: 61
12, 5, {1}: 61
12, {5}, 7: 67
12, 5, {7}: 67
12, {5}, 11: 71
12, 5, {11}: 71
12, {6}, 1: 73
12, 6, {1}: 73
12, {6}, 5: 941
12, 6, {5}: 929
12, {6}, 7: 79
12, 6, {7}: 79
12, {6}, 11: 83
12, 6, {11}: 83
12, {7}, 1: 1093
12, 7, {1}: 1021
12, {7}, 5: 89
12, 7, {5}: 89
12, {7}, 11: 1103
12, 7, {11}: 1151
12, {8}, 1: 97
12, 8, {1}: 97
12, {8}, 5: 101
12, 8, {5}: 101
12, {8}, 7: 103
12, 8, {7}: 103
12, {8}, 11: 107
12, 8, {11}: 107
12, {9}, 1: 109
12, 9, {1}: 109
12, {9}, 5: 113
12, 9, {5}: 113
12, {9}, 7: 16963
12, 9, {7}: 16651
12, {9}, 11: 203591
12, 9, {11}: 1439
12, {10}, 1: 226201
12, 10, {1}: 1453
12, {10}, 7: 127
12, 10, {7}: 127
12, {10}, 11: 131
12, 10, {11}: 131
12, {11}, 1: 248821
12, 11, {1}: 1597
12, {11}, 5: 137
12, 11, {5}: 137
12, {11}, 7: 139
12, 11, {7}: 139
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: 0
13, 1, {5}: 239
13, {1}, 6: 19
13, 1, {6}: 19
13, {1}, 7: 0
13, 1, {7}: 253217502498750291800692183145337720992638880271493569431738157631027569095215561
13, {1}, 8: 0
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, {9}: 0
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: 0
13, 3, {7}: 2923035083
13, {3}, 8: 47
13, 3, {8}: 47
13, {3}, 10: 0
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, 4, {11}: 0
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: 0
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: 0
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, {1}: 0
13, {7}, 2: 2815633
13, 7, {2}: 425554642597531069476088349319265344520247822549860785329030452393239592223053102610453
13, {7}, 3: 1277
13, 7, {3}: 0
13, {7}, 4: 16657
13, 7, {4}: 16111
13, {7}, 5: 1279
13, 7, {5}: 0
13, {7}, 6: 97
13, 7, {6}: 97
13, {7}, 8: 16661
13, 7, {8}: 16843
13, {7}, 9: 1283
13, 7, {9}: 0
13, {7}, 10: 101
13, 7, {10}: 101
13, {7}, 11: 176677145512151
13, 7, {11}: 0
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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]

 sweety439 2020-12-13 10:49

[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)

 sweety439 2020-12-15 14:04

1 Attachment(s)
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)

 sweety439 2020-12-15 16:06

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

 sweety439 2020-12-15 16:12

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]

 sweety439 2020-12-17 08:40

1 Attachment(s)
[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

 sweety439 2020-12-26 03:14

[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.

 sweety439 2020-12-26 03:20

[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.

 sweety439 2020-12-26 03:23

[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)

* 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.

 sweety439 2020-12-26 03:51

[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)

 sweety439 2020-12-26 03:54

* 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.

 sweety439 2020-12-29 00:50

[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)

 sweety439 2020-12-29 01:12

* 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)

 sweety439 2020-12-29 02:44

[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]

 sweety439 2020-12-29 02:58

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

 sweety439 2020-12-29 10:37

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

 sweety439 2020-12-30 02:36

Minimal set of prime-strings with â‰¥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]

 sweety439 2020-12-30 02:46

[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).

 sweety439 2020-12-30 04:52

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]

 sweety439 2020-12-30 05:04

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)

 sweety439 2020-12-30 05:35

* 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

 sweety439 2020-12-30 05:48

* 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

 sweety439 2020-12-30 10:21

[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.

 sweety439 2020-12-30 10:41

* 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.

 sweety439 2020-12-30 10:48

* 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.

 sweety439 2020-12-30 10:51

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]

 sweety439 2020-12-30 10:52

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.

 sweety439 2020-12-30 10:55

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.

 sweety439 2020-12-30 10:59

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.

 sweety439 2021-01-01 03:57

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]

 sweety439 2021-01-01 03:58

Base 11 5{7} family searched to around 15000 digits, without finding any (probable) primes

 sweety439 2021-01-01 15:39

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]

 sweety439 2021-01-01 15:45

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]

 sweety439 2021-01-01 16:07

[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

 sweety439 2021-01-01 16:10

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)

 sweety439 2021-01-02 03:15

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.

 sweety439 2021-01-02 04:34

* 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)

 sweety439 2021-01-02 04:51

* 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://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://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]

* [URL="https://primes.utm.edu/primes/lists/all.txt"]Top proven primes[/URL]

 sweety439 2021-01-03 05:22

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.

 sweety439 2021-01-03 10:35

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.

 sweety439 2021-01-03 10:41

[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

 sweety439 2021-01-03 11:07

[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).

 sweety439 2021-01-03 11:10

[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

[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, 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16: {11, 13, 17, 1D, 1F, 25, 29, 2B, 2F, 35, 3B, 3D, 43, 47, 49, 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, 2E01, 2E0D, 2ECD, 3023, 3079, 3109, 3263, 3341, 36AF, 3941, 3991, 39AF, 3E41, 3E81, 3EE1, 3EE7, 3F79, 4021, 40DB, 440B, 444B, 44A1, 44AB, 44DB, 4541, 45BB, 4A41, 4B0B, 4BBB, 4C4B, 4D41, 4DED, 5045, 50A1, 50ED, 540D, 5441, 555B, 556F, 5585, 560F, 56FF, 5705, 574D, 580D, 582D, 5855, 588D, 5A01, 5AA1, 5B01, 5B4B, 5B87, 5BB1, 5BEB, 5C4D, 5CDD, 5CED, 5DD7, 5DDD, 5E0D, 5E2D, 5EBB, 68FF, 6A69, 6AC9, 6C8F, 6CA9, 6CAF, 6F8F, 6FAF, 7033, 7063, 7075, 7087, 70A5, 70AB, 7303, 7393, 74DD, 754D, 7603, 7633, 7663, 7669, 7705, 772D, 775D, 77D5, 7807, 7877, 7885, 7939, 7969, 7993, 79AB, 7A05, 7A69, 7A9B, 7AA5, 7B77, 7BA9, 7D4D, 7D75, 7D77, 8077, 808D, 80D7, 80E7, 8587, 86CF, 8777, 8785, 8885, 88CF, 88ED, 88FD, 8C6F, 8C8F, 8E8D, 8EE7, 8F2D, 8F8D, 9031, 9041, 90AF, 90B9, 9221, 9319, 9401, 944B, 9881, 9931, 9941, 9991, 99AF, 9A0F, 9A1B, 9A4B, 9AFF, 9BA1, 9BB1, 9CAF, 9E81, 9EA1, 9FAF, A001, A05B, A0C9, A105, A10B, A4CB, A55B, A6C9, A88F, A91B, A9B1, A9BB, AA15, AB01, AB0B, AB19, ABBB, AC09, AF09, B041, 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, ...}
[/CODE]

 sweety439 2021-01-03 11:13

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
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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]

 sweety439 2021-01-03 15:49

[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, {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
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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

 sweety439 2021-01-03 15:51

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))

 sweety439 2021-01-04 04:16

[QUOTE=sweety439;568170]start searching 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, 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, 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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, 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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, ...}
[/CODE]

 LaurV 2021-01-04 08:19

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.

 sweety439 2021-01-04 13:51

[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?

 LaurV 2021-01-04 13:57

[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.

 sweety439 2021-01-04 14:02

[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)

 sweety439 2021-01-04 15:20

[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, 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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, ...}
[/CODE]

 sweety439 2021-01-04 16:21

[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}

 sweety439 2021-01-05 05:17

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

 sweety439 2021-01-05 06:21

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)

 sweety439 2021-01-05 16:09

[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, 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A91B, A9B1, A9BB, AA15, AB01, AB0B, AB19, ABBB, AC09, AF09, B041, 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, 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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, 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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]

 sweety439 2021-01-05 16:12

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]

 sweety439 2021-01-05 16:26

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.

 sweety439 2021-01-05 16:40

* 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.

 sweety439 2021-01-05 17:11

* 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.

 sweety439 2021-01-06 08:17

* 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.

 sweety439 2021-01-06 08:49

* 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]

 sweety439 2021-01-06 08:54

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]

 sweety439 2021-01-06 08:56

There are totally 77 minimal primes (start with 2 digits) in base 10, there are 75 such primes in base 8

 sweety439 2021-01-06 10:36

[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.

 sweety439 2021-01-06 12:48

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, 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, 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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]

 sweety439 2021-01-06 12:55

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]

 sweety439 2021-01-06 12:57

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)

 sweety439 2021-01-06 17:11

[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]

 LaurV 2021-01-07 04:34

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:)

 sweety439 2021-01-07 07:53

[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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