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Old 2021-01-04, 15:20   #78
sweety439
 
Nov 2016

1011000001002 Posts
Default

Newest status for bases 13~16:

Code:
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Old 2021-01-04, 16:21   #79
sweety439
 
Nov 2016

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Quote:
Originally Posted by sweety439 View Post
Largest minimal primes in simple families for bases 13 to 16 (written in decimal):
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 4D19698, 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}

Last fiddled with by sweety439 on 2021-01-05 at 05:15 Reason: Gratuitous quoting
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Old 2021-01-05, 05:17   #80
sweety439
 
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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
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Old 2021-01-05, 06:21   #81
sweety439
 
Nov 2016

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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 https://github.com/curtisbright/mepn...ee/master/data (base 2~30, "minimal n" for primes, "unsolved n" for unsolved families) and https://github.com/RaymondDevillers/primes (base 28~50, "kernel n" for primes, "left n" for unsolved families)

Base 5:

109313

Base 7:

3161

Base 8:

7121
51325
42207

Base 9:

5411
20117
56136
763292

Base 10:

5111
502827

Base 11:

117
14121111
414111
A1415
A9156 (already minimal when primes <=b are not excluded)
8173
1418 (already minimal when primes <=b are not excluded)
7181
4015A041 (already minimal when primes <=b are not excluded)
A921 (already minimal when primes <=b are not excluded)
A4251 (already minimal when primes <=b are not excluded)
150257
402641 (already minimal when primes <=b are not excluded)
440271 (already minimal when primes <=b are not excluded)
9321 (already minimal when primes <=b are not excluded)
A4741
4441 (already minimal when primes <=b are not excluded)
5{7} (unsolved family)

Base 12:

B0279B

Base 13:

31785
402023
57230
93081 (already minimal when primes <=b are not excluded)
8B343
B7486
B563C
1B576
9968B
913625
715041
93015511 (found by CRUS)
39062661 (found by CRUS)
8032017111 (already minimal when primes <=b are not excluded)
9{5} (unsolved family)

Base 14:

1015D
8299 (already minimal when primes <=b are not excluded)
8536
93689
A593
46309 (already minimal when primes <=b are not excluded)
408349 (already minimal when primes <=b are not excluded)
886B
4D19698 (found by CRUS)

Base 15:

714B
DE14
D16B
7B20
B221
EB31
1487 (*** not minimal)
9610408 (already minimal when primes <=b are not excluded)

Base 16:

D149
3241
7D25
2327
E34B
4D39
8537
EB145
8888F201 (already minimal when primes <=b are not excluded)
88F545 (already minimal when primes <=b are not excluded)
D91052
F81517F (already minimal when primes <=b are not excluded)
90354291 (already minimal when primes <=b are not excluded)
D{B} (unsolved family)

Last fiddled with by sweety439 on 2021-01-05 at 06:23
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Old 2021-01-05, 16:09   #82
sweety439
 
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Newest status:

Code:
13: {14, 16, 1A, 23, 25, 2B, 32, 34, 38, 41, 47, 49, 52, 56, 58, 61, 65, 6B, 76, 7A, 7C, 83, 85, 89, 9A, A1, A7, A9, B6, B8, C1, C7, CB, 10C, 119, 11B, 122, 133, 155, 157, 173, 179, 17B, 188, 197, 1B1, 1B5, 1CC, 209, 212, 218, 229, 272, 274, 281, 287, 292, 296, 298, 29C, 2C9, 311, 313, 331, 33B, 355, 371, 373, 379, 397, 3A6, 3AA, 3B3, 3B9, 3BB, 3CA, 43C, 445, 44B, 45A, 463, 4A3, 4A5, 4B2, 4B4, 4BA, 50C, 511, 515, 533, 54A, 551, 559, 571, 575, 57B, 595, 599, 5B3, 5B9, 5CC, 607, 629, 63A, 643, 674, 704, 715, 724, 728, 731, 737, 739, 742, 751, 75B, 773, 775, 779, 782, 784, 791, 793, 797, 7B1, 812, 818, 874, 878, 8AB, 8B4, 902, 919, 922, 926, 92C, 937, 93B, 946, 95B, 962, 968, 971, 977, 979, 982, 98C, 9B3, 9B5, A03, A3C, A45, A4B, A54, AA3, AAB, B02, B0C, B11, B15, B17, B24, B33, B39, B42, B57, B59, B71, B93, B9B, BA4, BAA, BB1, BB9, BC2, BCC, C29, C43, C98, CA3, 1013, 1031, 1037, 105B, 1075, 10B7, 10BB, 1105, 1112, 1121, 1127, 113C, 1172, 1187, 1208, 1211, 1277, 12C8, 1307, 1309, 131C, 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Old 2021-01-05, 16:12   #83
sweety439
 
Nov 2016

22·3·5·47 Posts
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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, ...}
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Old 2021-01-05, 16:26   #84
sweety439
 
Nov 2016

1011000001002 Posts
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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):

** 11 is prime, and thus the only minimal prime in this family.

* Case (1,3):

** 13 is prime, and thus the only minimal prime in this family.

* Case (1,7):

** 17 is prime, and thus the only minimal prime in this family.

* Case (1,9):

** 19 is prime, and thus the only minimal prime in this family.

* Case (2,1):

** Since 23, 29, 11, 31, 41, 61, 71, 251, 281 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 2221 and 20201 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 20021

**** The smallest prime of the form 22{0}1 is 22000001

* Case (2,3):

** 23 is prime, and thus the only minimal prime in this family.

* Case (2,7):

** Since 23, 29, 17, 37, 47, 67, 97 227, 257, 277 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 2087 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):

** 29 is prime, and thus the only minimal prime in this family.

Last fiddled with by sweety439 on 2021-01-06 at 11:05
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Old 2021-01-05, 16:40   #85
sweety439
 
Nov 2016

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* Case (3,1):

** 31 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):

** 37 is prime, and thus the only minimal prime in this family.

* Case (3,9):

** Since 31, 37, 19, 29, 59, 79, 89, 349 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):

** 41 is prime, and thus the only minimal prime in this family.

* Case (4,3):

** 43 is prime, and thus the only minimal prime in this family.

* Case (4,7):

** 47 is prime, and thus the only minimal prime in this family.

* Case (4,9):

** Since 41, 43, 47, 19, 29, 59, 79, 89, 409, 449, 499 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.
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Old 2021-01-05, 17:11   #86
sweety439
 
Nov 2016

22·3·5·47 Posts
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* Case (5,1):

** Since 53, 59, 11, 31, 41, 61, 71, 521 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 5051 and 5501 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 555555555551

**** For the 5{0,5}8{0,5}1 family, since 5081, 5581, 5801, 5851 are primes, we only need to consider the number 581

***** 581 is not prime.

* Case (5,3):

** 53 is prime, and thus the only minimal prime in this family.

* Case (5,7):

** Since 53, 59, 17, 37, 47, 67, 97, 557, 577, 587 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 50207 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 5000000000000000000000000000027

**** The smallest prime of the form 52{0}7 is 5200007

* Case (5,9):

** 59 is prime, and thus the only minimal prime in this family.

Last fiddled with by sweety439 on 2021-01-06 at 11:00
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Old 2021-01-06, 08:17   #87
sweety439
 
Nov 2016

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* Case (6,1):

** 61 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):

** 67 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, 6469, 499 are primes, we only need to consider the family 6{0,3,6,9}49

***** Since 349, 6949 are primes, we only need to consider the family 6{0,6}49

****** Since 60649 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 666649

******** 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 60000049

********* The smallest prime of the form 66{0}49 is 66000049

********* The smallest prime of the form 666{0}49 is 66600049

* Case (7,1):

** 71 is prime, and thus the only minimal prime in this family.

* Case (7,3):

** 73 is prime, and thus the only minimal prime in this family.

* Case (7,7):

** Since 71, 73, 79, 17, 37, 47, 67, 97, 727, 757, 787 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):

** 79 is prime, and thus the only minimal prime in this family.
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Old 2021-01-06, 08:49   #88
sweety439
 
Nov 2016

54048 Posts
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* Case (8,1):

** Since 83, 89, 11, 31, 41, 61, 71, 821, 881 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 8501 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 80051 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 80555551

* Case (8,3):

** 83 is prime, and thus the only minimal prime in this family.

* Case (8,7):

** Since 83, 89, 17, 37, 47, 67, 97, 827, 857, 877, 887 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):

** 89 is prime, and thus the only minimal prime in this family.

* Case (9,1):

** Since 97, 11, 31, 41, 61, 71, 991 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, 9001, 9221, 9551, 9851 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):

** 97 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 9049, 349, 9649, 9949 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 946669

Last fiddled with by sweety439 on 2021-01-06 at 10:50
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