Quasi-minimal prime

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A quasi-minimal prime is a prime number for which there is no shorter subsequence of its digits in a given base b that form a prime > b. For example, 857 is a quasi-minimal prime in decimal because there is no prime > 10 among the shorter subsequences of the digits: 8, 5, 7, 85, 87, 57. The subsequence does not have to consist of consecutive digits, so 149 is not a quasi-minimal prime in decimal (because 19 is prime and 19 > 10). But it does have to be in the same order; so, for example, 991 is still a quasi-minimal prime in decimal even though a subset of the digits can form the shorter prime 19 > 10 by changing the order.

(using A−Z to represent digit values 10 to 35)

For the quasi-minimal primes in bases up to 16, I have only solved (found all quasi-minimal primes and proved that these are all such primes) bases 2, 3, 4, 5, 6, 7, 8, 9, 10, 12. For the remain bases 11, 13, 14, 15, 16, I only found all quasi-minimal primes up to certain limit (about 232) and some larger quasi-minimal primes.

I left as a challenge to readers the task of solving (finding all quasi-minimal primes and proving that these are all such primes) bases 11, 13, 14, 15, 16, and bases 17 through 36 (this will be a hard problem, e.g. base 23 has a quasi-minimal prime 9E800873, and base 30 has a quasi-minimal prime OT34205).

Proving the set of the quasi-minimal primes in base b is S, is equivalent to:

  • Prove that all elements in S, when read as base b representation, are primes > b.
  • Prove that all proper subsequence of all elements in S, when read as base b representation, which are > b, are composite.
  • Prove that all primes > b, when written in base b, contain at least one element in S as subsequence (equivalently, prove that all strings not containing any element in S as subsequence, when read as base b representation, which are > b, are composite).

e.g. proving the set of the quasi-minimal primes in base b = 10 is {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}, is equivalent to:

  • Prove that all of 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 are primes > 10.
  • Prove that all proper subsequence of all elements in {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} which are > 10 are composite.
  • Prove that all primes > 10 contain at least one element in {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} as subsequence (equivalently, prove that all numbers > 10 not containing any element in {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} as subsequence are composite).

Condensed table[edit | edit source]

b number of quasi-minimal primes base b base-b form of largest known quasi-minimal prime base b length of largest known quasi-minimal prime base b algebraic ((a×bn+c)/d) form of largest known quasi-minimal prime base b
2 1 11 2 3
3 3 111 3 13
4 5 221 3 41
5 22 109313 96 595+8
6 11 40041 5 5209
7 71 3161 17 (717−5)/2
8 75 42207 221 (4×8221+17)/7
9 151 30115811 1161 3×91160+10
10 77 502827 31 5×1030+27
11 ≥914 5571011 1013 (607×111011−7)/10
12 106 403977 42 4×1241+91
13 ≥2497 8032017111 32021 8×1332020+183
14 ≥606 4D19698 19699 5×1419698−1
15 ≥1212 715597 157 (15157+59)/2
16 ≥2045 DB32234 32235 (206×1632234−11)/15

Data for quasi-minimal primes[edit | edit source]

Base 2[edit | edit source]

11

Base 3[edit | edit source]

12, 21, 111

Base 4[edit | edit source]

11, 13, 23, 31, 221

Base 5[edit | edit source]

12, 21, 23, 32, 34, 43, 104, 111, 131, 133, 313, 401, 414, 3101, 10103, 14444, 30301, 33001, 33331, 44441, 300031, 100000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000013

Base 6[edit | edit source]

11, 15, 21, 25, 31, 35, 45, 51, 4401, 4441, 40041

Base 7[edit | edit source]

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

Base 8[edit | edit source]

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

Base 9[edit | edit source]

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, 7270000000007, 88888888833335, 100000000000507, 5111111111111161, 7277777777777777707, 8888888888888888888335, 30000000000000000000051, 1000000000000000000000000057, 56111111111111111111111111111111111111, 7666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666662, 27777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777707, 300000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000011

Base 10[edit | edit source]

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

Base 11[edit | edit source]

(test limit: 1500000001)

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, ..., 83000000006, ..., A0000000001, ..., A0014444444, ..., 100000000057, ..., 370000000007, ..., 40000000A0041, ..., A000000014444, ..., A044444444441, ..., A144444444411, ..., 40000000000401, ..., A0000044444441, ..., A00000000444441, ..., 1000000000000073, ..., 6000000000000083, ..., 11111111111111111, ..., 14444444444441111, ..., 44444444444444111, ..., 70000000000000004, ..., A1444444444444444, ..., A9999999999999996, ..., 320000000000000002, ..., 888888888888888883, ..., 1444444444444444444, ..., 7777777777777777771, ..., 4000000000000000A041, ..., 45AAAAAAAAAAAAAAAAAAAA, ..., 7900000000000000000005, ..., 9777777777777777777707, ..., A999999999999999999999, ..., 10000000000000000000747, ..., 3577777777777777777777777, ..., 10000000000000000000000044, ..., 77700000000000000000000008, ..., A44444444444444444444444441, ..., 1500000000000000000000000007, ..., 40000000000000000000000000041, ..., 440000000000000000000000000001, ..., 999999999999999999999999999999991, ..., 1900000000000000000000000000000000001, ..., 7777777777777777777777777777777777704, ..., 5900000000000000000000000000000000000000003, ..., A477777777777777777777777777777777777777777, ..., 444444444444444444444444444444444444444444441, ..., 9700000000000000000000000000000000000000000000000007, ..., 8055555555555555555555555555555555555555555555555555555555555, ..., 44777777777777777777777777777777777777777777777777777777777777777, ..., 99777777777777777777777777777777777777777777777777777777777777777, ..., 100000000000000000000000000000000000000000000000000000000000000000000000000035, ..., 10000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000037, ..., 10000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000051, ..., 500000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000057, ..., 555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555552A, ..., 5077777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777, ..., 77777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777744, ..., 55777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777, ...

Base 12[edit | edit source]

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

Base 13[edit | edit source]

(test limit: 1010008001)

14, 16, 1A, 23, 25, 2B, 32, 34, 38, 41, 47, 49, 52, 56, 58, 61, 65, 6B, 76, 7A, 7C, 83, 85, 89, 9A, A1, A7, A9, B6, B8, C1, C7, CB, 10C, 119, 11B, 122, 133, 155, 157, 173, 179, 17B, 188, 197, 1B1, 1B5, 1CC, 209, 212, 218, 229, 272, 274, 281, 287, 292, 296, 298, 29C, 2C9, 311, 313, 331, 33B, 355, 371, 373, 379, 397, 3A6, 3AA, 3B3, 3B9, 3BB, 3CA, 43C, 445, 44B, 45A, 463, 4A3, 4A5, 4B2, 4B4, 4BA, 50C, 511, 515, 533, 54A, 551, 559, 571, 575, 57B, 595, 599, 5B3, 5B9, 5CC, 607, 629, 63A, 643, 674, 704, 715, 724, 728, 731, 737, 739, 742, 751, 75B, 773, 775, 779, 782, 784, 791, 793, 797, 7B1, 812, 818, 874, 878, 8AB, 8B4, 902, 919, 922, 926, 92C, 937, 93B, 946, 95B, 962, 968, 971, 977, 979, 982, 98C, 9B3, 9B5, A03, A3C, A45, A4B, A54, AA3, AAB, B02, B0C, B11, B15, B17, B24, B33, B39, B42, B57, B59, B71, B93, B9B, BA4, BAA, BB1, BB9, BC2, BCC, C29, C43, C98, CA3, 1013, 1031, 1037, 105B, 1075, 10B7, 10BB, 1105, 1112, 1121, 1127, 113C, 1172, 1187, 1208, 1211, 1277, 12C8, 1307, 1309, 131C, 139C, 151C, 1721, 1727, 1787, 1901, 1909, 1912, 1918, 193C, 1981, 198B, 199C, 19B2, 19C3, 1B29, 1BB2, 1BBC, 1C28, 1C39, 2021, 2078, 2117, 2201, 2221, 2267, 2278, 2627, 2678, 2711, 2771, 2788, 3037, 3053, 306A, 3077, 3091, 309B, 30AC, 3305, 353C, 35AB, 35BA, 35BC, 3677, 3905, 390B, 39C5, 3A0C, 3AB5, 3B5C, 3C35, 3C59, 3C95, 403A, 40AB, 4333, 435B, 4403, 44C3, 4535, 4544, 454C, 45B5, 45BB, 480B, 4B35, 4B5B, 4C36, 5057, 5077, 509B, 50A4, 5107, 5305, 530B, 539C, 53AB, 53C9, 5444, 5455, 54C4, 5503, 5545, 55AB, 5774, 5794, 590B, 594B, 5974, 59B4, 5A4C, 5A53, 5AA4, 5AB5, 5ABB, 5ACA, 5B4B, 5B5A, 5BA5, 5CA4, 6227, 6278, 6667, 6698, 6733, 6872, 6928, 6944, 694C, 6973, 6986, 6997, 69C8, 6AC3, 6C92, 6C94, 7019, 7057, 70B5, 7103, 710B, 7118, 7127, 7129, 7172, 7178, 7192, 7211, 7217, 7219, 7271, 7303, 7408, 7433, 7444, 7505, 7507, 7574, 770B, 7774, 7778, 7787, 7871, 7877, 7888, 794B, 7994, 79B4, 7B43, 7B74, 7B94, 7BB2, 8027, 8072, 8081, 80BA, 8171, 8207, 821C, 848B, 8687, 8711, 8722, 87BB, 8867, 88B2, 88BA, 8B22, 8B2A, 8BAC, 9004, 9017, 9031, 9053, 9055, 9073, 9091, 90BB, 90C8, 9107, 9118, 913C, 9181, 91C3, 9284, 935C, 93C5, 9424, 9428, 9448, 9509, 959C, 96C4, 9703, 9743, 9745, 974B, 97B2, 9811, 981B, 987B, 98B1, 991C, 9967, 9998, 9B12, 9B74, 9B92, 9BBC, 9C55, 9C86, 9CC4, A0BA, A306, A436, A535, A5B5, A636, A6C3, A80B, AB04, AB22, AB35, AB3B, AB4C, AB55, ABAC, ABB5, AC36, ACA5, B044, B04A, B0B7, B129, B1B2, B219, B222, B291, B299, B2CA, B35A, B3A5, B404, B44C, B45B, B4B3, B501, B51C, B55A, B5A5, B5AB, B5C3, B707, B792, B794, B905, B912, B9C5, BA5B, BAB3, BB03, BB45, BB72, BBA5, BBB2, BC44, BC53, BC95, BC99, C30A, C36A, C395, C454, C535, C553, C593, C944, C953, C964, CC94, 10015, 10051, 10099, 10118, 10291, 10712, 10772, 10811, 10877, 10921, 10B92, 11111, 11135, 11171, 111C8, 11531, 11C03, 13001, 13177, 13777, 13915, 13951, 13991, 159BB, 17018, 17102, 17111, 17117, 17171, 17177, 17708, 17711, 17801, 18071, 18101, 18271, 18B27, 19003, 19153, 19315, 19351, 19591, 19913, 19951, 1C099, 20171, 20177, 20207, 20227, 20777, 21011, 21077, 2111C, 21707, 22207, 30017, 300B5, 301C9, 3033A, 303A3, 303C5, 3050B, 305C9, 3095C, 30B05, 31007, 3159B, 31999, 31C09, 3330A, 33353, 33593, 33757, 33C5C, 33CC5, 35003, 3591B, 39353, 39539, 39935, 39995, 3ACCC, 3C5C3, 3CC53, 40043, 40306, 405C4, 408BC, 40BBB, 40C54, 43066, 4366A, 4443A, 45055, 45505, 45554, 4555C, 455BC, 455C3, 45C04, 488BC, 4B03B, 4B0B5, 4B55C, 4BB0B, 4C003, 4C054, 4C5C4, 50053, 500B1, 5035A, 504B5, 5053A, 50554, 505B4, 50A35, 50B07, 50BBA, 5139B, 519BB, 51BB7, 535AC, 53A5C, 53AC5, 53BAC, 54004, 54035, 5403B, 545C3, 54B05, 54B5C, 54BBC, 54C53, 55357, 5535B, 553AC, 554BC, 55537, 55544, 5554C, 55577, 555A4, 555BB, 55A5C, 55B04, 55B55, 55B77, 55BB5, 55BC4, 55C54, 55C5A, 57403, 591BB, 59443, 59BB7, 5A044, 5AC04, 5AC35, 5B001, 5B007, 5B0AB, 5B0B4, 5B4C5, 5B544, 5B555, 5B5BB, 5B744, 5B777, 5BA0B, 5BB44, 5BB55, 5BBC4, 5BC54, 5C039, 5C35A, 5C53A, 60098, 60964, 60988, 60A63, 66094, 66377, 66692, 66694, 669C2, 669C4, 66A36, 67022, 67099, 67222, 67277, 67772, 68627, 69088, 690C4, 69808, 69994, 6A663, 7007B, 70181, 70222, 70277, 70772, 70808, 70B0B, 70B29, 71113, 71711, 71908, 71999, 7199B, 71BB7, 71BBB, 74035, 74305, 7430B, 74503, 75443, 75454, 75535, 77072, 77108, 77177, 77717, 77BBB, 78011, 79BBB, 7B007, 7B7B7, 7B7BB, 7BBB3, 7BBB7, 80117, 80221, 80771, 80777, 80807, 8084B, 80B7B, 80BBB, 81107, 8400B, 86267, 87107, 87277, 87727, 87B27, 88111, 88201, 88702, 88771, 8888B, 88B77, 88BBB, 8B1BC, 8B727, 90035, 90059, 90088, 90095, 9009B, 90101, 90103, 90305, 90488, 904CC, 90574, 90644, 9064C, 90806, 908B7, 9090B, 90994, 90B09, 90C35, 90C59, 90C64, 91111, 91135, 91315, 9180B, 92008, 92408, 92488, 93359, 93395, 944C2, 944CC, 94505, 9455C, 94804, 94888, 94C0C, 94C33, 94C4C, 95045, 95504, 95573, 955C4, 95C54, 96044, 97BBB, 98066, 98408, 98444, 98804, 98848, 99001, 99005, 9900B, 99074, 990BC, 99113, 99175, 99278, 99335, 99454, 994C3, 99517, 99593, 9984B, 99881, 99904, 99917, 99935, 99955, 99973, 999BB, 999C2, 999C4, 99B99, 9B00B, 9B04B, 9B0B4, 9B1BB, 9BB04, 9C059, 9C244, 9C404, 9C44C, 9C488, 9C503, 9C5C9, 9C644, 9C664, 9CC88, 9CCC2, A00B4, A05BB, A08B2, A08BC, A0BC4, A3336, A3633, A443A, A4443, A50BB, A55C5, A5AAC, A5BBA, A5C53, A5C55, AACC5, AB05B, AB0BB, AB40A, ABBBC, ABC4A, ACC5A, ACCC3, B0053, B0075, B010B, B0455, B0743, B0774, B0909, B0BB4, B2277, B2A2C, B3005, B351B, B37B5, B3A0B, B3ABC, B3B0A, B400A, B4035, B403B, B4053, B4305, B4BC5, B4C0A, B504B, B50BA, B530A, B5454, B54BC, B54C5, B5544, B55B5, B5B44, B5B4C, B5BB5, B7403, B7535, B77BB, B7955, B7B7B, B9207, B9504, B9999, BA055, BA305, BABC5, BAC35, BB054, BB05A, BB207, BB3B5, BB4C3, BB504, BB544, BB54C, BB5B5, BB753, BB7B7, BBABC, BBB04, BBB4C, BBB55, BBBAC, BC035, BC455, C0353, C0359, C03AC, C0904, C0959, C0A5A, C0CC5, C3059, C335C, C5A0A, C5A44, C5AAC, C6692, C69C2, C904C, C9305, C9905, C995C, C99C5, C9C04, C9C59, C9CC2, CA50A, CA5AC, CAA05, CAA5A, CC335, CC544, CC5AA, CC935, CC955, 100039, 100178, 100718, 100903, 101177, 101708, 101711, 101777, 102017, 102071, 103999, 107081, 107777, 108217, 109111, 109151, 110078, 110108, 110717, 111017, 111103, 1111C3, 111301, 111707, 113501, 115103, 117017, 117107, 117181, 117701, 120701, 13C999, 159103, 170717, 177002, 177707, 180002, 187001, 18C002, 19111C, 199903, 1B0007, 1BB077, 1BBB07, 1C0903, 1C8002, 1C9993, 200027, 207107, 217777, 219991, 220027, 222227, 270008, 271007, 277777, 290444, 300059, 300509, 303359, 303995, 309959, 30B50A, 3336AC, 333707, 33395C, 335707, 3360A3, 350009, 36660A, 3666AC, 370007, 377B07, 39001C, 399503, 3BC005, 400366, 400555, 400B3B, 400B53, 400BB5, 400CC3, 4030B5, 40B053, 40B30B, 40B505, 43600A, 450004, 4A088B, 4B0503, 4B5C05, 4BBBB5, 4BC505, 500039, 50045B, 50405B, 504B0B, 50555B, 5055B5, 505B0A, 509003, 50A50B, 50B045, 50B054, 539B01, 550054, 5500BA, 55040B, 553BC5, 5553C5, 55550B, 5555C3, 555C04, 55B00A, 55BB0B, 570007, 5A500B, 5A555B, 5AC505, 5B055B, 5B0B5B, 5B5B5C, 5B5BC5, 5BB05B, 5BBB0B, 5BBB54, 5BBBB4, 5BBC0A, 5BC405, 5C5A5A, 5CA5A5, 600694, 6060A3, 609992, 637777, 6606A3, 6660A3, 667727, 667808, 668777, 669664, 670088, 679988, 696064, 69C064, 6A6333, 700727, 700811, 700909, 70098B, 700B92, 701117, 701171, 701717, 707027, 707111, 707171, 707201, 707801, 70788B, 7080BB, 708101, 70881B, 70887B, 70B227, 710012, 710177, 711002, 711017, 711071, 717707, 718001, 718111, 720077, 722002, 727777, 74BB3B, 74BB53, 770102, 770171, 770801, 777112, 777202, 777727, 777772, 778801, 77B772, 780008, 78087B, 781001, 788B07, 79088B, 794555, 7B000B, 7B0535, 7B077B, 7B2777, 7B4BBB, 7BB4BB, 800021, 800717, 801077, 80BB07, 811117, 870077, 8777B7, 877B77, 880177, 88071B, 88077B, 8808BC, 887017, 88707B, 888227, 88877B, 8887B7, 888821, 888827, 888BB7, 8B001B, 8B00BB, 8BBB77, 8BBBB7, 900097, 900BC9, 901115, 903935, 904033, 90440C, 908008, 908866, 909359, 909C05, 90B944, 90C95C, 90CC95, 91008B, 91115C, 911503, 920888, 930335, 933503, 935903, 940033, 94040C, 940808, 94CCCC, 950005, 950744, 95555C, 9555C5, 95C003, 95C005, 96400C, 96440C, 96664C, 966664, 966994, 969094, 969964, 97008B, 97080B, 975554, 97800B, 97880B, 980006, 980864, 980B07, 984884, 986006, 986606, 986644, 988006, 988088, 988664, 988817, 988886, 988B0B, 98B007, 990115, 990151, 990694, 990B44, 990C5C, 991501, 993059, 99408B, 994555, 995404, 995435, 996694, 9978BB, 998087, 999097, 999103, 99944C, 999503, 9995C3, 999754, 999901, 99990B, 999B09, 99B4C4, 99C0C5, 99C539, 99CC05, 9B9444, 9B9909, 9C0484, 9C0808, 9C2888, 9C400C, 9C4CCC, 9C6994, 9C90C5, 9C9C5C, 9CC008, 9CC5C3, 9CC905, 9CCC08, A0055B, A005AC, A0088B, A00B2C, A00BBB, A0555C, A05CAA, A0A5AC, A0A5CA, A0AC05, A0AC5A, A0B50B, A0BB0B, A0BBB4, A0C5AC, A3660A, A5050B, A555AC, A5B00B, AA0C05, AAA05C, AAA0C5, AAC05C, AB4444, ABB00B, AC050A, AC333A, B0001B, B00099, B0030B, B004B5, B00A35, B00B54, B030BA, B05043, B0555B, B05B0A, B05B5B, B07B53, B09074, B09755, B09975, B09995, B0AB0B, B0B04B, B0B535, B0BB53, B4C055, B50003, B5003A, B500A3, B50504, B50B04, B53BC5, B54BBB, B550BB, B555BC, B55C55, B5B004, B5B0BB, B5B50B, B5B554, B5B55C, B5B5B4, B5BBB4, B5BBBC, B5BC0A, B5C045, B5C054, B70995, B70B3B, B74555, B74B55, B99921, B99945, BAC505, BB0555, BB077B, BB0B5B, BB0BB5, BB500A, BB53BC, BB53C5, BB5505, BB55BC, BB5BBA, BB5C0A, BB7BB4, BBB00A, BBB74B, BBBB54, BBBBAB, BC5054, BC5504, C00094, C00694, C009C4, C00C05, C03035, C050AA, C05309, C05404, C0544C, C05AC4, C05C39, C06092, C06694, C09035, C094CC, C09992, C09994, C09C4C, C09C95, C0CC3A, C0CC92, C33539, C35009, C4C555, C50309, C50AAA, C53009, C550A5, C555CA, C55A5A, C55CA5, C5AC55, C60094, C60694, C93335, C95405, C99094, CA05CA, CA0AC5, CA555C, CAC5CA, CC05A4, CC0AA5, CC0C05, CC3509, CC4555, CC5039, CC5554, CC555A, CC6092, CCC0C5, CCC353, CCC959, CCC9C2, 1000271, 1000802, 1000871, 1001771, 1001801, 1007078, 1008002, 1008107, 1008701, 1010117, 1027001, 1070771, 1077107, 1077701, 1080107, 1101077, 1110008, 1111078, 1115003, 1117777, 1170008, 1170101, 1700078, 1700777, 1800017, 1877017, 18B7772, 18BBB0B, 1999391, 1999931, 1BBBB3B, 2011001, 2107001, 2110001, 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...

Base 14[edit | edit source]

(test limit: 108000000D)

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4DDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDD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...

Base 15[edit | edit source]

(test limit: 1000000000)

12, 14, 18, 1E, 21, 27, 2B, 2D, 32, 38, 3E, 41, 47, 4B, 4D, 54, 58, 5E, 67, 6B, 6D, 72, 74, 78, 87, 8B, 92, 94, 9E, A1, A7, AD, B2, B8, BE, C1, CB, CD, D2, D4, E1, ED, 111, 11B, 131, 137, 13B, 13D, 157, 15B, 15D, 171, 177, 197, 19D, 1B7, 1BB, 1D1, 1DB, 1DD, 234, 298, 311, 31B, 337, 33D, 344, 351, 357, 35B, 364, 377, 391, 39B, 39D, 3A4, 3BD, 3C4, 3D7, 3DB, 3DD, 452, 51B, 51D, 531, 53B, 551, 55D, 562, 571, 577, 5A2, 5B1, 5B7, 5BB, 5BD, 5C2, 5D1, 5D7, 634, 652, 681, 698, 717, 71B, 731, 737, 757, 75D, 77D, 79B, 79D, 7B1, 7B7, 7BD, 7D7, 7DD, 801, 852, 88D, 8D8, 91D, 93B, 93D, 95B, 95D, 971, 977, 97B, 97D, 988, 991, 9BD, 9C8, 9D1, A98, AAB, B1D, B31, B3B, B44, B51, B57, B7B, B7D, B97, B9B, BB7, BC4, BD1, BD7, BDD, C07, C34, C52, C7E, C98, CC7, CE7, D0E, D1D, D31, D51, D5B, D68, D77, D7B, D91, D97, DA8, DAE, DCE, DD1, EB4, EEB, 107B, 1091, 10B1, 1107, 110D, 1561, 1651, 1691, 1B01, 2052, 2502, 2522, 303B, 307D, 3097, 30BB, 30D1, 3107, 3361, 3701, 3907, 3B01, 3B0B, 3C97, 4434, 4498, 4834, 4898, 49A8, 4E34, 5037, 507D, 5091, 509B, 5107, 5161, 5202, 53C7, 5552, 570B, 590B, 590D, 59C7, 5A5B, 5C97, 5D0D, 5DAB, 6061, 6151, 6191, 6511, 6601, 6911, 707B, 7091, 7097, 70AE, 70BB, 70CE, 70DB, 7561, 760E, 7691, 76CE, 7907, 7961, 7A0E, 7A3B, 7AEE, 7B0B, 7BAB, 7C0E, 7C77, 7CAE, 7D0B, 7D61, 7DAB, 7E5B, 7E6E, 7E7B, 7EBB, 8098, 811D, 8191, 835D, 853D, 8881, 8908, 8951, 8968, 899D, 8D3D, 8D5D, 8D6E, 8DDD, 8E98, 9011, 9037, 9097, 90D7, 9301, 93C7, 95C7, 9611, 9631, 96A8, 9811, 9851, 989D, 990B, 990D, 998D, 99AB, 99C7, 99D8, 9A08, 9A9B, 9AA8, 9ABB, 9B61, 9BC7, 9D0B, 9DAB, 9DC7, 9DD8, A052, A304, A502, A55B, A9BB, AB04, AB64, B09D, B107, B10B, B161, B1AB, B1C7, B30D, B3C7, B50B, B664, B691, B6A4, B707, B761, B90D, B961, BA5B, BABB, BBAB, BBB4, BC37, BC77, C777, C937, C997, D011, D03D, D05D, D09B, D0B1, D0BD, D101, D10B, D30D, D3AB, D507, D50D, D66E, D761, D7DE, D811, D85D, D86E, D89D, D8C8, D8E8, D9AB, D9D8, DA3B, DA9B, DABB, DB01, DB61, DBAB, DC88, DD07, DD0B, DD7E, DD8D, DDE7, DE6E, E252, E33B, E522, E57B, E7AE, E7CE, E898, E997, E9A8, E9BB, EA34, EB5B, EE98, EEC7, 10017, 10B0D, 170AB, 17A0B, 19001, 19601, 1A09B, 1D0C7, 22E52, 2EA52, 30017, 3001D, 300B1, 301C7, 30334, 30631, 307AB, 3300B, 3333B, 36031, 36301, 37A0B, 37BBB, 39997, 3A30B, 3B0C7, 3D001, 3D601, 40034, 40968, 43334, 49668, 49998, 50022, 5009D, 501C7, 50222, 50507, 505C7, 50611, 50C57, 53007, 53997, 55537, 5555B, 5557B, 5599B, 56101, 56691, 56961, 5700D, 5755B, 59001, 59557, 59997, 5999D, 599DB, 59DDD, 5D99B, 5DD3D, 5DD9D, 60931, 63031, 65691, 66951, 69031, 69361, 69561, 70011, 70051, 7005B, 7006E, 7030D, 703AB, 70501, 70701, 707C7, 71601, 71951, 7300D, 7333B, 75001, 7555B, 75911, 76011, 76051, 766EE, 76EEE, 7700B, 77191, 77661, 7776E, 77771, 777BB, 77911, 77BBB, 79001, 7A05B, 7A66E, 7AA6E, 7AAAE, 7ACCE, 7C6EE, 7CCEE, 7CECE, 7CEEE, 7D3BB, 7E7C7, 7EECE, 80034, 80304, 80434, 809DD, 80A34, 84A34, 850DD, 85961, 86661, 88151, 88331, 88511, 88591, 88898, 890DD, 89998, 89D0D, 8D90D, 8E434, 90017, 90051, 900A8, 900DB, 901C7, 90C57, 90D8D, 91007, 91061, 9199B, 95997, 96068, 96561, 99397, 99537, 9999B, 999B7, 999D7, 999DB, 999DD, 99BBB, 99DBB, 99DD7, 99DDD, 9B007, 9B00B, 9B0AB, 9BB11, 9BBBB, 9D007, 9D08D, 9D537, 9D9BB, 9D9DB, 9DD57, 9DDB7, 9DDDB, 9DDDD, A0A34, A0B5B, A0BBB, A0E34, A2E52, A330B, A8434, A8834, A8E34, A909B, AAA34, AAE52, AB0BB, AB334, ABB34, AE034, AE834, AE99B, AEA52, AEE52, B0011, B0071, B0077, B00B1, B0611, B0A64, B500D, B599D, B6101, B7771, B7911, BA064, BAAA4, BAB34, BB061, BB304, BB53D, BB601, BBB91, BBB9D, BBBBD, BDA0B, BDBBB, D0088, D00D7, D0307, D05C7, D070D, D0888, D0B07, D0BC7, D0C08, D0DC7, D0DD8, D1661, D59DD, D5D3D, D5DDD, D6611, D700D, D8D0D, D900B, D9908, D999D, D9BBB, D9D9D, D9DDB, DB007, DB00D, DB1B1, DB53D, DB59D, DB99D, DBBB1, DD0D8, DD33B, DD3B7, DD3BB, DD57D, DD898, DD9DD, DDB37, DDBDB, DDD08, DDD3D, DDD5D, DDD7D, DDD88, DDD9D, DDDB7, DDDC8, DDDD7, DDE98, DE037, DE998, DEB07, E0098, E00C7, E0537, E0557, E077B, E0834, E0968, E3334, E37AB, E39C7, E4034, E5307, E55AB, E705B, E750B, E766E, E76EE, E8304, E8434, E9608, E9C37, EAE52, EBB0B, EC557, EC597, EC957, 1000BD, 1009AB, 10A90B, 1900AB, 190661, 19099B, 190A0B, 1A900B, 222A52, 2AAA52, 31000D, 330331, 333334, 3733AB, 373ABB, 3BBB61, 430004, 490068, 490608, 5000DB, 500D0B, 505557, 505A0B, 50D00B, 50DDDB, 50DDDD, 522222, 5500AB, 5500C7, 550957, 550A0B, 555A9B, 559057, 560011, 590661, 633331, 666331, 666591, 666661, 7050AB, 705A0B, 706101, 70A50B, 7300AB, 761661, 76666E, 777011, 777101, 77750B, 777A5B, 777CEE, 779051, 791501, 7E7797, 7ECCCE, 7EEE97, 800D9D, 808834, 836631, 83D661, 843004, 856611, 884034, 884304, 888E34, 88A434, 88AE34, 8A4034, 8AEE34, 8E8034, 8E8E34, 8EEE34, 9000BB, 9001AB, 900B07, 900D98, 903661, 905661, 906651, 9080DD, 9099A8, 909D9B, 90A668, 90DD9B, 90DDBB, 910001, 9100AB, 91A00B, 930007, 950001, 956661, 9909A8, 995907, 999068, 999507, 999907, 9B0B1B, 9B0BB1, 9BB01B, 9C5597, 9C5957, 9D09DD, 9D0D9D, 9D800D, 9DB307, 9DD09D, A00034, A0033B, A033B4, A2A252, AAAA52, ABBBBB, B00004, B0001B, B0003D, B00A04, B0555B, B07191, B07711, B07777, B0B911, B0BDBB, B77011, B777C7, BB0001, BB0034, BB035D, BB055B, BB0BDB, BB9101, BBB0DB, BBB50D, BBBB01, BBD0BB, C55397, C55557, C55597, D0003B, D00057, D0007D, D000B7, D000C8, D008DD, D00DAB, D0333B, D05537, D099DD, D09DDD, D0DDBB, D555C7, D5C537, D88008, D88088, D888EE, D909DD, D9D0DD, D9DD0D, DB0BBB, DBBB0B, DBBB0D, DC0008, DC5537, DDDDD8, DDDEBB, DDE99B, DE0808, DE0C57, DE300B, DE5537, DE8888, DEE088, DEE307, DEE888, DEEE37, DEEE57, DEEEC8, E0000B, E007BB, E00A52, E03BC7, E07ABB, E09B07, E0A99B, E0C397, E0E76E, E50057, E55007, E55597, E55937, E730AB, E73A0B, E80E34, E88834, E8E034, E90008, E95557, EA099B, EE4304, EE5057, EE5507, EE8E34, EE9307, EEE434, 100001D, 1000A9B, 1000DC7, 22AA252, 3000BC7, 3033301, 3076661, 333B304, 33B3034, 3B33304, 3D66661, 50007AB, 5005957, 5500597, 5550057, 5559007, 5559597, 5595007, 5966661, 5DDDDDB, 6366631, 7010001, 7066651, 7100061, 733BBBB, 766A6AE, 77505AB, 7776501, 777775B, 777AACE, 777ECCE, 777EEAE, 7CCCCCE, 7E30A0B, 7EEEEAE, 8300004, 8363331, 8693331, 880E834, 8833304, 8888034, 8888434, 888A034, 88A3334, 88E8834, 88EE034, 88EE304, 8AA3334, 8D0009D, 8EE8834, 9000361, 9000668, 9003331, 9005557, 9006008, 9008D0D, 9083331, 9090968, 90BBB01, 90D0908, 9500661, 9555597, 9555957, 9660008, 9900968, 9995597, 9996008, 9999557, 9999597, 9999908, 9A66668, A003B34, A003BB4, AA22252, B00B034, B00B35D, B033334, B0B6661, B0BB01B, B100001, B333304, B777777, B99999D, BA60004, BAA0334, BBB001B, BBB6611, BBBBB11, BBBD00B, BD000AB, D0000DB, D009098, D00CCC8, D00D908, D00D99D, D03000B, D0BB0BB, D0D9008, D0D9998, D1000C7, D800008, D8DDEEE, D90080D, DBBBBBB, DD09998, DDD5557, DDDDBBB, DDDDDBD, DDDE8EE, DECC008, DECCCC8, DEE0CC8, DEEC0C8, E000397, E0003BB, E000434, E00076E, E000937, E007A5B, E00909B, E0090B7, E009307, E00B077, E00E434, E00E797, E00E937, E05999B, E09009B, E0900B7, E0E0937, E0E7E97, E0EAA52, E0EEA52, E555057, E5555C7, E7777C7, E77E797, E88EE34, E999998, EA5999B, EB000BB, EB0BBBB, EE00434, EE0E797, EEE076E, EEE706E, EEE8834, EEEE557, EEEE797, 30333331, 30B66661, 33000034, 33030004, 33B33004, 500575AB, 55000007, 5500075B, 55500907, 55555057, 55555907, 55559507, 60003301, 60033001, 60330001, 7000003D, 70106661, 70666611, 77000001, 7777770B, 777777C7, 77777ACE, 77777EAE, 777E30AB, 777E3A0B, 7CCCC66E, 800005DD, 88AA0834, 90000008, 900008DD, 90099668, 90500557, 90555007, 90666668, 90909998, 90990998, 90996668, 9099999D, 90D00098, 90D90998, 95500057, 99099098, 99555057, 99900998, 99966608, 99966668, 99999668, 99999998, 9D009008, 9D090998, A0803334, A2222252, AAA52222, B00005AB, B000B55B, B0BBBB5B, B3330034, BB0BBB1B, BBAA3334, BBB0BB1B, BBB0BB5B, BBDB000B, D000BBBB, D00100C7, D8888888, D900008D, D9000098, DBB000BB, DC0CCCC8, DCC0CCC8, DCCCC008, DD000908, DD09009D, DDDDDDAB, DDDDDEEE, DDDEEE8E, DDDEEEE8, DEE80008, E0777E97, E0E0E397, E0E77797, E0EE0397, E7777797, E9066668, EE00E397, EE077797, EE0E0397, EEE00797, EEE07E97, EEE0AA52, EEE55397, EEE55557, EEEAAA52, EEEEE834, EEEEEA52, 300003331, 300007661, 300330031, 333000004, 333300001, 333B00034, 3700000AB, 3B3300034, 500000057, 555555007, 555555557, 5DDDDDDDD, 600000331, 7500000AB, 75000A00B, 75A00000B, 761000001, 77000E0C7, 777700EC7, 7777730AB, 7777777AE, 77777EE97, 7777E7E97, 777999997, 7A500000B, 7BBBBBB5B, 88888A834, 900000031, 900666608, 909990098, 90D009998, 950000557, 966666008, 990000007, 990555507, 999999997, A000000B4, A0005999B, AAEEEEE34, B000AA334, BBBBB005B, BBBBBBB5B, D09999998, D0D90009D, D800000DD, D90009998, DCCCC0CC8, DE88EEEEE, DEEEEEE88, E000B7777, E000BBBBB, E003ABBBB, EE0000797, EE0EEE397, EE5555557, EE777EE97, EEEEEE537, EEEEEE937, ..., 2222222252, ..., 3000000071, ..., 500000007B, ..., 7000000071, ..., 8888888834, ..., 900000009B, ..., 900000009D, ..., 9000099998, ..., D300000007, ..., DDDDDDDDDE, ..., DDDEEEEEEE, ..., E000000797, ..., 9999999999D, ..., 5000000000DD, ..., 9000000000B7, ..., D000000001C7, ..., DCCCCCCCCCC8, ..., 3333333333331, ..., 3BBBBBBBBBBBB, ..., 9555555555557, ..., 71000000000001, ..., 77777777777777B, ..., DEEEEEEEEEEEEEE, ..., DDDDDDDDDDDDDDDDB, ..., E9666666666666668, ..., E55555555555555557, ..., 3330000000000000031, ..., 7BBBBBBBBBBBBBBBBBBBB, ..., BBBBBBBBBBBBBBBBBBBBBB1, ..., B00000000000000000000005B, ..., B70000000000000000000000001, ..., 633000000000000000000000000001, ..., EBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBB, ..., 500000000000000000000000000000000017, ..., 96666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666608, ..., EEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEE397, ..., 7777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777797, ...

Base 16[edit | edit source]

(test limit: 1000000000)

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, 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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, B00000881, B00009801, B00090081, B00BBBABB, B0EB0000B, B4444444D, B77777777, B7E777777, BB00000BD, BB0C0000D, BBBBBA00B, BBBBBBABB, BE0EEEE0B, BE7777777, C00000CAF, C00006AAF, C000082CD, C00063AFF, C000820CD, C00F00023, C0444444D, C66666AFF, CCCD99999, CF0000023, CF66666AF, D00000009, D0000044D, D0044000D, D040E000D, D0440000D, D0DD000D9, DAAAAAA45, E004044DD, E004444DD, E044400DD, E0C00008D, E0C08000D, E0EAAAAA1, E2000000D, E400044DD, EAAA4AAA1, EAAAAEAA1, EAAAEA041, EBBBBC00D, EEEE00CCB, F00000545, F02600003, F066AAAAF, F0FF5666F, F3FFF3F23, F60AAAA0F, F77777777, FFEEEEEE7, FFFF33323, FFFF5666F, FFFFF2CC3, FFFFF7777, FFFFFEEE7, FFFFFFF77, ..., 2666666663, ..., 400000000D, ..., 500000006F, ..., 700000077B, ..., 8000000AA1, ..., 800AAAAA01, ..., 8886888AAF, ..., 88888888AF, ..., 8888888A8F, ..., 888AAFFFFF, ..., 9000000019, ..., 9000000109, ..., 908AAAAA01, ..., AAAAAAAAA1, ..., AAAAAAAE41, ..., C000CC866F, ..., C00CCCCCAF, ..., C6666666AF, ..., CCCCCCCAAF, ..., CFFFFFFAAF, ..., E444444441, ..., E4444444DD, ..., EAAAAAA4A1, ..., 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5FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF6F, ..., 88FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF, ..., 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..., 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BBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBB, ...

Base 17 (current data)[edit | edit source]

12, 16, 1C, 1E, 23, 27, 29, 2D, 32, 38, 3A, 3G, 43, 45, 4B, 4F, 54, 5C, 5G, 61, 65, 67, 6B, 78, 7C, 81, 83, 8D, 8F, 94, 9A, 9E, A3, A9, AB, B4, B6, BA, BC, C7, D2, D6, D8, DC, E1, E3, ED, F2, F8, FE, FG, G5, G9, GB, 104, 111, 115, 117, 11B, 137, 139, 13D, 14A, 14G, 155, 159, 15F, 171, 17B, 17D, 188, 191, 197, 19F, 1A4, 1A8, 1B3, 1BB, 1BF, 1DB, 1DD, 1F3, 1FD, 1G8, 1GA, 1GG, 20F, 214, 221, 225, 241, 25A, 25E, 285, 2B8, 2C5, 2CF, 2E5, 2EB, 2F6, 30E, 313, 331, 33B, 346, 34C, 351, 35F, 36E, 375, 37B, 391, 39B, 39D, 3B7, 3B9, 3BF, 3D3, 3D5, 3D9, 3DF, 3E4, 3EC, 3F1, 3F7, 407, 418, 447, 44D, 472, 474, 47E, 47G, 489, 49C, 4A1, 4C1, 4CD, 4D4, 4G1, 502, 506, 508, 50E, 519, 522, 528, 52A, 52E, 533, 53F, 551, 55D, 562, 566, 573, 577, 57F, 582, 593, 599, 59B, 59F, 5A6, 5B5, 5D1, 5D3, 5EA, 5EE, 5F9, 60D, 62F, 634, 649, 689, 692, 6CD, 6EF, 6F4, 6FA, 704, 706, 70G, 71D, 726, 737, 739, 73D, 73F, 753, 755, 764, 766, 76G, 771, 77B, 793, 7AA, 7AE, 7B3, 7BB, 7D7, 7E6, 7F3, 7F9, 7FF, 7G2, 7GE, 7GG, 825, 82B, 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AFFFFFFF4, B000000EE, B000000F5, B00007D0E, B0000DFBB, B0000FBBB, B000E0EF7, B000F0BBB, B00BBB991, B00BBBBB1, B00BF1999, B0300000B, B03000333, B0770000E, B07D0000E, B0B0000BE, B0B000EEF, B0BBBB30B, B0BBBBB0D, B0BBBBEBF, B0BBBE999, B0FBB0BBB, B33333333, B3F333333, B55355555, B55555539, B5BBB3BDB, B7000700E, B7700000E, B93900003, BB0000B33, BB0BBBBE9, BB3FBBBBD, BB8BB0BBB, BBB0000FB, BBB003333, BBB030003, BBB3BBBD1, BBB8B0BBB, BBBB3DBBB, BBBBB22EE, BBBBB3333, BBBBB3BBB, BBBBBB10G, BBBBBB1B9, BBBBBBB11, BBBBBBB22, BBBBBBB33, BBBBBBE09, BBBBBFB99, BBBBBFBBB, BBBBDBB0G, BBBBF9991, BBBD0000F, BDB0E000E, BF7700005, BFBBBBB33, C0049DDDD, C00999585, C00CE9666, C00EEE999, C022B000E, C028BBBBB, C04DEE00E, C09FBBBBB, C0BBB2EEE, C130FFFFF, C22BE000E, C31000009, C3C40DDDD, C3D400DDD, C500000FF, C500FF0FF, C5550F00F, C55555505, C555F0FFF, C5AAAA0AA, C5AAAAAE2, ...

Base 18 (current data)[edit | edit source]

11, 15, 1B, 1D, 21, 25, 27, 2B, 2H, 35, 37, 3D, 3H, 41, 47, 4B, 4H, 57, 5B, 5D, 5H, 61, 65, 71, 75, 7B, 7D, 85, 87, 8D, 91, 95, 9B, 9H, A1, AB, AD, AH, B1, BD, C7, CB, CD, CH, D5, D7, DH, E5, EB, EH, F1, F7, FB, FD, G5, H1, H5, H7, HB, 107, 167, 16H, 177, 17H, 1G7, 1HH, 20D, 24D, 26D, 29D, 30B, 36B, 381, 3BB, 405, 445, 44D, 49D, 4A5, 4DD, 4F5, 4GD, 501, 545, 5E1, 607, 62D, 64D, 66B, 66H, 67H, 68B, 697, 6A7, 6BB, 6E7, 6G7, 6GB, 6HH, 767, 76H, 77H, 797, 7HH, 801, 80H, 831, 83B, 86B, 88H, 8BB, 8FH, 8GH, 94D, 96D, 977, 9DD, 9ED, 9GD, A77, AC5, AE7, B07, B0H, B55, B77, B8B, B97, BB5, BB7, BBH, BE7, BFH, BGB, C01, C31, CA5, CG1, D2D, D4D, D81, DBB, DD1, DDB, DGD, E0D, E17, E31, E4D, E67, E6D, EA7, EDD, EE1, EED, EG7, F0H, F45, F8H, FC5, FFH, G0D, G17, G2D, G6B, G6H, GBB, GBH, GD1, GDD, GE1, GE7, GED, GFH, GG7, GGB, GHD, GHH, H0D, H2D, H8H, H9D, HGH, HHD, 100H, 19E7, 1A97, 1EE7, 1G8H, 1GGH, 22ED, 22GD, 2DED, 2E2D, 3001, 3031, 30C1, 30E1, 3331, 33G1, 3CC1, 40ED, 45C5, 46ED, 4CC5, 5331, 5551, 55G1, 5C05, 608H, 60ED, 60FH, 60HD, 666D, 66ED, 699D, 6B67, 6BGH, 6D0D, 6DDD, 6E9D, 6EGD, 6G0H, 6G9D, 6HGD, 700H, 70A7, 7A07, 7FGH, 7G77, 808B, 8881, 88G1, 88GB, 8BHH, 8EG1, 8GC1, 8H6H, 900D, 90E7, 90G7, 9667, 9907, 999D, 99E7, 9A67, 9A97, 9E97, 9EE7, 9G07, 9G67, 9GA7, AA45, AA97, AGA7, B005, B03B, B06B, B0C5, B60B, B63B, BAA5, BAA7, BCC5, BFA5, BG8H, C045, C055, C555, C5C1, C5F5, CC05, CC81, CCC5, D06D, D09D, D0ED, D38B, D3E1, D60D, D6DD, D8GB, DD6D, DE9D, DG01, E001, E097, E0G1, E8C1, EDC1, EE97, EGC1, EGG1, EGGD, FH6H, G007, G00B, G00H, G03B, G067, G097, G0C1, G0G1, G1GH, G33B, G38B, G3G1, G70H, G777, G88B, GA67, GAA7, GG81, GGC1, GGGH, H0FH, H66D, HEGD, HFHH, 1AAA7, 222DD, 30GG1, 3388B, 33E01, 38G8B, 3G3C1, 3GGG1, 4002D, 500C5, 50C55, 50CF5, 53GG1, 558C1, 55CC5, 55CF5, 58GG1, 5C8C1, 5CFF5, 5G881, 5GG31, 6000H, 6003B, 6006D, 600DB, 6033B, 606GD, 60D0B, 66GGD, 6D03B, 6D33B, 6H6DD, 6HD6D, 6HDED, 70G07, 70GGH, 777A7, 7AAG7, 7G0GH, 80G0B, 8888B, 8CCE1, 90067, 90097, 9022D, 99967, 99997, 9A007, 9A0A7, 9AA07, 9AAA7, 9E007, A0045, A0455, A0667, A09G7, A0A07, A0G07, A0G97, A9997, AA0A7, AAG67, B0AF5, B6GGH, B7GGH, B8HHH, BA045, BAF05, BG667, C0F05, C5005, C5581, C88C1, C8CC1, C8CE1, CCF55, D03C1, D060B, D080B, D0CC1, D0G0B, D0G8B, D3G3B, D600B, DDDED, DG331, DG80B, E8G81, E9007, F6GGH, G018H, G0301, G0331, G466D, G6667, G66GD, GD08B, GG18H, GG6GD, GGG4D, H060H, HGGGD, HHH6H, 199AA7, 40006D, 40600D, 46600D, 5055C5, 5505C5, 55CCC1, 588CC1, 58CCC1, 60000D, 60009D, 7077G7, 7707G7, 777G07, 88000B, 9099A7, A000A7, A009A7, A09067, A099A7, A0AAA7, A90AA7, A99AA7, AA0007, AA6667, AAAG07, BFFF05, BFFFF5, C0FFF5, CCECC1, CECCC1, CF0FF5, CFF005, D0008B, D0033B, D0088B, D0333B, D033GB, D03G31, D0633B, DD990D, DGGG31, FHHHHH, G00081, G6GGGD, G8GGG1, GGG001, GGG331, GGGGG1, GGGGGD, H0006H, H00H6H, HH600H, 222222D, 22DDDDD, 333333B, 5CCCCC1, 70007G7, 88CCCC1, 9000007, 9000A07, A000G67, AAAA667, BBBB33B, C000CF5, C000FF5, CCCCCE1, CCCCEC1, D00063B, D00GG31, D63333B, DCCCCC1, DDDDD9D, DGCCCC1, GCCCCC1, GG00031, 4022222D, 6000GGGD, 66666667, 770000G7, AAAAA007, B6666667, BBBBBB3B, CFFFFF55, D00000C1, D0000EC1, 455555555, 5555550C5, 667777777, A00000967, A00009097, A00009967, A45555555, AAAAAAA07, BHHHHHHHH, CCCCCCCC1, CF0000005, CFFFFFF05, D00000G3B, E0CCCCCC1, G00000031, ...

Known large quasi-minimal primes[edit | edit source]

Base 17[edit | edit source]

  • (6^4661)E9
  • 53(0^4867)1
  • 7(4^4904)
  • 1(F^7092)
  • 1(0^9019)1F
  • 57(0^51310)1
  • 4(9^111333)
  • 97(0^166047)1
  • F7(0^186767)1

Base 18[edit | edit source]

  • 7(0^7)G7
  • A(0^7)97
  • D(0^9)1
  • 6(7^11)
  • 8(H^11)
  • 2(D^12)
  • (5^11)C5
  • (A^14)7
  • D(0^13)B
  • C(F^30)5
  • GG(0^30)1
  • H(D^93)
  • C(0^116)F5
  • 8(0^298)B

Known unsolved families[edit | edit source]

Base 11[edit | edit source]

  • 5{7}

Base 13[edit | edit source]

  • 9{5}
  • A{3}A

Base 16[edit | edit source]

  • {3}AF
  • {4}DD

Base 17[edit | edit source]

  • 15{0}D
  • 1{7}
  • 1F{0}7
  • 51{0}D
  • 73{0}B
  • 9D{0}5
  • B3{0}D
  • B{0}B3
  • B{0}DB
  • F1{9}

Base 18[edit | edit source]

  • C{0}C5

Proof[edit | edit source]

Base 2[edit | edit source]

The possible (first digit,last digit) combo for a quasi-minimal prime with ≥3 digits are:

(1,1)

  • Case (1,1):
    • 11 is prime, and thus the only minimal prime in this family.

Base 3[edit | edit source]

The possible (first digit,last digit) combo for a quasi-minimal prime with ≥3 digits are:

(1,1), (1,2), (2,1), (2,2)

  • Case (1,1):
    • Since 12, 21, 111 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):
    • 12 is prime, and thus the only minimal prime in this family.
  • Case (2,1):
    • 21 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.

Base 4[edit | edit source]

The possible (first digit,last digit) combo for a quasi-minimal prime with ≥3 digits are:

(1,1), (1,3), (2,1), (2,3), (3,1), (3,3)

  • 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 (2,1):
    • Since 23, 11, 31, 221 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):
    • 23 is prime, and thus the only minimal prime in this family.
  • Case (3,1):
    • 31 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.

Base 5[edit | edit source]

The possible (first digit,last digit) combo for a quasi-minimal prime with ≥3 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, 111, 131 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):
    • 12 is prime, and thus the only minimal prime in this family.
  • Case (1,3):
    • Since 12, 23, 43, 133 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 10103 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 100000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000013, which can be written as 1(0^93)13 and equal the prime 5^95+8 (factordb)
          • All numbers of the form 11{0}3 are divisible by 3, thus cannot be prime.
  • Case (1,4):
    • Since 12, 34, 104 are primes, we only need to consider the family 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 families 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 14444.
        • All numbers of the form 11{4}4 are divisible by 2, thus cannot be prime.
  • Case (2,1):
    • 21 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):
    • 23 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, 3101 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 30301, 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 300031.
          • 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 33001.
          • For the 33{0,3}31 family, we have the prime 33331, 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.
        • All numbers of the form 3{0,3}11 are divisible by 3, thus cannot be prime.
  • Case (3,2):
    • 32 is prime, and thus the only minimal prime in this family.
  • Case (3,3):
    • Since 32, 34, 23, 43, 313 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):
    • 34 is prime, and thus the only minimal prime in this family.
  • Case (4,1):
    • Since 43, 21, 401 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 families 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 44441.
        • 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):
    • 43 is prime, and thus the only minimal prime in this family.
  • Case (4,4):
    • Since 43, 34, 414 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.

Base 6[edit | edit source]

The possible (first digit,last digit) combo for a quasi-minimal prime with ≥3 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):
    • 11 is prime, and thus the only minimal prime in this family.
  • Case (1,5):
    • 15 is prime, and thus the only minimal prime in this family.
  • Case (2,1):
    • 21 is prime, and thus the only minimal prime in this family.
  • Case (2,5):
    • 25 is prime, and thus the only minimal prime in this family.
  • Case (3,1):
    • 31 is prime, and thus the only minimal prime in this family.
  • Case (3,5):
    • 35 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 4401 and 4441 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 40041
  • Case (4,5):
    • 45 is prime, and thus the only minimal prime in this family.
  • Case (5,1):
    • 51 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.

Base 7[edit | edit source]

The possible (first digit,last digit) combo for a quasi-minimal prime with ≥3 digits are:

(1,1), (1,2), (1,3), (1,4), (1,5), (1,6), (2,1), (2,2), (2,3), (2,4), (2,5), (2,6), (3,1), (3,2), (3,3), (3,4), (3,5), (3,6), (4,1), (4,2), (4,3), (4,4), (4,5), (4,6), (5,1), (5,2), (5,3), (5,4), (5,5), (5,6), (6,1), (6,2), (6,3), (6,4), (6,5), (6,6)

  • Case (1,1):
    • Since 14, 16, 41, 61, 131 are primes, we only need to consider the family 1{0,1,2,5}1 (since any digits 3, 4, 6 between them will produce smaller primes)
      • Since the digit sum of primes must be odd (otherwise the number will be divisible by 2, thus cannot be prime), there is an odd total number of 1 and 5 in the {}
        • If there are >=3 number of 1 and 5 in the {}:
          • If there is 111 in the {}, then we have the prime 11111
          • If there is 115 in the {}, then the prime 115 is a subsequence
          • If there is 151 in the {}, then the prime 115 is a subsequence
          • If there is 155 in the {}, then the prime 155 is a subsequence
          • If there is 511 in the {}, then the current number is 15111, which has digit sum = 12, but digit sum divisible by 3 will cause the number divisible by 3 and cannot be prime, and we cannot add more 1 or 5 to this number (to avoid 11111, 155, 515, 551 as subsequence), thus we must add at least one 2 to this number, but then the number has both 2 and 5, and will have either 25 or 52 as subsequence, thus cannot be minimal prime
          • If there is 515 in the {}, then the prime 515 is a subsequence
          • If there is 551 in the {}, then the prime 551 is a subsequence
          • If there is 555 in the {}, then the prime 551 is a subsequence
        • Thus there is only one 1 (and no 5) or only one 5 (and no 1) in the {}, i.e. we only need to consider the families 1{0,2}1{0,2}1 and 1{0,2}5{0,2}1
          • For the 1{0,2}1{0,2}1 family, since 1211 is prime, we only need to consider the family 1{0}1{0,2}1
            • Since all numbers of the form 1{0}1{0}1 are divisible by 3 and cannot be prime, we only need to consider the family 1{0}1{0}2{0}1
              • Since 11201 is prime, we only need to consider the family 1{0}1{0}21
                • The smallest prime of the form 11{0}21 is 1100021
                • All numbers of the form 101{0}21 are divisible by 5, thus cannot be prime
                • The smallest prime of the form 1001{0}21 is 100121
                  • Since this prime has no 0 between 1{0}1 and 21, we do not need to consider more families
          • For the 1{0,2}5{0,2}1 family, since 25 and 52 are primes, we only need to consider the family 1{0}5{0}1
            • Since 1051 is prime, we only need to consider the family 15{0}1
              • The smallest prime of the form 15{0}1 is 150001
  • Case (1,2):
    • Since 14, 16, 32, 52 are primes, we only need to consider the family 1{0,1,2}2 (since any digits 3, 4, 5, 6 between them will produce smaller primes)
      • Since 1112 and 1222 are primes, there is at most one 1 and at most one 2 in {}
        • If there are one 1 and one 2 in {}, then the digit sum is 6, and the number will be divisible by 6 and cannot be prime.
        • If there is one 1 but no 2 in {}, then the digit sum is 4, and the number will be divisible by 2 and cannot be prime.
        • If there is no 1 but one 2 in {}, then the form is 1{0}2{0}2
          • Since 1022 and 1202 are primes, we only need to consider the number 122
            • 122 is not prime.
        • If there is no 1 and no 2 in {}, then the digit sum is 3, and the number will be divisible by 3 and cannot be prime.
  • Case (1,3):
    • Since 14, 16, 23, 43, 113, 133 are primes, we only need to consider the family 1{0,5}3 (since any digits 1, 2, 3, 4, 6 between them will produce smaller primes)
      • Since 155 is prime, we only need to consider the family 1{0}3 and 1{0}5{0}3
        • All numbers of the form 1{0}3 are divisible by 2, thus cannot be prime.
        • All numbers of the form 1{0}5{0}3 are divisible by 3, thus cannot be prime.
  • Case (1,4):
    • 14 is prime, and thus the only minimal prime in this family.
  • Case (1,5):
    • Since 14, 16, 25, 65, 115, 155 are primes, we only need to consider the family 1{0,3}5 (since any digits 1, 2, 4, 5, 6 between them will produce smaller primes)
      • All numbers of the form 1{0,3}5 are divisible by 3, thus cannot be prime.
  • Case (1,6):
    • 16 is prime, and thus the only minimal prime in this family.
  • Case (2,1):
    • Since 23, 25, 41, 61, 221 are primes, we only need to consider the family 2{0,1}1 (since any digits 2, 3, 4, 5, 6 between them will produce smaller primes)
      • Since 2111 is prime, we only need to consider the families 2{0}1 and 2{0}1{0}1
        • All numbers of the form 2{0}1 are divisible by 3, thus cannot be prime.
        • All numbers of the form 2{0}1{0}1 are divisible by 2, thus cannot be prime.
  • Case (2,2):
    • Since 23, 25, 32, 52, 212 are primes, we only need to consider the family 2{0,2,4,6}2 (since any digits 1, 3, 5 between them will produce smaller primes)
      • All numbers of the form 2{0,2,4,6}2 are divisible by 2, thus cannot be prime.
  • Case (2,3):
    • 23 is prime, and thus the only minimal prime in this family.
  • Case (2,4):
    • Since 23, 25, 14 are primes, we only need to consider the family 2{0,2,4,6}4 (since any digits 1, 3, 5 between them will produce smaller primes)
      • All numbers of the form 2{0,2,4,6}4 are divisible by 2, thus cannot be prime.
  • Case (2,5):
    • 25 is prime, and thus the only minimal prime in this family.
  • Case (2,6):
    • Since 23, 25, 16, 56 are primes, we only need to consider the family 2{0,2,4,6}6 (since any digits 1, 3, 5 between them will produce smaller primes)
      • All numbers of the form 2{0,2,4,6}6 are divisible by 2, thus cannot be prime.
  • Case (3,1):
    • Since 32, 41, 61 are primes, we only need to consider the family 3{0,1,3,5}1 (since any digits 2, 4, 6 between them will produce smaller primes)
      • Since 551 is prime, we only need to consider the family 3{0,1,3}1 and 3{0,1,3}5{0,1,3}1 (since any digits combo 55 between (3,1) will produce smaller primes)
        • For the 3{0,1,3}1 family, since 3031 and 131 are primes, we only need to consider the families 3{0,1}1 and 3{3}3{0,1}1 (since any digits combo 03, 13 between (3,1) will produce smaller primes, thus for the digits between (3,1), all 3's must be before all 0's and 1's, and thus we can let the red 3 in 3{3}3{0,1}1 be the rightmost 3 between (3,1), all digits before this 3 must be 3's, and all digits after this 3 must be either 0's or 1's)
          • For the 3{0,1}1 family:
            • If there are >=2 0's and >=1 1's between (3,1), then at least one of 30011, 30101, 31001 will be a subsequence.
            • If there are no 1's between (3,1), then the form will be 3{0}1
              • All numbers of the form 3{0}1 are divisible by 2, thus cannot be prime.
            • If there are no 0's between (3,1), then the form will be 3{1}1
              • The smallest prime of the form 3{1}1 is 31111
            • If there are exactly 1 0's between (3,1), then there must be <3 1's between (3,1), or 31111 will be a subsequence.
              • If there are 2 1's between (3,1), then the digit sum is 6, thus the number is divisible by 6 and cannot be prime.
              • If there are 1 1's between (3,1), then the number can only be either 3101 or 3011
                • Neither 3101 nor 3011 is prime.
              • If there are no 1's between (3,1), then the number must be 301
                • 301 is not prime.
          • For the 3{3}3{0,1}1 family:
            • If there are at least one 3 between (3,3{0,1}1) and at least one 1 between (3{3}3,1), then 33311 will be a subsequence.
            • If there are no 3 between (3,3{0,1}1), then the form will be 33{0,1}1
              • If there are at least 3 1's between (33,1), then 31111 will be a subsequence.
              • If there are exactly 2 1's between (33,1), then the digit sum is 12, thus the number is divisible by 3 and cannot be prime.
              • If there are exactly 1 1's between (33,1), then the digit sum is 11, thus the number is divisible by 2 and cannot be prime.
              • If there are no 1's between (33,1), then the form will be 33{0}1
                • The smallest prime of the form 33{0}1 is 33001
            • If there are no 1 between (3{3}3,1), then the form will be 3{3}3{0}1
              • If there are at least 2 0's between (3{3}3,1), then 33001 will be a subsequence.
              • If there are exactly 1 0's between (3{3}3,1), then the form is 3{3}301
                • The smallest prime of the form 3{3}301 is 33333301
              • If there are no 0's between (3{3}3,1), then the form is 3{3}31
                • The smallest prime of the form 3{3}31 is 33333333333333331
        • For the 3{0,1,3}5{0,1,3}1 family, since 335 is prime, we only need to consider the family 3{0,1}5{0,1,3}1
          • Numbers containing 3 between (3{0,1}5,1):
            • The form is 3{0,1}5{0,1,3}3{0,1,3}1
              • Since 3031 and 131 are primes, we only need to consider the family 35{3}3{0,1,3}1 (since any digits combo 03, 13 between (3,1) will produce smaller primes)
                • Since 533 is prime, we only need to consider the family 353{0,1}1 (since any digits combo 33 between (35,1) will produce smaller primes)
                  • Since 5011 is prime, we only need to consider the family 353{1}{0}1 (since any digits combo 01 between (353,1) will produce smaller primes)
                    • If there are at least 3 1's between (353,{0}1), then 31111 will be a subsequence.
                    • If there are exactly 2 1's between (353,{0}1), then the digit sum is 20, thus the number is divisible by 2 and cannot be prime.
                    • If there are exactly 1 1's between (353,{0}1), then the form is 3531{0}1
                      • The smallest prime of the form 3531{0}1 is 3531001, but it is not minimal prime since 31001 is prime.
                    • If there are no 1's between (353,{0}1), then the digit sum is 15, thus the number is divisible by 6 and cannot be prime.
          • Numbers not containing 3 between (3{0,1}5,1):
            • The form is 3{0,1}5{0,1}1
              • If there are >=2 0's and >=1 1's between (3,1), then at least one of 30011, 30101, 31001 will be a subsequence.
              • If there are no 1's between (3,1), then the form will be 3{0}5{0}1
                • All numbers of the form 3{0}5{0}1 are divisible by 3, thus cannot be prime.
              • If there are no 0's between (3,1), then the form will be 3{1}5{1}1
                • If there are >=3 1's between (3,1), then 31111 will be a subsequence.
                • If there are exactly 2 1's between (3,1), then the number can only be 31151, 31511, 35111
                  • None of 31151, 31511, 35111 are primes.
                • If there are exactly 1 1's between (3,1), then the digit sum is 13, thus the number is divisible by 2 and cannot be prime.
                • If there are no 1's between (3,1), then the number is 351
                  • 351 is not prime.
              • If there are exactly 1 0's between (3,1), then the form will be 3{1}0{1}5{1}1 or 3{1}5{1}0{1}1
                • No matter 3{1}0{1}5{1}1 or 3{1}5{1}0{1}1, if there are >=3 1's between (3,1), then 31111 will be a subsequence.
                • If there are exactly 2 1's between (3,1), then the number can only be 311051, 310151, 310511, 301151, 301511, 305111, 311501, 315101, 315011, 351101, 351011, 350111
                  • Of these numbers, 311051, 301151, 311501, 351101, 350111 are primes.
                    • However, 311051, 301151, 311501 have 115 as subsequence, and 350111 has 5011 as subsequence, thus only 351101 is minimal prime.
                • No matter 3{1}0{1}5{1}1 or 3{1}5{1}0{1}1, if there are exactly 1 1's between (3,1), then the digit sum is 13, thus the number is divisible by 2 and cannot be prime.
                • If there are no 1's between (3,1), then the number is 3051 for 3{1}0{1}5{1}1 or 3501 for 3{1}5{1}0{1}1
                  • Neither 3051 nor 3501 is prime.
  • Case (3,2):
    • 32 is prime, and thus the only minimal prime in this family.
  • Case (3,3):
    • Since 32, 23, 43, 313 are primes, we only need to consider the family 3{0,3,5,6}3 (since any digits 1, 2, 4 between them will produce smaller primes)
      • If there are >=2 5's in {}, then 553 will be a subsequence.
      • If there are no 5's in {}, then the family will be 3{0,3,6}3
        • All numbers of the form 3{0,3,6}3 are divisible by 3, thus cannot be prime.
      • If there are exactly 1 5's in {}, then the family will be 3{0,3,6}5{0,3,6}3
        • Since 335, 65, 3503, 533, 56 are primes, we only need to consider the family 3{0}53 (since any digit 3, 6 between (3,5{0,3,6}3) and any digit 0, 3, 6 between (3{0,3,6}5,3) will produce smaller primes)
          • The smallest prime of the form 3{0}53 is 300053
  • Case (3,4):
    • Since 32, 14, 304, 344, 364 are primes, we only need to consider the family 3{3,5}4 (since any digits 0, 1, 2, 4, 6 between them will produce smaller primes)
      • Since 3334 and 335 are primes, we only need to consider the family 3{5}4 and 3{5}34 (since any digits combo 33, 35 between them will produce smaller primes)
        • The smallest prime of the form 3{5}4 is 35555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555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with 9234 5's, which can be written as 3(5^9234)4 and equal the prime (23*7^9235-11)/6 (factordb) (not minimal prime, since 35555 and 5554 are primes)
        • The smallest prime of the form 3{5}34 is 355555555555555555555555555555555555555555555555555555555555555534 (not minimal prime, since 35555, 553, and 5554 are primes)
  • Case (3,5):
    • Since 32, 25, 65, 335 are primes, we only need to consider the family 3{0,1,4,5}5 (since any digits 2, 3, 6 between them will produce smaller primes)
      • If there are at least one 1's and at least one 5's in {}, then either 155 or 515 will be a subsequence.
      • If there are at least one 1's and at least one 4's in {}, then either 14 or 41 will be a subsequence.
      • If there are at least two 1's in {}, then 115 will be a subsequence.
      • If there are exactly one 1's and no 4's or 5's in {}, then the family will be 3{0}1{0}5
        • All numbers of the form 3{0}1{0}5 are divisible by 3, thus cannot be prime.
      • If there is no 1's in {}, then the family will be 3{0,4,5}5
        • If there are at least to 4's in {}, then 344 and 445 will be subsequences.
        • If there is no 4's in {}, then the family will be 3{0,5}5
          • Since 3055 and 3505 are primes, we only need to consider the families 3{0}5 and 3{5}5
            • All numbers of the form 3{0}5 are divisible by 2, thus cannot be prime.
            • The smallest prime of the form 3{5}5 is 35555
        • If there is exactly one 4's in {}, then the family will be 3{0,5}4{0,5}5
          • Since 304, 3545 are primes, we only need to consider the families 34{0,5}5 (since any digits 0 or 5 between (3,4{0,5}5) will produce small primes)
            • All numbers of the form 34{0,5}5 are divisible by 5, thus cannot be prime.
  • Case (3,6):
    • Since 32, 16, 56, 346 are primes, we only need to consider the family 3{0,3,6}6 (since any digits 1, 2, 4, 5 between them will produce smaller primes)
      • All numbers of the form 3{0,3,6}6 are divisible by 3, thus cannot be prime.
  • Case (4,1):
    • 41 is prime, and thus the only minimal prime in this family.
  • Case (4,2):
    • Since 41, 43, 32, 52 are primes, we only need to consider the family 4{0,2,4,6}2 (since any digits 1, 3, 5 between them will produce smaller primes)
      • All numbers of the form 4{0,2,4,6}2 are divisible by 2, thus cannot be prime.
  • Case (4,3):
    • 43 is prime, and thus the only minimal prime in this family.
  • Case (4,4):
    • Since 41, 43, 14 are primes, we only need to consider the family 4{0,2,4,5,6}4 (since any digits 1, 3 between them will produce smaller primes)
      • If there is no 5's in {}, then the family will be 4{0,2,4,6}4
        • All numbers of the form 4{0,2,4,6}4 are divisible by 2, thus cannot be prime.
      • If there is at least one 5's in {}, then there cannot be 2 in {} (since if so, then either 25 or 52 will be a subsequence) and there cannot be 6 in {} (since if so, then either 65 or 56 will be a subsequence), thus the family is 4{0,4,5}5{0,4,5}4
        • Since 445, 4504, 544 are primes, we only need to consider the family 4{0,5}5{5}4 (since any digit 4 between (4,5{0,4,5}4) and any digit 0, 4 between (4{0,4,5}5,4) will produce smaller primes)
          • If there are at least two 0's between (4,5{0,4,5}4), then 40054 will be a subsequence.
          • If there is no 0's between (4,5{0,4,5}4), then the family will be 4{5}5{5}4, which is equivalent to 4{5}4
            • The smallest prime of the form 4{5}4 is 45555555555555554 (not minimal prime, since 4555 and 5554 are primes)
          • If there is exactly one 0's between (4,5{0,4,5}4), then the family will be 4{5}0{5}5{5}4
            • Since 4504 is prime, we only need to consider the family 40{5}5{5}4 (since any digit 5 between (4,0{5}5{5}4) will produce small primes), which is equivalent to 40{5}4
              • The smallest prime of the form 40{5}4 is 405555555555555554 (not minimal prime, since 4555 and 5554 are primes)
  • Case (4,5):
    • Since 41, 43, 25, 65, 445 are primes, we only need to consider the family 4{0,5}5 (since any digits 1, 2, 3, 4, 6 between them will produce smaller primes)
      • If there are at least two 5's in {}, then 4555 will be a subsequence.
      • If there is exactly one 5's in {}, then the digit sum is 20, and the number will be divisible by 2 and cannot be prime.
      • If there is no 5's in {}, then the family will be 4{0}5
        • All numbers of the form 4{0}5 are divisible by 3, thus cannot be prime.
  • Case (4,6):
    • Since 41, 43, 16, 56 are primes, we only need to consider the family 4{0,2,4,6}6 (since any digits 1, 3, 5 between them will produce smaller primes)
      • All numbers of the form 4{0,2,4,6}6 are divisible by 2, thus cannot be prime.
  • Case (5,1):
    • Since 52, 56, 41, 61, 551 are primes, we only need to consider the family 5{0,1,3}1 (since any digits 2, 4, 5, 6 between them will produce smaller primes)
      • If there are at least two 3's in {}, then 533 will be a subsequence.
      • If there is no 3's in {}, then the family will be 5{0,1}1
        • Since 5011 is prime, we only need to consider the family 5{1}{0}1
          • Since 11111 is prime, we only need to consider the families 5{0}1, 51{0}1, 511{0}1, 5111{0}1 (since any digits combo 1111 between (5,1) will produce small primes)
            • All numbers of the form 5{0}1 are divisible by 6, thus cannot be prime.
            • The smallest prime of the form 51{0}1 is 5100000001
            • All numbers of the form 511{0}1 are divisible by 2, thus cannot be prime.
            • All numbers of the form 5111{0}1 are divisible by 3, thus cannot be prime.
      • If there is exactly one 3's in {}, then the family will be 5{0,1}3{0,1}1
        • If there is at least one 1's between (5,3{0,1}1), then 131 will be a subsequence.
          • Thus we only need to consider the family 5{0}3{0,1}1
            • If there are no 1's between (5{0}3,1), then the digit sum is 12, and the number will be divisible by 3 and cannot be prime.
            • If there are exactly one 1's between (5{0}3,1), then the digit sum is 13, and the number will be divisible by 2 and cannot be prime.
            • If there are exactly three 1's between (5{0}3,1), then the digit sum is 15, and the number will be divisible by 6 and cannot be prime.
            • If there are at least four 1's between (5{0}3,1), then 11111 will be a subsequence.
            • If there are exactly two 1's between (5{0}3,1), then the family will be 5{0}3{0}1{0}1{0}1
              • Since 5011 is prime, we only need to consider the family 5311{0}1 (since any digit 0 between (5,1{0}1) will produce small primes, this includes the leftmost three {} in 5{0}3{0}1{0}1{0}1, and thus only the rightmost {} can contain 0)
                • The smallest prime of the form 5311{0}1 is 531101
  • Case (5,2):
    • 52 is prime, and thus the only minimal prime in this family.
  • Case (5,3):
    • Since 52, 56, 23, 43, 533, 553 are primes, we only need to consider the family 5{0,1}3 (since any digits 2, 3, 4, 5, 6 between them will produce smaller primes)
      • If there are at least two 1's in {}, then 113 will be a subsequence.
      • If there is exactly one 1's in {}, then the digit sum is 12, and the number will be divisible by 3 and cannot be prime.
      • If there is no 1's in {}, then the digit sum is 11, and the number will be divisible by 2 and cannot be prime.
  • Case (5,4):
    • Since 52, 56, 14, 544 are primes, we only need to consider the family 5{0,3,5}4 (since any digits 1, 2, 4, 6 between them will produce smaller primes)
      • If there are no 5's in {}, then the family will be 5{0,3}4
        • All numbers of the form 5{0,3}4 are divisible by 3, thus cannot be prime.
      • If there are at least one 5's and at least one 3's in {}, then either 535 or 553 will be a subsequence.
      • If there are exactly one 5's and no 3's in {}, then the digit sum is 20, and the number will be divisible by 2 and cannot be prime.
      • If there are at least two 5's in {}, then 5554 will be a subsequence.
  • Case (5,5):
    • Since 52, 56, 25, 65, 515, 535 are primes, we only need to consider the family 5{0,4,5}5 (since any digits 1, 2, 3, 6 between them will produce smaller primes)
      • If there are no 4's in {}, then the family will be 5{0,5}5
        • All numbers of the form 5{0,5}5 are divisible by 5, thus cannot be prime.
      • If there are no 5's in {}, then the family will be 5{0,4}5
        • All numbers of the form 5{0,4}5 are divisible by 2, thus cannot be prime.
      • If there are at least one 4's and at least one 5's in {}, then either 5455 or 5545 will be a subsequence.
  • Case (5,6):
    • 56 is prime, and thus the only minimal prime in this family.
  • Case (6,1):
    • 61 is prime, and thus the only minimal prime in this family.
  • Case (6,2):
    • Since 61, 65, 32, 52 are primes, we only need to consider the family 6{0,2,4,6}2 (since any digits 1, 3, 5 between them will produce smaller primes)
      • All numbers of the form 6{0,2,4,6}2 are divisible by 2, thus cannot be prime.
  • Case (6,3):
    • Since 61, 65, 23, 43 are primes, we only need to consider the family 6{0,3,6}3 (since any digits 1, 2, 4, 5 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 (6,4):
    • Since 61, 65, 14 are primes, we only need to consider the family 6{0,2,3,4,6}4 (since any digits 1, 5 between them will produce smaller primes)
      • If there is no 3's in {}, then the family will be 6{0,2,4,6}4
        • All numbers of the form 6{0,2,4,6}4 are divisible by 2, thus cannot be prime.
      • If there are exactly two 3's in {}, then the family will be 6{0,2,4,6}3{0,2,4,6}3{0,2,4,6}4
        • All numbers of the form 6{0,2,4,6}3{0,2,4,6}3{0,2,4,6}4 are divisible by 2, thus cannot be prime.
      • If there are at least three 3's in {}, then 3334 will be a subsequence.
      • If there is exactly one 3's in {}, then the family will be 6{0,2,4,6}3{0,2,4,6}4
        • If there is 0 between (6,3{0,2,4,6}4), then 6034 will be a subsequence.
        • If there is 2 between (6,3{0,2,4,6}4), then 23 will be a subsequence.
        • If there is 4 between (6,3{0,2,4,6}4), then 43 will be a subsequence.
        • If there is 6 between (6,3{0,2,4,6}4), then 6634 will be a subsequence.
        • If there is 0 between (6{0,2,4,6}3,4), then 304 will be a subsequence.
        • If there is 2 between (6{0,2,4,6}3,4), then 32 will be a subsequence.
        • If there is 4 between (6{0,2,4,6}3,4), then 344 will be a subsequence.
        • If there is 6 between (6{0,2,4,6}3,4), then 364 will be a subsequence.
        • Thus the number can only be 634
          • 634 is not prime.
  • Case (6,5):
    • 65 is prime, and thus the only minimal prime in this family.
  • Case (6,6):
    • Since 61, 65, 16, 56 are primes, we only need to consider the family 6{0,2,3,4,6}6 (since any digits 1, 5 between them will produce smaller primes)
      • If there is no 3's in {}, then the family will be 6{0,2,4,6}6
        • All numbers of the form 6{0,2,4,6}6 are divisible by 2, thus cannot be prime.
      • If there is no 2's and no 4's in {}, then the family will be 6{0,3,6}6
        • All numbers of the form 6{0,3,6}6 are divisible by 3, thus cannot be prime.
      • If there is at least one 3's and at least one 2's in {}, then either 32 or 23 will be a subsequence.
      • If there is at least one 3's and at least one 4's in {}, then either 346 or 43 will be a subsequence.

Base 8[edit | edit source]

The possible (first digit,last digit) combo for a quasi-minimal prime with ≥3 digits are:

(1,1), (1,3), (1,5), (1,7), (2,1), (2,3), (2,5), (2,7), (3,1), (3,3), (3,5), (3,7), (4,1), (4,3), (4,5), (4,7), (5,1), (5,3), (5,5), (5,7), (6,1), (6,3), (6,5), (6,7), (7,1), (7,3), (7,5), (7,7)

  • Case (1,1):
    • Since 13, 15, 21, 51, 111, 141, 161 are primes, we only need to consider the family 1{0,7}1 (since any digits 1, 2, 3, 4, 5, 6 between them will produce smaller primes)
      • Since 107, 177, 701 are primes, we only need to consider the number 171 and the family 1{0}1 (since any digits combo 07, 70, 77 between them will produce smaller primes)
        • 171 is not prime.
        • All numbers of the form 1{0}1 factored as 10^n+1 = (2^n+1) * (4^n-2^n+1) (n≥1) (and since if n≥1, 2^n+1 ≥ 2^1+1 = 3 > 1, 4^n-2^n+1 ≥ 4^1-2^1+1 = 3 > 1, this factorization is nontrivial), thus cannot be prime.
  • Case (1,3):
    • 13 is prime, and thus the only minimal prime in this family.
  • Case (1,5):
    • 15 is prime, and thus the only minimal prime in this family.
  • Case (1,7):
    • Since 13, 15, 27, 37, 57, 107, 117, 147, 177 are primes, we only need to consider the family 1{6}7 (since any digits 0, 1, 2, 3, 4, 5, 7 between them will produce smaller primes)
      • The smallest prime of the form 1{6}7 is 16667 (not minimal prime, since 667 is prime)
  • Case (2,1):
    • 21 is prime, and thus the only minimal prime in this family.
  • Case (2,3):
    • 23 is prime, and thus the only minimal prime in this family.
  • Case (2,5):
    • Since 21, 23, 27, 15, 35, 45, 65, 75, 225, 255 are primes, we only need to consider the family 2{0}5 (since any digits 1, 2, 3, 4, 5, 6, 7 between them will produce smaller primes)
      • All numbers of the form 2{0}5 are divisible by 7, thus cannot be prime.
  • Case (2,7):
    • 27 is prime, and thus the only minimal prime in this family.
  • Case (3,1):
    • Since 35, 37, 21, 51, 301, 361 are primes, we only need to consider the family 3{1,3,4}1 (since any digits 0, 2, 5, 6, 7 between them will produce smaller primes)
      • Since 13, 343, 111, 131, 141, 431, 3331, 3411 are primes, we only need to consider the families 3{3}11, 33{1,4}1, 3{3,4}4{4}1 (since any digits combo 11, 13, 14, 33, 41, 43 between them will produce smaller primes)
        • All numbers of the form 3{3}11 are divisible by 3, thus cannot be prime.
        • For the 33{1,4}1 family, since 111 and 141 are primes, we only need to consider the families 33{4}1 and 33{4}11 (since any digits combo 11, 14 between them will produce smaller primes)
          • The smallest prime of the form 33{4}1 is 3344441
          • All numbers of the form 33{4}11 are divisible by 301, thus cannot be prime.
        • For the 3{3,4}4{4}1 family, since 3331 and 3344441 are primes, we only need to consider the families 3{4}1, 3{4}31, 3{4}341, 3{4}3441, 3{4}34441 (since any digits combo 33 or 34444 between (3,1) will produce smaller primes)
          • All numbers of the form 3{4}1 are divisible by 31, thus cannot be prime.
          • Since 4443 is prime, we only need to consider the numbers 3431, 34431, 34341, 344341, 343441, 3443441, 3434441, 34434441 (since any digit combo 444 between (3,3{4}1) will produce smaller primes)
            • None of 3431, 34431, 34341, 344341, 343441, 3443441, 3434441, 34434441 are primes.
  • Case (3,3):
    • Since 35, 37, 13, 23, 53, 73, 343 are primes, we only need to consider the family 3{0,3,6}3 (since any digits 1, 2, 4, 5, 7 between them will produce smaller primes)
      • All numbers of the form 3{0,3,6}3 are divisible by 3, thus cannot be prime.
  • Case (3,5):
    • 35 is prime, and thus the only minimal prime in this family.
  • Case (3,7):
    • 37 is prime, and thus the only minimal prime in this family.
  • Case (4,1):
    • Since 45, 21, 51, 401, 431, 471 are primes, we only need to consider the family 4{1,4,6}1 (since any digits 0, 2, 3, 5, 7 between them will produce smaller primes)
      • Since 111, 141, 161, 661, 4611 are primes, we only need to consider the families 4{4}11, 4{4,6}4{1,4,6}1, 4{4}6{4}1 (since any digits combo 11, 14, 16, 61, 66 between them will produce smaller primes)
        • The smallest prime of the form 4{4}11 is 44444444444444411 (not minimal prime, since 444444441 is prime)
        • For the 4{4,6}4{1,4,6}1 family, we can separate this family to 4{4,6}41, 4{4,6}411, 4{4,6}461
          • For the 4{4,6}41 family, since 661 and 6441 are primes, we only need to consider the families 4{4}41 and 4{4}641 (since any digits combo 64 or 66 between (4,41) will produce smaller primes)
            • The smallest prime of the form 4{4}41 is 444444441
            • The smallest prime of the form 4{4}641 is 444641
          • For the 4{4,6}411 family, since 661 and 6441 are primes, we only need to consider the families 4{4}411 and 4{4}6411 (since any digits combo 64 or 66 between (4,411) will produce smaller primes)
            • The smallest prime of the form 4{4}411 is 444444441
            • The smallest prime of the form 4{4}6411 is 4444444444444446411 (not minimal prime, since 444444441 and 444641 are primes)
          • For the 4{4,6}461 family, since 661 is prime, we only need to consider the family 4{4}461
            • The smallest prime of the form 4{4}461 is 4444444461 (not minimal prime, since 444444441 is prime)
        • For the 4{4}6{4}1 family, since 6441 is prime, we only need to consider the families 4{4}61 and 4{4}641 (since any digits combo 44 between (4{4}6,1) will produce smaller primes)
          • The smallest prime of the form 4{4}61 is 4444444461 (not minimal prime, since 444444441 is prime)
          • The smallest prime of the form 4{4}641 is 444641
  • Case (4,3):
    • Since 45, 13, 23, 53, 73, 433, 463 are primes, we only need to consider the family 4{0,4}3 (since any digits 1, 2, 3, 5, 6, 7 between them will produce smaller primes)
      • Since 4043 and 4443 are primes, we only need to consider the families 4{0}3 and 44{0}3 (since any digits combo 04, 44 between them will produce smaller primes)
        • All numbers of the form 4{0}3 are divisible by 7, thus cannot be prime.
        • All numbers of the form 44{0}3 are divisible by 3, thus cannot be prime.
  • Case (4,5):
    • 45 is prime, and thus the only minimal prime in this family.
  • Case (4,7):
    • Since 45, 27, 37, 57, 407, 417, 467 are primes, we only need to consider the family 4{4,7}7 (since any digits 0, 1, 2, 3, 5, 6 between them will produce smaller primes)
      • Since 747 is prime, we only need to consider the families 4{4}7, 4{4}77, 4{7}7, 44{7}7 (since any digits combo 74 between (4,7) will produce smaller primes)
        • The smallest prime of the form 4{4}7 is 44444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444447, with 220 4's, which can be written as (4^220)7 and equal the prime (4*8^221+17)/7 (factordb)
        • The smallest prime of the form 4{4}77 is 4444477
        • The smallest prime of the form 4{7}7 is 47777
        • The smallest prime of the form 44{7}7 is 4477777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777, with 851 7's, which can be written as 44(7^851) and equal the prime 37*8^851-1 (factordb) (not minimal prime, since 47777 is prime)
  • Case (5,1):
    • 51 is prime, and thus the only minimal prime in this family.
  • Case (5,3):
    • 53 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, 5205 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 500025 and 505525 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 555555555555525
          • The smallest prime of the form 5{5}025 is 55555025
          • The smallest prime of the form 5{5}0025 is 5555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555550025, with 184 5's, which can be written as (5^183)0025 and equal the prime (5*8^187-20333)/7 (factordb) (not minimal prime, since 55555025 and 555555555555525 are primes)
          • The smallest prime of the form 5{5}0525 is 5550525
          • The smallest prime of the form 5{5}00525 is 5500525
          • The smallest prime of the form 5{5}05025 is 5555555555555555555555505025 (not minimal prime, since 5550525, 55555025, and 555555555555525 are primes)
  • Case (5,7):
    • 57 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)
      • Numbers containing 4: (note that the number cannot contain two or more 4's, or 6441 will be a subsequence)
        • The form is 6{0,1,7}4{0,1,7}1
          • Since 141, 401, 471 are primes, we only need to consider the family 6{0,7}4{1}1
            • Since 111 is prime, we only need to consider the families 6{0,7}41 and 6{0,7}411
              • For the 6{0,7}41 family, since 60741 is prime, we only need to consider the family 6{7}{0}41
                • Since 6777 is prime, we only need to consider the families 6{0}41, 67{0}41, 677{0}41
                  • All numbers of the form 6{0}41 are divisible by 3, thus cannot be prime.
                  • All numbers of the form 67{0}41 are divisible by 13, thus cannot be prime.
                  • All numbers of the form 677{0}41 are divisible by 3, thus cannot be prime.
              • For the 6{0,7}411 family, since 60411 is prime, we only need to consider the family 6{7}411
                • The smallest prime of the form 6{7}411 is 67777411 (not minimal prime, since 6777 is prime)
      • Numbers not containing 4:
        • The form is 6{0,1,7}1
          • Since 111 is prime, we only need to consider the families 6{0,7}1 and 6{0,7}1{0,7}1
            • All numbers of the form 6{0,7}1 are divisible by 7, thus cannot be prime.
            • For the 6{0,7}1{0,7}1 family, since 711 and 6101 are primes, we only need to consider the family 6{0}1{7}1
              • Since 60171 is prime, we only need to consider the families 6{0}11 and 61{7}1
                • All numbers of the form 6{0}11 are divisible by 3, thus cannot be prime.
                • The smallest prime of the form 61{7}1 is 617771 (not minimal prime, since 6777 is prime)
  • Case (6,3):
    • Since 65, 13, 23, 53, 73, 643 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):
    • 65 is prime, and thus the only minimal prime in this family.
  • Case (6,7):
    • Since 65, 27, 37, 57, 667 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, 6007, 6477, 6707, 6777 are primes, there cannot be digits combo 00, 10, 11, 14, 17, 40, 41, 47, 70, 71, 74, 77 between them
        • If there is 1 between them, then there cannot be 1, 4, 7 before it and cannot be 0, 1, 4, 7 after it, thus the form will be 6{0}17
          • All numbers of the form 6{0}17 are divisible by 3, thus cannot be prime.
        • If there is 7 between them, then there cannot be 1, 4, 7 before it and cannot be 0, 1, 4, 7 after it, thus the form will be 6{0}77
          • All numbers of the form 6{0}77 are divisible by 3, thus cannot be prime.
        • If there is neither 1 nor 7 between them, then the form is 6{0,4}7
          • Since 6007, 407 at primes, we only need to consider the families 6{4}7 and 60{4}7 (since any digits combo 00, 40 between them 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, 701, 711 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, 7461, 7641 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 7777777777771
          • The smallest prime of the form 7{7}41 is 777777777777777777777777777777777777777777777777777777777777777777777777777777741, with 79 7's, which can be written as (7^79)41 and equal the prime 8^81-31 (factordb) (not minimal prime, since 7777777777771 is prime)
          • The smallest prime of the form 7{7}441 is 777777777777777777777777777777777777777777777777777777777777777777777777777777777777441, with 84 7's, which can be written as (7^84)441 and equal the prime 8^87-223 (factordb) (not minimal prime, since 7777777777771 is prime)
          • The smallest prime of the form 7{7}4441 is 777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777774441, with 233 7's, which can be written as (7^233)4441 and equal the prime 8^237-1759 (factordb) (not minimal prime, since 7777777777771 is prime)
          • The smallest prime of the form 7{7}44441 is 7777777777777777777777777777777777777777777777777777777744441, with 56 7's, which can be written as (7^56)44441 and equal the prime 8^61-14047 (factordb) (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 77774444441
            • 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 744444441
              • 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):
    • 73 is prime, and thus the only minimal prime in this family.
  • Case (7,5):
    • 75 is prime, and thus the only minimal prime in this family.
  • Case (7,7):
    • Since 73, 75, 27, 37, 57, 717, 747, 767 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.

Base 10[edit | edit source]

The possible (first digit,last digit) combo for a quasi-minimal prime with ≥3 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.
  • 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.
  • 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.
  • 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.
  • 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

Base 12[edit | edit source]

The possible (first digit,last digit) combo for a quasi-minimal prime with ≥3 digits are:

(1,1), (1,5), (1,7), (1,B), (2,1), (2,5), (2,7), (2,B), (3,1), (3,5), (3,7), (3,B), (4,1), (4,5), (4,7), (4,B), (5,1), (5,5), (5,7), (5,B), (6,1), (6,5), (6,7), (6,B), (7,1), (7,5), (7,7), (7,B), (8,1), (8,5), (8,7), (8,B), (9,1), (9,5), (9,7), (9,B), (A,1), (A,5), (A,7), (A,B), (B,1), (B,5), (B,7), (B,B)

  • Case (1,1):
    • 11 is prime, and thus the only minimal prime in this family.
  • Case (1,5):
    • 15 is prime, and thus the only minimal prime in this family.
  • Case (1,7):
    • 17 is prime, and thus the only minimal prime in this family.
  • Case (1,B):
    • 1B is prime, and thus the only minimal prime in this family.
  • Case (2,1):
    • Since 25, 27, 11, 31, 51, 61, 81, 91, 221, 241, 2A1, 2B1 are primes, we only need to consider the family 2{0}1 (since any digits 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B between them will produce smaller primes)
      • The smallest prime of the form 2{0}1 is 2001
  • Case (2,5):
    • 25 is prime, and thus the only minimal prime in this family.
  • Case (2,7):
    • 27 is prime, and thus the only minimal prime in this family.
  • Case (2,B):
    • Since 25, 27, 1B, 3B, 4B, 5B, 6B, 8B, AB, 2BB are primes, we only need to consider the family 2{0,2,9}B (since any digits 1, 3, 4, 5, 6, 7, 8, A, B between them will produce smaller primes)
      • Since 90B, 200B, 202B, 222B, 229B, 292B, 299B are primes, we only need to consider the numbers 20B, 22B, 29B, 209B, 220B (since any digits combo 00, 02, 22, 29, 90, 92, 99 between them will produce smaller primes)
        • None of 20B, 22B, 29B, 209B, 220B are primes.
  • Case (3,1):
    • 31 is prime, and thus the only minimal prime in this family.
  • Case (3,5):
    • 35 is prime, and thus the only minimal prime in this family.
  • Case (3,7):
    • 37 is prime, and thus the only minimal prime in this family.
  • Case (3,B):
    • 3B is prime, and thus the only minimal prime in this family.
  • Case (4,1):
    • Since 45, 4B, 11, 31, 51, 61, 81, 91, 401, 421, 471 are primes, we only need to consider the family 4{4,A}1 (since any digit 0, 1, 2, 3, 5, 6, 7, 8, 9, B between them will produce smaller primes)
      • Since A41 and 4441 are primes, we only need to consider the families 4{A}1 and 44{A}1 (since any digit combo 44, A4 between them will produce smaller primes)
        • All numbers of the form 4{A}1 are divisible by 5, thus cannot be prime.
        • The smallest prime of the form 44{A}1 is 44AAA1
  • Case (4,5):
    • 45 is prime, and thus the only minimal prime in this family.
  • Case (4,7):
    • Since 45, 4B, 17, 27, 37, 57, 67, 87, A7, B7, 447, 497 are primes, we only need to consider the family 4{0,7}7 (since any digit 1, 2, 3, 4, 5, 6, 8, 9, A, B between them will produce smaller primes)
      • Since 4707 and 4777 are primes, we only need to consider the families 4{0}7 and 4{0}77 (since any digit combo 70, 77 between them will produce smaller primes)
        • All numbers of the form 4{0}7 are divisible by B, thus cannot be prime.
        • The smallest prime of the form 4{0}77 is 400000000000000000000000000000000000000077
  • Case (4,B):
    • 4B is prime, and thus the only minimal prime in this family.
  • Case (5,1):
    • 51 is prime, and thus the only minimal prime in this family.
  • Case (5,5):
    • Since 51, 57, 5B, 15, 25, 35, 45, 75, 85, 95, B5, 565 are primes, we only need to consider the family 5{0,5,A}5 (since any digits 1, 2, 3, 4, 6, 7, 8, 9, B between them will produce smaller primes)
      • All numbers of the form 5{0,5,A}5 are divisible by 5, thus cannot be prime.
  • Case (5,7):
    • 57 is prime, and thus the only minimal prime in this family.
  • Case (5,B):
    • 5B is prime, and thus the only minimal prime in this family.
  • Case (6,1):
    • 61 is prime, and thus the only minimal prime in this family.
  • Case (6,5):
    • Since 61, 67, 6B, 15, 25, 35, 45, 75, 85, 95, B5, 655, 665 are primes, we only need to consider the family 6{0,A}5 (since any digits 1, 2, 3, 4, 5, 6, 7, 8, 9, B between them will produce smaller primes)
      • Since 6A05 and 6AA5 are primes, we only need to consider the families 6{0}5 and 6{0}A5 (since any digit combo A0, AA between them will produce smaller primes)
        • All numbers of the form 6{0}5 are divisible by B, thus cannot be prime.
        • The smallest prime of the form 6{0}A5 is 600A5
  • Case (6,7):
    • 67 is prime, and thus the only minimal prime in this family.
  • Case (6,B):
    • 6B is prime, and thus the only minimal prime in this family.
  • Case (7,1):
    • Since 75, 11, 31, 51, 61, 81, 91, 701, 721, 771, 7A1 are primes, we only need to consider the family 7{4,B}1 (since any digits 0, 1, 2, 3, 5, 6, 7, 8, 9, A between them will produce smaller primes)
      • Since 7BB, 7441 and 7B41 are primes, we only need to consider the numbers 741, 7B1, 74B1
        • None of 741, 7B1, 74B1 are primes.
  • Case (7,5):
    • 75 is prime, and thus the only minimal prime in this family.
  • Case (7,7):
    • Since 75, 17, 27, 37, 57, 67, 87, A7, B7, 747, 797 are primes, we only need to consider the family 7{0,7}7 (since any digits 1, 2, 3, 4, 5, 6, 8, 9, A, B 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,B):
    • Since 75, 1B, 3B, 4B, 5B, 6B, 8B, AB, 70B, 77B, 7BB are primes, we only need to consider the family 7{2,9}B (since any digits 0, 1, 3, 4, 5, 6, 7, 8, A, B between them will produce smaller primes)
      • Since 222B, 729B is prime, we only need to consider the families 7{9}B, 7{9}2B, 7{9}22B (since any digits combo 222, 29 between them will produce smaller primes)
        • The smallest prime of the form 7{9}B is 7999B
        • The smallest prime of the form 7{9}2B is 79992B (not minimal prime, since 992B and 7999B are primes)
        • The smallest prime of the form 7{9}22B is 79922B (not minimal prime, since 992B is prime)
  • Case (8,1):
    • 81 is prime, and thus the only minimal prime in this family.
  • Case (8,5):
    • 85 is prime, and thus the only minimal prime in this family.
  • Case (8,7):
    • 87 is prime, and thus the only minimal prime in this family.
  • Case (8,B):
    • 8B is prime, and thus the only minimal prime in this family.
  • Case (9,1):
    • 91 is prime, and thus the only minimal prime in this family.
  • Case (9,5):
    • 95 is prime, and thus the only minimal prime in this family.
  • Case (9,7):
    • Since 91, 95, 17, 27, 37, 57, 67, 87, A7, B7, 907 are primes, we only need to consider the family 9{4,7,9}7 (since any digit 0, 1, 2, 3, 5, 6, 8, A, B between them will produce smaller primes)
      • Since 447, 497, 747, 797, 9777, 9947, 9997 are primes, we only need to consider the numbers 947, 977, 997, 9477, 9977 (since any digits combo 44, 49, 74, 77, 79, 94, 99 between them will produce smaller primes)
        • None of 947, 977, 997, 9477, 9977 are primes.
  • Case (9,B):
    • Since 91, 95, 1B, 3B, 4B, 5B, 6B, 8B, AB, 90B, 9BB are primes, we only need to consider the family 9{2,7,9}B (since any digit 0, 1, 3, 4, 5, 6, 8, A, B between them will produce smaller primes)
      • Since 27, 77B, 929B, 992B, 997B are primes, we only need to consider the families 9{2,7}2{2}B, 97{2,9}B, 9{7,9}9{9}B (since any digits combo 27, 29, 77, 92, 97 between them will produce smaller primes)
        • For the 9{2,7}2{2}B family, since 27 and 77B are primes, we only need to consider the families 9{2}2{2}B and 97{2}2{2}B (since any digits combo 27, 77 between (9,2{2}B) will produce smaller primes)
          • The smallest prime of the form 9{2}2{2}B is 9222B (not minimal prime, since 222B is prime)
          • The smallest prime of the form 97{2}2{2}B is 9722222222222B (not minimal prime, since 222B is prime)
        • For the 97{2,9}B family, since 729B and 929B are primes, we only need to consider the family 97{9}{2}B (since any digits combo 29 between (97,B) will produce smaller primes)
          • Since 222B is prime, we only need to consider the families 97{9}B, 97{9}2B, 97{9}22B (since any digit combo 222 between (97,B) will produce smaller primes)
            • All numbers of the form 97{9}B are divisible by 11, thus cannot be prime.
            • The smallest prime of the form 97{9}2B is 979999992B (not minimal prime, since 9999B is prime)
            • All numbers of the form 97{9}22B are divisible by 11, thus cannot be prime.
        • For the 9{7,9}9{9}B family, since 77B and 9999B are primes, we only need to consider the numbers 99B, 999B, 979B, 9799B, 9979B
          • None of 99B, 999B, 979B, 9799B, 9979B are primes.
  • Case (A,1):
    • Since A7, AB, 11, 31, 51, 61, 81, 91, A41 are primes, we only need to consider the family A{0,2,A}1 (since any digits 1, 3, 4, 5, 6, 7, 8, 9, B between them will produce smaller primes)
      • Since 221, 2A1, A0A1, A201 are primes, we only need to consider the families A{A}{0}1 and A{A}{0}21 (since any digits combo 0A, 20, 22, 2A between them will produce smaller primes)
        • For the A{A}{0}1 family:
          • All numbers of the form A{0}1 are divisible by B, thus cannot be prime.
          • The smallest prime of the form AA{0}1 is AA000001
          • The smallest prime of the form AAA{0}1 is AAA0001
          • The smallest prime of the form AAAA{0}1 is AAAA1
            • Since this prime has no 0's, we do not need to consider the families {A}1, {A}01, {A}001, etc.
        • All numbers of the form A{A}{0}21 are divisible by 5, thus cannot be prime.
  • Case (A,5):
    • Since A7, AB, 15, 25, 35, 45, 75, 85, 95, B5 are primes, we only need to consider the family A{0,5,6,A}5 (since any digits 1, 2, 3, 4, 7, 8, 9, B between them will produce smaller primes)
      • Since 565, 655, 665, A605, A6A5, AA65 are primes, we only need to consider the families A{0,5,A}5 and A{0}65 (since any digits combo 56, 60, 65, 66, 6A, A6 between them will produce smaller primes)
        • All numbers of the form A{0,5,A}5 are divisible by 5, thus cannot be prime.
        • The smallest prime of the form A{0}65 is A00065
  • Case (A,7):
    • A7 is prime, and thus the only minimal prime in this family.
  • Case (A,B):
    • AB is prime, and thus the only minimal prime in this family.
  • Case (B,1):
    • Since B5, B7, 11, 31, 51, 61, 81, 91, B21 are primes, we only need to consider the family B{0,4,A,B}1 (since any digits 1, 2, 3, 5, 6, 7, 8, 9 between them will produce smaller primes)
      • Since 4B, AB, 401, A41, B001, B0B1, BB01, BB41 are primes, we only need to consider the families B{A}0{4,A}1, B{0,4}4{4,A}1, B{0,4,A,B}A{0,A}1, B{B}B{A,B}1 (since any digits combo 00, 0B, 40, 4B, A4, AB, B0, B4 between them will produce smaller primes)
        • For the B{A}0{4,A}1 family, since A41 is prime, we only need consider the families B0{4}{A}1 and B{A}0{A}1
          • For the B0{4}{A}1 family, since B04A1 is prime, we only need to consider the families B0{4}1 and B0{A}1
            • The smallest prime of the form B0{4}1 is B04441 (not minimal prime, since 4441 is prime)
            • The smallest prime of the form B0{A}1 is B0AAAAA1 (not minimal prime, since AAAA1 is prime)
          • For the B{A}0{A}1 family, since A0A1 is prime, we only need to consider the families B{A}01 and B0{A}1
            • The smallest prime of the form B{A}01 is BAA01
            • The smallest prime of the form B0{A}1 is B0AAAAA1 (not minimal prime, since AAAA1 is prime)
        • For the B{0,4}4{4,A}1 family, since 4441 is prime, we only need to consider the families B{0}4{4,A}1 and B{0,4}4{A}1
          • For the B{0}4{4,A}1 family, since B001 is prime, we only need to consider the families B4{4,A}1 and B04{4,A}1
            • For the B4{4,A}1 family, since A41 is prime, we only need to consider the family B4{4}{A}1
              • Since 4441 and BAAA1 are primes, we only need to consider the numbers B41, B441, B4A1, B44A1, B4AA1, B44AA1
                • None of B41, B441, B4A1, B44A1, B4AA1, B44AA1 are primes.
            • For the B04{4,A}1 family, since B04A1 is prime, we only need to consider the family B04{4}1
              • The smallest prime of the form B04{4}1 is B04441 (not minimal prime, since 4441 is prime)
          • For the B{0,4}4{A}1 family, since 401, 4441, B001 are primes, we only need to consider the families B4{A}1, B04{A}1, B44{A}1, B044{A}1 (since any digits combo 00, 40, 44 between (B,4{A}1) will produce smaller primes)
            • The smallest prime of the form B4{A}1 is B4AAA1 (not minimal prime, since BAAA1 is prime)
            • The smallest prime of the form B04{A}1 is B04A1
            • The smallest prime of the form B44{A}1 is B44AAAAAAA1 (not minimal prime, since BAAA1 is prime)
            • The smallest prime of the form B044{A}1 is B044A1 (not minimal prime, since B04A1 is prime)
        • For the B{0,4,A,B}A{0,A}1 family, since all numbers in this family with 0 between (B,1) are in the B{A}0{4,A}1 family, and all numbers in this family with 4 between (B,1) are in the B{0,4}4{4,A}1 family, we only need to consider the family B{A,B}A{A}1
          • Since BAAA1 is prime, we only need to consider the families B{A,B}A1 and B{A,B}AA1
            • For the B{A,B}A1 family, since AB and BAAA1 are primes, we only need to consider the families B{B}A1 and B{B}AA1
              • All numbers of the form B{B}A1 are divisible by B, thus cannot be prime.
              • The smallest prime of the form B{B}AA1 is BBBAA1
            • For the B{A,B}AA1 family, since BAAA1 is prime, we only need to consider the families B{B}AA1
              • The smallest prime of the form B{B}AA1 is BBBAA1
        • For the B{B}B{A,B}1 family, since AB and BAAA1 are primes, we only need to consider the families B{B}B1, B{B}BA1, B{B}BAA1 (since any digits combo AB or AAA between (B{B}B,1) will produce smaller primes)
          • The smallest prime of the form B{B}B1 is BBBB1
          • All numbers of the form B{B}BA1 are divisible by B, thus cannot be prime.
          • The smallest prime of the form B{B}BAA1 is BBBAA1
  • Case (B,5):
    • B5 is prime, and thus the only minimal prime in this family.
  • Case (B,7):
    • B7 is prime, and thus the only minimal prime in this family.
  • Case (B,B):
    • Since B5, B7, 1B, 3B, 4B, 5B, 6B, 8B, AB, B2B are primes, we only need to consider the family B{0,9,B}B (since any digits 1, 2, 3, 4, 5, 6, 7, 8, A between them will produce smaller primes)
      • Since 90B and 9BB are primes, we only need to consider the families B{0,B}{9}B
        • Since 9999B is prime, we only need to consider the families B{0,B}B, B{0,B}9B, B{0,B}99B, B{0,B}999B
          • All numbers of the form B{0,B}B are divisible by B, thus cannot be prime.
          • For the B{0,B}9B family:
            • Since B0B9B and BB09B are primes, we only need to consider the families B{0}9B and B{B}9B (since any digits combo 0B, B0 between (B,9B) will produce smaller primes)
              • The smallest prime of the form B{0}9B is B0000000000000000000000000009B
              • All numbers of the from B{B}9B is either divisible by 11 (if totally number of B's is even) or factored as 10^(2*n)-21 = (10^n-5) * (10^n+5) (if totally number of B's is odd number 2*n-1 (n≥1)) (and since if n≥1, 10^n-5 ≥ 10^1-5 = 7 > 1, 10^n+5 ≥ 10^1+5 = 15 > 1, this factorization is nontrivial), thus cannot be prime.
          • For the B{0,B}99B family:
            • Since B0B9B and BB09B are primes, we only need to consider the families B{0}99B and B{B}99B (since any digits combo 0B, B0 between (B,99B) will produce smaller primes)
              • The smallest prime of the form B{0}99B is B00099B
              • The smallest prime of the form B{B}99B is BBBBBB99B
          • For the B{0,B}999B family:
            • Since B0B9B and BB09B are primes, we only need to consider the families B{0}999B and B{B}999B (since any digits combo 0B, B0 between (B,999B) will produce smaller primes)
              • The smallest prime of the form B{0}999B is B0000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000999B, with 1765 0's, which can be written as B(0^1765)999B and equal the prime 11*12^1769+16967 (factordb) (primality certificate) (not minimal prime, since B00099B and B0000000000000000000000000009B are primes)
              • The smallest prime of the form B{B}999B is BBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBB999B, with 245 B's, which can be written as (B^244)999B and equal the prime 12^248-3769 (factordb) (not minimal prime, since BBBBBB99B is prime)

Examples of families which can be ruled out as contain no primes > b[edit | edit source]

It is not known if this problem is solvable:

Problem: Given strings x, y, z, and a base b, does there exist a prime number whose base-b expansion is of the form x{y}z?

It will be necessary for our algorithm to determine if families of the form x{y}z contain a prime > b or not. We use two different heuristic strategies to show that such families contain no primes > b.

In the first strategy, we mimic the well-known technique of “covering congruences”, by finding some finite set S of primes p such that every number in a given family is divisible by some element of S. In the second strategy, we attempt to find an algebraic factorization, such as difference-of-squares factorization, difference-of-cubes factorization, and Aurifeuillian factorization for numbers of the form x4+4y4.

Examples of first strategy: (we can show that the corresponding numbers are > all elements in S, if n makes corresponding numbers > b (i.e. n≥1 for 5{1} in base 9 and 2{5} in base 11 and {4}D in base 16 and {8}F in base 16, n≥0 for other examples), thus these factorizations are nontrivial)

  • In base 10, all numbers of the form 4{6}9 are divisible by 7
  • In base 6, all numbers of the form 4{0}1 are divisible by 5
  • In base 15, all numbers of the form 9{6}8 are divisible by 11
  • In base 9, all numbers of the form 5{1} are divisible by some element of {2, 5}
  • In base 11, all numbers of the form 2{5} are divisible by some element of {2, 3}
  • In base 14, all numbers of the form B{0}1 are divisible by some element of {3, 5}
  • In base 8, all numbers of the form 6{4}7 are divisible by some element of {3, 5, 13}
  • In base 13, all numbers of the form 3{0}95 are divisible by some element of {5, 7, 17}
  • In base 16, all numbers of the form {4}D are divisible by some element of {3, 7, 13}
  • In base 16, all numbers of the form {8}F are divisible by some element of {3, 7, 13}

Examples of second strategy: (we can show that both factors are > 1, if n makes corresponding numbers > b (i.e. n≥2 for {1} in base 9, n≥0 for 1{0}1 in base 8 and B{4}1 in base 16, n≥1 for other examples), thus these factorizations are nontrivial)

  • In base 9, all numbers of the form {1} factored as difference of squares
  • In base 8, all numbers of the form 1{0}1 factored as sum of cubes
  • In base 9, all numbers of the form 3{8} factored as difference of squares
  • In base 16, all numbers of the form 8{F} factored as difference of squares
  • In base 16, all numbers of the form {F}7 factored as difference of squares
  • In base 9, all numbers of the form 3{1} factored as difference of squares
  • In base 16, all numbers of the form {4}1 factored as difference of squares
  • In base 16, all numbers of the form 1{5} factored as difference of squares
  • In base 16, all numbers of the from {C}D factored as x4+4y4
  • In base 16, all numbers of the form B{4}1 factored as difference of squares

Examples of combine of the two strategies: (we can show that for the part of the first strategy, the corresponding numbers are > all elements in S, and for the part of the second strategy, both factors are > 1, if n makes corresponding numbers > b, thus these factorizations are nontrivial)

  • In base 14, numbers of the form 8{D} are divisible by 5 if n is odd and factored as difference of squares if n is even
  • In base 12, numbers of the form {B}9B are divisible by 13 if n is odd and factored as difference of squares if n is even
  • In base 14, numbers of the form {D}5 are divisible by 5 if n is even and factored as difference of squares if n is odd
  • In base 17, numbers of the form 1{9} are divisible by 2 if n is odd and factored as difference of squares if n is even
  • In base 19, numbers of the form 1{6} are divisible by 5 if n is odd and factored as difference of squares if n is even

Bases 2≤b≤1024 such that these families can be ruled out as contain no primes > b[edit | edit source]

(using A−Z to represent digit values 10 to 35, z−a to represent digit values b−1 to b−26 (e.g. "z" means 1 in base 2, 2 in base 3, 3 in base 4, ..., 8 in base 9, 9 in base 10, A in base 11, B in base 12, ..., Y in base 35, Z in base 36, ...), only consider bases which these families are interpretable, e.g. digit "7" is only interpretable for bases ≥8, and digit "u" (means b−6) is only interpretable for bases ≥7)

1{0}1[edit | edit source]

  • b == 1 mod 2: Finite covering set {2}
  • b = mr with odd r>1: Sum-of-rth-powers factorization

1{0}2[edit | edit source]

  • b == 0 mod 2: Finite covering set {2}
  • b == 1 mod 3: Finite covering set {3}

1{0}3[edit | edit source]

  • b == 1 mod 2: Finite covering set {2}
  • b == 0 mod 3: Finite covering set {3}

1{0}4[edit | edit source]

  • b == 0 mod 2: Finite covering set {2}
  • b == 1 mod 5: Finite covering set {5}
  • b == 14 mod 15: Finite covering set {3, 5}
  • b = m4: Aurifeuillian factorization of x4+4y4

1{0}5[edit | edit source]

  • b == 1 mod 2: Finite covering set {2}
  • b == 1 mod 3: Finite covering set {3}
  • b == 0 mod 5: Finite covering set {5}

1{0}6[edit | edit source]

  • b == 0 mod 2: Finite covering set {2}
  • b == 0 mod 3: Finite covering set {3}
  • b == 1 mod 7: Finite covering set {7}

1{0}7[edit | edit source]

  • b == 1 mod 2: Finite covering set {2}
  • b == 0 mod 7: Finite covering set {7}

1{0}z[edit | edit source]

(none)

1{0}11 (not quasi-minimal prime if there is smaller prime of the form 1{0}1)[edit | edit source]

  • b == 1 mod 3: Finite covering set {3}

10{z} (not quasi-minimal prime if there is smaller prime of the form 1{z})[edit | edit source]

(none)

11{0}1 (not quasi-minimal prime if there is smaller prime of the form 1{0}1)[edit | edit source]

  • b == 1 mod 3: Finite covering set {3}

{1}0z (not quasi-minimal prime if there is smaller prime of the form {1} or {1}z)[edit | edit source]

  • b such that b and 2b−1 are both squares: Difference-of-squares factorization (such bases are 25, 841)

{1}[edit | edit source]

  • b = mr with r>1: Difference-of-rth-powers factorization (some bases still have primes, since for the corresponding length this factorization is trivial, but they only have this prime, they are 4 (length 2), 8 (length 3), 16 (length 2), 27 (length 3), 36 (length 2), 100 (length 2), 128 (length 7), 196 (length 2), 256 (length 2), 400 (length 2), 512 (length 3), 576 (length 2), 676 (length 2))

{1}2 (not quasi-minimal prime if there is smaller prime of the form {1})[edit | edit source]

  • b == 0 mod 2: Finite covering set {2}

1{2}[edit | edit source]

  • b == 0 mod 2: Finite covering set {2}
  • b such that b and 2(b+1) are both squares: Difference-of-squares factorization (such bases are 49)

1{3}[edit | edit source]

  • b == 0 mod 3: Finite covering set {3}
  • b such that b and 3(b+2) are both squares: Difference-of-squares factorization (such bases are 25, 361)
  • b == 1 mod 2 such that 3(b+2) is square: Combine of finite covering set {2} (when length is even) and difference-of-squares factorization (when length is odd) (such bases are 25, 73, 145, 241, 361, 505, 673, 865)

1{4}[edit | edit source]

  • b == 0 mod 2: Finite covering set {2}
  • b such that b and 4(b+3) are both squares: Difference-of-squares factorization

1{z}[edit | edit source]

(none)

2{0}1[edit | edit source]

  • b == 1 mod 3: Finite covering set {3}

2{0}3[edit | edit source]

  • b == 0 mod 3: Finite covering set {3}
  • b == 1 mod 5: Finite covering set {5}

2{1} (not quasi-minimal prime if there is smaller prime of the form {1})[edit | edit source]

  • b such that b and 2b−1 are both squares: Difference-of-squares factorization (such bases are 25, 841)

{2}1[edit | edit source]

  • b such that b and 2(b+1) are both squares: Difference-of-squares factorization (such bases are 49)

2{z}[edit | edit source]

  • b == 1 mod 2: Finite covering set {2}

3{0}1[edit | edit source]

  • b == 1 mod 2: Finite covering set {2}

3{0}2[edit | edit source]

  • b == 0 mod 2: Finite covering set {2}
  • b == 1 mod 5: Finite covering set {5}

3{0}4[edit | edit source]

  • b == 0 mod 2: Finite covering set {2}
  • b == 1 mod 7: Finite covering set {7}

{3}1[edit | edit source]

  • b such that b and 3(2b+1) are both squares: Difference-of-squares factorization (such bases are 121)

3{z}[edit | edit source]

  • b == 1 mod 3: Finite covering set {3}
  • b == 14 mod 15: Finite covering set {3, 5}
  • b = m2: Difference-of-squares factorization
  • b == 4 mod 5: Combine of finite covering set {5} (when length is even) and difference-of-squares factorization (when length is odd)

4{0}1[edit | edit source]

  • b == 1 mod 5: Finite covering set {5}
  • b == 14 mod 15: Finite covering set {3, 5}
  • b = m4: Aurifeuillian factorization of x4+4y4

4{0}3[edit | edit source]

  • b == 0 mod 3: Finite covering set {3}
  • b == 1 mod 7: Finite covering set {7}

{4}1[edit | edit source]

  • b such that b and 4(3b+1) are both squares: Difference-of-squares factorization (such bases are 16, 225)

4{z}[edit | edit source]

  • b == 1 mod 2: Finite covering set {2}

5{0}1[edit | edit source]

  • b == 1 mod 2: Finite covering set {2}
  • b == 1 mod 3: Finite covering set {3}

5{z}[edit | edit source]

  • b == 1 mod 5: Finite covering set {5}
  • b == 34 mod 35: Finite covering set {5, 7}
  • b = 6m2 with m == 2 or 3 mod 5: Combine of finite covering set {5} (when length is odd) and difference-of-squares factorization (when length is even) (such bases are 24, 54, 294, 384, 864, 1014)

6{0}1[edit | edit source]

  • b == 1 mod 7: Finite covering set {7}
  • b == 34 mod 35: Finite covering set {5, 7}

6{z}[edit | edit source]

  • b == 1 mod 2: Finite covering set {2}
  • b == 1 mod 3: Finite covering set {3}

7{0}1[edit | edit source]

  • b == 1 mod 2: Finite covering set {2}

7{z}[edit | edit source]

  • b == 1 mod 7: Finite covering set {7}
  • b == 20 mod 21: Finite covering set {3, 7}
  • b == 83, 307 mod 455: Finite covering set {5, 7, 13} (such bases are 83, 307, 538, 762, 993)
  • b = m3: Difference-of-cubes factorization

8{0}1[edit | edit source]

  • b == 1 mod 3: Finite covering set {3}
  • b == 20 mod 21: Finite covering set {3, 7}
  • b == 47, 83 mod 195: Finite covering set {3, 5, 13} (such bases are 47, 83, 242, 278, 437, 473, 632, 668, 827, 863, 1022)
  • b = 467: Finite covering set {3, 5, 7, 19, 37}
  • b = 722: Finite covering set {3, 5, 13, 73, 109}
  • b = m3: Sum-of-cubes factorization
  • b = 128: Cannot have primes since 7n+3 cannot be power of 2

8{z}[edit | edit source]

  • b == 1 mod 2: Finite covering set {2}
  • b = m2: Difference-of-squares factorization
  • b == 4 mod 5: Combine of finite covering set {5} (when length is even) and difference-of-squares factorization (when length is odd)

9{0}1[edit | edit source]

  • b == 1 mod 2: Finite covering set {2}
  • b == 1 mod 5: Finite covering set {5}

9{z}[edit | edit source]

  • b == 1 mod 3: Finite covering set {3}
  • b == 32 mod 33: Finite covering set {3, 11}

A{0}1[edit | edit source]

  • b == 1 mod 11: Finite covering set {11}
  • b == 32 mod 33: Finite covering set {3, 11}

A{z}[edit | edit source]

  • b == 1 mod 2: Finite covering set {2}
  • b == 1 mod 5: Finite covering set {5}
  • b == 14 mod 15: Finite covering set {3, 5}

B{0}1[edit | edit source]

  • b == 1 mod 2: Finite covering set {2}
  • b == 1 mod 3: Finite covering set {3}
  • b == 14 mod 15: Finite covering set {3, 5}

B{z}[edit | edit source]

  • b == 1 mod 11: Finite covering set {11}
  • b == 142 mod 143: Finite covering set {11, 13}
  • b = 307: Finite covering set {5, 11, 29}
  • b = 901: Finite covering set {7, 11, 13, 19}

C{0}1[edit | edit source]

  • b == 1 mod 13: Finite covering set {13}
  • b == 142 mod 143: Finite covering set {11, 13}
  • b = 296, 901: Finite covering set {7, 11, 13, 19}
  • b = 562, 828, 900: Finite covering set {7, 13, 19}
  • b = 563: Finite covering set {5, 7, 13, 19, 29}
  • b = 597: Finite covering set {5, 13, 29}

{#}$ (for bases b == 1 mod 3, # = (b−1)/3, $ = (b+2)/3)[edit | edit source]

(none)

{#}$ (for odd bases b, # = (b−1)/2, $ = (b+1)/2)[edit | edit source]

  • b = mr with odd r>1: Sum-of-rth-power factorization

#{z} (for even bases b, # = b/2−1)[edit | edit source]

(none)

y{z}[edit | edit source]

(none)

{y}z[edit | edit source]

(none)

z{0}1[edit | edit source]

(none)

{z0}z1 (almost cannot be quasi-minimal prime, since this is not simple family)[edit | edit source]

  • b = mr with odd r>1: Sum-of-rth-power factorization (some bases still have primes, since for the corresponding length this factorization is trivial, but they only have this prime, they are 128 (length 7), 216 (length 3), 343 (length 3), 729 (length 3))
  • b = 4m4: Aurifeuillian factorization of x4+4y4 (base 4 still have primes, since for the corresponding length this factorization is trivial, but it only have this prime, at length 2)

{z}yz (not quasi-minimal prime if there is smaller prime of the form {z}y)[edit | edit source]

(none)

{z}1[edit | edit source]

(none)

{z}t[edit | edit source]

  • b == 1 mod 2: Finite covering set {2}
  • b == 1 mod 3: Finite covering set {3}
  • b == 0 mod 7: Finite covering set {7}

{z}u[edit | edit source]

  • b == 0 mod 2: Finite covering set {2}
  • b == 0 mod 3: Finite covering set {3}
  • b == 1 mod 5: Finite covering set {5}
  • b == 34 mod 35: Finite covering set {5, 7}

{z}v[edit | edit source]

  • b == 1 mod 2: Finite covering set {2}
  • b == 0 mod 5: Finite covering set {5}

{z}w[edit | edit source]

  • b == 0 mod 2: Finite covering set {2}
  • b == 1 mod 3: Finite covering set {3}
  • b == 14 mod 15: Finite covering set {3, 5}
  • b = m2: Difference-of-squares factorization
  • b == 4 mod 5: Combine of finite covering set {5} (when length is even) and difference-of-squares factorization (when length is odd)

{z}x[edit | edit source]

  • b == 1 mod 2: Finite covering set {2}
  • b == 0 mod 3: Finite covering set {3}

{z}y[edit | edit source]

  • b == 0 mod 2: Finite covering set {2}

Large known (probable) primes (length ≥10000) in these families (for bases 2≤b≤1024)[edit | edit source]

Format: base (length)

(using A−Z to represent digit values 10 to 35, z−a to represent digit values b−1 to b−26 (e.g. "z" means 1 in base 2, 2 in base 3, 3 in base 4, ..., 8 in base 9, 9 in base 10, A in base 11, B in base 12, ..., Y in base 35, Z in base 36, ...), only consider bases which these families are interpretable, e.g. digit "7" is only interpretable for bases ≥8, and digit "u" (means b−6) is only interpretable for bases ≥7)

1{0}1[edit | edit source]

(none)

1{0}2[edit | edit source]

(none)

1{0}3[edit | edit source]

(none)

1{0}4[edit | edit source]

53 (13403)

113 (10647)

1{0}z[edit | edit source]

113 (20089)

123 (64371)

1{0}11 (not quasi-minimal prime if there is smaller prime of the form 1{0}1)[edit | edit source]

(none)

10{z} (not quasi-minimal prime if there is smaller prime of the form 1{z})[edit | edit source]

208 (26682)

607 (11032)

828 (19659)

11{0}1 (not quasi-minimal prime if there is smaller prime of the form 1{0}1)[edit | edit source]

201 (31276)

222 (52727)

227 (36323)

327 (135983)

425 (11231)

710 (24112)

717 (37508)

719 (13420)

{1}[edit | edit source]

152 (270217)

184 (16703)

200 (17807)

311 (36497)

326 (26713)

331 (25033)

371 (15527)

485 (99523)

629 (32233)

649 (43987)

670 (18617)

684 (22573)

691 (62903)

693 (41189)

731 (15427)

752 (32833)

872 (10093)

932 (20431)

{1}2 (not quasi-minimal prime if there is smaller prime of the form {1})[edit | edit source]

(none)

1{z}[edit | edit source]

107 (21911)

170 (166429)

278 (43909)

303 (40175)

383 (20957)

515 (58467)

522 (62289)

578 (129469)

590 (15527)

647 (21577)

662 (16591)

698 (127559)

704 (62035)

845 (39407)

938 (40423)

969 (24097)

989 (26869)

2{0}1[edit | edit source]

101 (192276)

206 (46206)

218 (333926)

236 (161230)

257 (12184)

305 (16808)

467 (126776)

578 (44166)

626 (174204)

695 (94626)

752 (26164)

788 (72918)

869 (49150)

887 (27772)

899 (15732)

932 (13644)

2{z}[edit | edit source]

432 (16003)

3{0}1[edit | edit source]

(none)

3{z}[edit | edit source]

72 (1119850)

212 (34414)

218 (23050)

270 (89662)

303 (198358)

312 (51566)

422 (21738)

480 (93610)

513 (38032)

527 (46074)

566 (23874)

650 (498102)

686 (16584)

758 (15574)

783 (12508)

800 (33838)

921 (98668)

947 (10056)

4{0}1[edit | edit source]

107 (32587)

227 (13347)

257 (160423)

355 (10990)

410 (144079)

440 (56087)

452 (14155)

482 (30691)

542 (15983)

579 (67776)

608 (20707)

635 (11723)

650 (96223)

679 (69450)

737 (269303)

740 (58043)

789 (149140)

797 (468703)

920 (103687)

934 (101404)

962 (84235)

4{z}[edit | edit source]

14 (19699)

68 (13575)

254 (15451)

800 (20509)

5{0}1[edit | edit source]

326 (400786)

350 (20392)

554 (10630)

662 (13390)

926 (40036)

5{z}[edit | edit source]

258 (212135)

272 (148427)

299 (64898)

307 (26263)

354 (25566)

433 (283919)

635 (36163)

678 (40859)

692 (45447)

719 (20552)

768 (70214)

857 (23083)

867 (61411)

972 (36703)

6{0}1[edit | edit source]

108 (16318)

129 (16797)

409 (369833)

522 (52604)

587 (24120)

643 (164916)

762 (11152)

789 (27297)

986 (21634)

6{z}[edit | edit source]

68 (25396)

332 (15222)

338 (42868)

362 (146342)

488 (33164)

566 (164828)

980 (50878)

986 (12506)

1016 (23336)

7{0}1[edit | edit source]

398 (17473)

1004 (54849)

7{z}[edit | edit source]

97 (192336)

170 (15423)

194 (38361)

202 (155772)

282 (21413)

283 (164769)

332 (13205)

412 (29792)

560 (19905)

639 (10668)

655 (53009)

811 (31784)

814 (17366)

866 (108591)

908 (61797)

962 (31841)

992 (10605)

997 (15815)

8{0}1[edit | edit source]

23 (119216)

53 (227184)

158 (123476)

254 (67716)

320 (52004)

410 (279992)

425 (94662)

513 (19076)

518 (11768)

596 (148446)

641 (87702)

684 (23387)

695 (39626)

788 (11408)

893 (86772)

908 (243440)

920 (107822)

962 (47222)

998 (81240)

1013 (43872)

8{z}[edit | edit source]

138 (35686)

412 (12154)

788 (11326)

990 (23032)

9{0}1[edit | edit source]

248 (39511)

592 (96870)

9{z}[edit | edit source]

431 (43574)

446 (152028)

458 (126262)

599 (11776)

846 (12781)

A{0}1[edit | edit source]

173 (264235)

198 (47665)

311 (314807)

341 (106009)

449 (18507)

492 (42843)

605 (12395)

708 (17563)

710 (31039)

743 (285479)

744 (137056)

786 (68169)

800 (15105)

802 (149320)

879 (25004)

929 (13065)

977 (125873)

986 (48279)

1004 (10645)

A{z}[edit | edit source]

368 (10867)

488 (10231)

534 (80328)

662 (13307)

978 (14066)

B{0}1[edit | edit source]

710 (15272)

740 (33520)

878 (227482)

B{z}[edit | edit source]

153 (21660)

186 (112718)

439 (18752)

593 (16064)

602 (36518)

707 (10573)

717 (67707)

C{0}1[edit | edit source]

68 (656922)

219 (29231)

230 (94751)

312 (21163)

334 (83334)

353 (20262)

359 (61295)

457 (10024)

481 (45941)

501 (20140)

593 (42779)

600 (11242)

604 (17371)

641 (26422)

700 (91953)

887 (13961)

919 (45359)

923 (64365)

992 (10300)

{#}$ (for bases b == 1 mod 3, # = (b−1)/3, $ = (b+2)/3)[edit | edit source]

(none)

{#}$ (for odd bases b, # = (b−1)/2, $ = (b+1)/2)[edit | edit source]

(none)

#{z} (for even bases b, # = b/2−1)[edit | edit source]

(none)

y{z}[edit | edit source]

38 (136212)

83 (21496)

113 (286644)

188 (13508)

401 (103670)

417 (21003)

458 (46900)

494 (21580)

518 (129372)

527 (65822)

602 (17644)

608 (36228)

638 (74528)

663 (47557)

723 (24536)

758 (50564)

833 (12220)

904 (13430)

938 (50008)

950 (16248)

z{0}1[edit | edit source]

202 (46774)

251 (102979)

272 (16681)

297 (14314)

298 (60671)

326 (64757)

347 (69661)

363 (142877)

452 (71941)

543 (10042)

564 (38065)

634 (84823)

788 (13541)

869 (12289)

890 (37377)

953 (60995)

1004 (29685)

{z0}z1 (almost cannot be quasi-minimal prime, since this is not simple family)[edit | edit source]

53 (21942)

124 (16426)

175 (31626)

188 (22036)

316 (48538)

365 (25578)

373 (24006)

434 (10090)

530 (11086)

545 (12346)

560 (15072)

596 (12762)

701 (12576)

706 (10656)

821 (13536)

833 (17116)

966 (14820)

983 (11272)

{z}yz (not quasi-minimal prime if there is smaller prime of the form {z}y)[edit | edit source]

(none)

{z}1[edit | edit source]

(none)

{z}y[edit | edit source]

317 (13896)

Bases 2≤b≤1024 which have these families as unsolved families[edit | edit source]

Unsolved families are families which are neither primes (>b) found nor can be ruled out as contain no primes > b

(using A−Z to represent digit values 10 to 35, z−a to represent digit values b−1 to b−26 (e.g. "z" means 1 in base 2, 2 in base 3, 3 in base 4, ..., 8 in base 9, 9 in base 10, A in base 11, B in base 12, ..., Y in base 35, Z in base 36, ...), only consider bases which these families are interpretable, e.g. digit "7" is only interpretable for bases ≥8, and digit "u" (means b−6) is only interpretable for bases ≥7)

1{0}1: 38, 50, 62, 68, 86, 92, 98, 104, 122, 144, 168, 182, 186, 200, 202, 212, 214, 218, 244, 246, 252, 258, 286, 294, 298, 302, 304, 308, 322, 324, 338, 344, 354, 356, 362, 368, 380, 390, 394, 398, 402, 404, 410, 416, 422, 424, 446, 450, 454, 458, 468, 480, 482, 484, 500, 514, 518, 524, 528, 530, 534, 538, 552, 558, 564, 572, 574, 578, 580, 590, 602, 604, 608, 620, 622, 626, 632, 638, 648, 650, 662, 666, 668, 670, 678, 684, 692, 694, 698, 706, 712, 720, 722, 724, 734, 744, 746, 752, 754, 762, 766, 770, 792, 794, 802, 806, 812, 814, 818, 836, 840, 842, 844, 848, 854, 868, 870, 872, 878, 888, 896, 902, 904, 908, 922, 924, 926, 932, 938, 942, 944, 948, 954, 958, 964, 968, 974, 978, 980, 988, 994, 998, 1002, 1006, 1014, 1016 (length limit: ≥8388608)

1{0}2: 167, 257, 323, 353, 383, 527, 557, 563, 623, 635, 647, 677, 713, 719, 803, 815, 947, 971, 1013 (length limit: 2000)

1{0}3: 646, 718, 998 (length limit: 2000)

1{0}4: 139, 227, 263, 315, 335, 365, 485, 515, 647, 653, 683, 773, 789, 797, 815, 857, 875, 893, 939, 995, 1007 (length limit: 2000)

1{0}5

1{0}6

1{0}7

1{0}8

1{0}9

1{0}A

1{0}B

1{0}C

1{0}D

1{0}E

1{0}F

1{0}G

1{0}z: 173, 179, 257, 277, 302, 333, 362, 392, 422, 452, 467, 488, 512, 527, 545, 570, 575, 614, 622, 650, 677, 680, 704, 707, 734, 740, 827, 830, 851, 872, 886, 887, 902, 904, 908, 929, 932, 942, 947, 949, 962, 973, 1022 (length limit: 2000)

1{0}11 (not quasi-minimal prime if there is smaller prime of the form 1{0}1): 198, 213, 318, 327, 353, 375, 513, 591, 647, 732, 734, 738, 759, 948, 951, 957, 1013, 1014 (length limit: 2000)

10{z} (not quasi-minimal prime if there is smaller prime of the form 1{z}): 575 (length limit: 247000)

11{0}1 (not quasi-minimal prime if there is smaller prime of the form 1{0}1): 813, 863, 962, 1017 (length limit: ≥100000)

{1}0z (not quasi-minimal prime if there is smaller prime of the form {1} or {1}z): 137, 161, 167, 217, 229, 232, 253, 261, 317, 325, 337, 347, 355, 375, 403, 411, 421, 427, 457, 479, 483, 505, 507, 537, 547, 577, 597, 599, 601, 613, 627, 631, 632, 641, 643, 649, 657, 679, 688, 697, 707, 711, 729, 733, 737, 742, 762, 773, 787, 793, 797, 817, 819, 841, 843, 853, 859, 861, 874, 877, 895, 899, 907, 913, 916, 917, 927, 957, 959, 997, 1003, 1009, 1015, 1017 (length limit: 2000)

{1}: 185, 269, 281, 380, 384, 385, 394, 452, 465, 511, 574, 601, 631, 632, 636, 711, 713, 759, 771, 795, 861, 866, 881, 938, 948, 951, 956, 963, 1005, 1015 (length limit: ≥100000)

11{z} (not quasi-minimal prime if there is smaller prime of the form 1{z})

{1}2 (not quasi-minimal prime if there is smaller prime of the form {1}): 31, 61, 91, 93, 143, 247, 253, 293, 313, 329, 371, 383, 391, 393, 403, 415, 435, 443, 451, 491, 493, 513, 523, 527, 537, 541, 553, 565, 581, 587, 601, 613, 615, 623, 627, 635, 663, 729, 735, 757, 763, 775, 783, 823, 843, 865, 873, 877, 883, 897, 931, 941, 943, 955, 983, 1013, 1015, 1021, 1023 (length limit: 2000)

{1}z

1{2}: 265, 355, 379, 391, 481, 649, 661, 709, 745, 811, 877, 977 (length limit: 2000)

1{3}: 107, 133, 179, 281, 305, 365, 473, 485, 487, 491, 535, 541, 601, 617, 665, 737, 775, 787, 802, 827, 905, 911, 928, 953, 955, 995

1{4}: 83, 143, 185, 239, 269, 293, 299, 305, 319, 325, 373, 383, 395, 431, 471, 503, 551, 577, 581, 593, 605, 617, 631, 659, 743, 761, 773, 781, 803, 821, 857, 869, 897, 911, 917, 923, 935, 983, 1019 (length limit: 2000)

1{z}: 581, 992, 1019 (length limit: ≥100000)

2{0}1: 365, 383, 461, 512, 542, 647, 773, 801, 836, 878, 908, 914, 917, 947, 1004 (length limit: ≥100000)

2{0}3: 79, 149, 179, 254, 359, 394, 424, 434, 449, 488, 499, 532, 554, 578, 664, 683, 694, 749, 794, 839, 908, 944, 982 (length limit: 2000)

2{1} (not quasi-minimal prime if there is smaller prime of the form {1}): 109, 117, 137, 147, 157, 175, 177, 201, 227, 235, 256, 269, 271, 297, 310, 331, 335, 397, 417, 427, 430, 437, 442, 451, 465, 467, 481, 502, 517, 547, 557, 567, 572, 577, 591, 597, 607, 627, 649, 654, 655, 667, 679, 687, 691, 697, 715, 727, 739, 759, 766, 782, 787, 796, 797, 808, 817, 821, 829, 841, 852, 877, 881, 899, 903, 907, 937, 947, 955, 1007, 1011, 1021 (length limit: 2000)

{2}1: 106, 238, 262, 295, 364, 382, 391, 397, 421, 458, 463, 478, 517, 523, 556, 601, 647, 687, 754, 790, 793, 832, 872, 898, 962, 1002, 1021 (length limit: 2000)

2{z}: 588, 972 (length limit: ≥100000)

3{0}1: 718, 912 (length limit: ≥100000)

3{0}2: 223, 283, 359, 489, 515, 529, 579, 619, 669, 879, 915, 997 (length limit: 2000)

3{0}4: 167, 391, 447, 487, 529, 653, 657, 797, 853, 913, 937 (length limit: 2000)

{3}1: 79, 101, 189, 215, 217, 235, 243, 253, 255, 265, 313, 338, 341, 378, 379, 401, 402, 413, 489, 498, 499, 508, 525, 535, 589, 591, 599, 611, 621, 635, 667, 668, 681, 691, 711, 717, 719, 721, 737, 785, 804, 805, 813, 831, 835, 837, 849, 873, 911, 915, 929, 933, 941, 948, 959, 999, 1013, 1019 (length limit: 2000)

3{z}: 275, 438, 647, 653, 812, 927, 968 (length limit: ≥100000)

4{0}1: 32, 53, 155, 174, 204, 212, 230, 332, 334, 335, 395, 467, 512, 593, 767, 803, 848, 875, 1024 (length limit: ≥100000)

4{0}3: 83, 88, 97, 167, 188, 268, 289, 293, 412, 419, 425, 433, 503, 517, 529, 548, 613, 620, 622, 650, 668, 692, 706, 727, 763, 818, 902, 913, 937, 947, 958 (length limit: 2000)

{4}1: 46, 77, 103, 107, 119, 152, 198, 203, 211, 217, 229, 257, 263, 291, 296, 305, 332, 371, 374, 407, 413, 416, 440, 445, 446, 464, 467, 500, 542, 545, 548, 557, 566, 586, 587, 605, 611, 614, 632, 638, 641, 653, 659, 698, 701, 731, 733, 736, 755, 786, 812, 820, 821, 827, 830, 887, 896, 899, 901, 922, 923, 935, 941, 953, 977, 983, 991, 1004 (length limit: 2000)

4{z}: 338, 998 (length limit: ≥100000)

5{0}1: 308, 512, 824 (length limit: ≥100000)

5{z}: 234, 412, 549, 553, 573, 619, 750, 878, 894, 954 (length limit: ≥100000)

6{0}1: 212, 509, 579, 625, 774, 794, 993, 999 (length limit: ≥100000)

6{z}: 308, 392, 398, 518, 548, 638, 662, 878 (length limit: ≥100000)

7{0}1: (none)

7{z}: 321, 328, 374, 432, 665, 697, 710, 721, 727, 728, 752, 800, 815, 836, 867, 957, 958, 972 (length limit: ≥100000)

8{0}1: 86, 140, 182, 263, 353, 368, 389, 395, 422, 426, 428, 434, 443, 488, 497, 558, 572, 575, 593, 606, 698, 710, 746, 758, 770, 773, 785, 824, 828, 866, 911, 930, 953, 957, 983, 993, 1014 (length limit: ≥100000)

8{z}: 378, 438, 536, 566, 570, 592, 636, 688, 718, 830, 852, 926, 1010 (length limit: ≥100000)

9{0}1: 724, 884 (length limit: ≥100000)

9{z}: 80, 233, 530, 551, 611, 899, 912, 980 (length limit: ≥100000)

A{0}1: 185, 338, 417, 432, 614, 668, 773, 863, 935, 1000 (length limit: ≥100000)

A{z}: 214, 422, 444, 452, 458, 542, 638, 668, 804, 872, 950, 962 (length limit: ≥100000)

B{0}1: 560, 770, 968 (length limit: ≥100000)

B{z}: 263, 615, 912, 978 (length limit: ≥100000)

C{0}1: 163, 207, 354, 362, 368, 480, 620, 692, 697, 736, 753, 792, 978, 998, 1019, 1022 (length limit: ≥100000)

C{z}

D{0}1

D{z}

E{0}1

E{z}

F{0}1

F{z}

G{0}1

{#}$ (for bases b == 1 mod 3, # = (b−1)/3, $ = (b+2)/3): 808, 829, 859, 1006 (length limit: 2000)

{#}$ (for odd bases b, # = (b−1)/2, $ = (b+1)/2): 31, 37, 55, 63, 67, 77, 83, 89, 91, 93, 97, 99, 107, 109, 117, 123, 127, 133, 135, 137, 143, 147, 149, 151, 155, 161, 177, 179, 183, 189, 193, 197, 207, 211, 213, 215, 217, 223, 225, 227, 233, 235, 241, 247, 249, 255, 257, 263, 265, 269, 273, 277, 281, 283, 285, 287, 291, 293, 297, 303, 307, 311, 319, 327, 347, 351, 355, 357, 359, 361, 367, 369, 377, 381, 383, 385, 387, 389, 393, 397, 401, 407, 411, 413, 417, 421, 423, 437, 439, 443, 447, 457, 465, 467, 469, 473, 475, 481, 483, 489, 493, 495, 497, 509, 511, 515, 533, 541, 547, 549, 555, 563, 591, 593, 597, 601, 603, 611, 615, 619, 621, 625, 627, 629, 633, 635, 637, 645, 647, 651, 653, 655, 659, 663, 667, 671, 673, 675, 679, 683, 687, 691, 693, 697, 707, 709, 717, 731, 733, 735, 737, 741, 743, 749, 753, 755, 757, 759, 765, 767, 771, 773, 775, 777, 783, 785, 787, 793, 797, 801, 807, 809, 813, 817, 823, 825, 849, 851, 853, 865, 867, 873, 877, 887, 889, 893, 897, 899, 903, 907, 911, 915, 923, 927, 933, 937, 939, 941, 943, 945, 947, 953, 957, 961, 967, 975, 977, 983, 987, 993, 999, 1003, 1005, 1009, 1017 (length limit: ≥262143)

#{z} (for even bases b, # = b/2−1): 108, 278, 296, 338, 386, 494, 626, 920 (length limit: 2000)

${#} (for odd bases b, # = (b−1)/2, $ = (b+1)/2)

x{z}

y{z}: 128, 233, 268, 383, 478, 488, 533, 554, 665, 698, 779, 863, 878, 932, 941, 1010 (length limit: ≥200000)

z{0}1: 123, 342, 362, 422, 438, 479, 487, 512, 542, 602, 757, 767, 817, 830, 872, 893, 932, 992, 997, 1005, 1007 (length limit: ≥100000)

{y}z: 143, 173, 176, 213, 235, 248, 253, 279, 327, 343, 353, 358, 373, 383, 401, 413, 416, 427, 439, 448, 453, 463, 481, 513, 522, 527, 535, 547, 559, 565, 583, 591, 598, 603, 621, 623, 653, 659, 663, 679, 691, 698, 711, 743, 745, 757, 768, 785, 793, 796, 801, 808, 811, 821, 835, 845, 847, 853, 856, 883, 898, 903, 927, 955, 961, 971, 973, 993, 1005, 1013, 1019, 1021 (length limit: 2000)

{z0}z1 (almost cannot be quasi-minimal prime, since this is not simple family): 97, 103, 113, 186, 187, 220, 304, 306, 309, 335, 414, 416, 428, 433, 445, 459, 486, 498, 539, 550, 557, 587, 592, 597, 598, 617, 624, 637, 659, 665, 671, 677, 696, 717, 726, 730, 740, 754, 766, 790, 851, 873, 890, 914, 923, 929, 943, 944, 965, 984, 985, 996, 1004, 1005 (length limit: ≥17326)

zy{z} (not quasi-minimal prime if there is smaller prime of the form y{z})

{z}yz (not quasi-minimal prime if there is smaller prime of the form {z}y): 215, 353, 517, 743, 852, 899, 913 (length limit: 2000)

{z}01 (not quasi-minimal prime if there is smaller prime of the form {z}1)

{z}1: 93, 113, 152, 158, 188, 217, 218, 226, 227, 228, 233, 240, 275, 278, 293, 312, 338, 350, 353, 383, 404, 438, 464, 471, 500, 533, 576, 614, 641, 653, 704, 723, 728, 730, 758, 779, 788, 791, 830, 878, 881, 899, 908, 918, 929, 944, 953, 965, 968, 978, 983, 986, 1013 (length limit: 2000)

{z}k

{z}l

{z}m

{z}n

{z}o

{z}p

{z}q

{z}r

{z}s

{z}t

{z}u

{z}v

{z}w: 207, 221, 293, 375, 387, 533, 633, 647, 653, 687, 701, 747, 761, 785, 863, 897, 905, 965, 1017 (length limit: 2000)

{z}x: (none)

{z}y: 305, 353, 397, 485, 487, 535, 539, 597, 641, 679, 731, 739, 755 (length limit: 2000)

List of lengths for quasi-minimal primes in some simple families[edit | edit source]

[1]

Only list simple families which must be quasi-minimal primes (when there exist primes of such families).

NB: this family is not interpretable in this base (e.g. family 7{0}1 and 7{z} in bases <=7, family {z}x in bases <=3) (including the case which this family has either leading zeros (leading zeros do not count) or ending zeros (numbers ending in zero cannot be prime > base) in this base)

RC: this family can be proven to only contain composite numbers (only count numbers > base)

unknown: this family has no primes or PRPs found, nor can this family be proven to only contain composite numbers (only count numbers > base)

Background color: red for title (bases or families), green for length > 10000, orange for 2500 < length ≤ 10000, white for length ≤ 2500, cyan for "RC", pink for "NB", yellow for "unknown".

Search limit for lengths: ≥8388608 for 1{0}1, ≥200000 for y{z}, ≥100000 for d{0}1 (d = one of digits in {2, 3, 4, 5, 6, 7, 8, 9, A, B, C}) and d{z} (d = one of digits in {1, 2, 3, 4, 5, 6, 7, 8, 9, A, B}) and z{0}1 and {1}, ≥5000 for 1{0}2, {z}y, 1{0}z, {z}1, {y}z, ≥2500 for other families.

References[edit | edit source]

Main reference: [2], also [3] for research for bases b≤16

[1] https://primes.utm.edu/glossary/xpage/MinimalPrime.html (article “minimal prime” in The Prime Glossary)

[2] https://en.wikipedia.org/wiki/Minimal_prime_(recreational_mathematics) (article “minimal prime” in Wikipedia)

[3] https://www.primepuzzles.net/puzzles/puzz_178.htm (the puzzle of minimal primes (when the restriction of prime>base is not required) in The Prime Puzzles & Problems Connection)

[4] https://www.primepuzzles.net/problems/prob_083.htm (the problem of minimal primes in The Prime Puzzles & Problems Connection)

[5] https://github.com/xayahrainie4793/non-single-digit-primes (my data for these M(Lb) sets for 2 ≤ b ≤ 16)

[6] http://www.cs.uwaterloo.ca/~shallit/Papers/minimal5.pdf (Shallit’s proof of base 10 minimal primes, when the restriction of prime>base is not required)

[7] https://scholar.colorado.edu/downloads/hh63sw661 (proofs of minimal primes in bases b≤10, when the restriction of prime>base is not required)

[8] https://cs.uwaterloo.ca/~cbright/reports/mepn.pdf (the article for this minimal prime problem in bases b≤30, when the restriction of prime>base is not required)

[9] https://cs.uwaterloo.ca/~cbright/talks/minimal-slides.pdf (the article for this minimal prime problem in bases b≤30, when the restriction of prime>base is not required)

[10] https://doi.org/10.1080/10586458.2015.1064048 (the article for this minimal prime problem in bases b≤30, when the restriction of prime>base is not required)

[11] https://github.com/curtisbright/mepn-data (data for these M(Lb) sets and unsolved families for 2 ≤ b ≤ 30, when the restriction of prime>base is not required, search limits of lengths: 1000000 for b=17, 707000 for b=19, 506000 for b=21, 292000 for b=25, 486000 for b=26, 543000 for b=28, 233000 for b=29)

[12] https://github.com/RaymondDevillers/primes (data for these M(Lb) sets and unsolved families for 2 ≤ b ≤ 50, when the restriction of prime>base is not required, search limits of lengths: 10000 for all b)

[13] http://www.bitman.name/math/article/730 (article for minimal primes, when the restriction of prime>base is not required)

[14] http://www.bitman.name/math/table/497 (data for minimal primes in bases 2 ≤ b ≤ 16, when the restriction of prime>base is not required)

[15] http://www.prothsearch.com/sierp.html (the Sierpinski problem)

[16] http://www.prothsearch.com/rieselprob.html (the Riesel problem)

[17] https://oeis.org/A076336/a076336c.html (the dual Sierpinski problem)

[18] http://www.noprimeleftbehind.net/crus/Sierp-conjectures.htm (generalized Sierpinski conjectures in bases b≤1030, some primes found in these conjectures are minimal primes in base b, especially, all primes for k<b (if exist for a (k,b) combo) are always minimal primes in the base b) (also some examples for simple families contain no primes > b)

[19] http://www.noprimeleftbehind.net/crus/Riesel-conjectures.htm (generalized Riesel conjectures in bases b≤1030, some primes found in these conjectures are minimal primes in base b, especially, all primes for k<b (if exist for a (k,b) combo) are always minimal primes in the base b) (also some examples for simple families contain no primes > b)

[20] http://www.noprimeleftbehind.net/crus/tab/CRUS_tab.htm (list for the status of the generalized Sierpinski conjectures and the generalized Riesel conjectures in bases b≤1030)

[21] https://www.utm.edu/staff/caldwell/preprints/2to100.pdf (article for generalized Sierpinski conjectures in bases b≤100)

[22] http://www.kurims.kyoto-u.ac.jp/EMIS/journals/INTEGERS/papers/i61/i61.pdf (article for the mixed (original+dual) Sierpinski problem)

[23] http://www.fermatquotient.com/PrimSerien/GenRepu.txt (generalized repunit primes (primes of the form (bn−1)/(b−1)) in bases b≤160, the smallest such prime for base b (if exists) is always minimal prime in base b)

[24] https://web.archive.org/web/20021111141203/http://www.users.globalnet.co.uk/~aads/primes.html (generalized repunit primes (primes of the form (bn−1)/(b−1)) in bases b≤1000, the smallest such prime for base b (if exists) is always minimal prime in base b)

[25] http://jeppesn.dk/generalized-fermat.html (generalized Fermat primes (primes of the form b2^n+1) in even bases b≤1000, the smallest such prime for base b (if exists) is always minimal prime in base b)

[26] http://www.noprimeleftbehind.net/crus/GFN-primes.htm (generalized Fermat primes (primes of the form b2^n+1) in even bases b≤1030, the smallest such prime for base b (if exists) is always minimal prime in base b)

[27] http://www.fermatquotient.com/PrimSerien/GenFermOdd.txt (list of generalized half Fermat primes (primes of the form (b2^n+1)/2) sorted by n, the smallest such prime for base b (if exists) is always minimal prime in base b)

[28] https://harvey563.tripod.com/wills.txt (primes of the form (b−1)*bn−1 for bases b≤2049, the smallest such prime for base b (if exists) is always minimal prime in base b)

[29] https://www.rieselprime.de/ziki/Williams_prime_MM_least (the smallest primes of the form (b−1)*bn−1 for bases b≤2049, these primes (if exists) is always minimal prime in base b)

[30] https://www.rieselprime.de/ziki/Williams_prime_MP_least (the smallest primes of the form (b−1)*bn+1 for bases b≤1024, these primes (if exists) is always minimal prime in base b)

[31] https://www.rieselprime.de/ziki/Riesel_prime_small_bases_least_n (the smallest primes of the form k*bn−1 for k≤12 and bases b≤1024, these primes (if exists) is always minimal prime in base b if b>k)

[32] https://www.rieselprime.de/ziki/Proth_prime_small_bases_least_n (the smallest primes of the form k*bn+1 for k≤12 and bases b≤1024, these primes (if exists) is always minimal prime in base b if b>k)

[33] https://docs.google.com/spreadsheets/d/e/2PACX-1vTKkSNKGVQkUINlp1B3cXe90FWPwiegdA07EE7-U7sqXntKAEQrynoI1sbFvvKriieda3LfkqRwmKME/pubhtml (list for the smallest primes in given simple family in bases b≤1024)

[34] https://www.rose-hulman.edu/~rickert/Compositeseq/ (a problem related to this project)

[35] http://www.worldofnumbers.com/Appending%201s%20to%20n.txt (a problem related to this project)

[36] https://stdkmd.net/nrr/prime/primecount.txt (near- and quasi- repdigit (probable) primes sorted by count)

[37] https://stdkmd.net/nrr/prime/primedifficulty.txt (near- and quasi- repdigit (probable) primes sorted by difficulty)

[38] http://www.prothsearch.com/fermat.html (factoring status of Fermat numbers)

[39] http://www.rieselprime.de/dl/CRUS_pack.zip (srsieve, sr1sieve, sr2sieve, pfgw, and llr softwares)

[40] https://www.bc-team.org/app.php/dlext/?cat=3 (srsieve, sr1sieve, sr2sieve, sr5sieve software)

[41] https://sourceforge.net/projects/openpfgw/ (pfgw software)

[42] http://jpenne.free.fr/index2.html (llr software)

[43] http://www.ellipsa.eu/public/primo/primo.html (PRIMO software)

[44] https://primes.utm.edu/prove/index.html (website for primality proving)

[45] https://primes.utm.edu/curios/page.php?number_id=22380 (the largest base 10 minimal prime in Prime Curios!)

[46] https://oeis.org/A071062 (OEIS sequence for base 10 minimal primes, when the restriction of prime>base is not required)

[47] https://oeis.org/A326609 (OEIS sequence for the largest base b minimal prime, when the restriction of prime>base is not required)

[48] https://primes.utm.edu/primes/lists/all.txt (top proven primes)

[49] http://www.primenumbers.net/prptop/prptop.php (top PRPs)

[50] http://factordb.com (online factor database, including many primes which are minimal primes in a small base)