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2021-01-05, 12:51
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#111 | |
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Nov 2016
22×3×5×47 Posts |
Quote:
Code:
30: "27J1"=60871 "27JT"=60899 "2B11"=63931 "2B71"=64111 "2JT7"=71977 "71DT"=190319 "71NT"=190619 "7J1T"=206159 "7N1T"=209759 "DBJ7"=361477 "J1T7"=514777 "NB11"=630931 "TDB7"=795037 36: 42: "3DNB"=246173 "B11H"=816791 48: "2fV5"=317141 "Hf5h"=1974811 54: 60: "2hrB"=589991 "3H11"=709261 "3JlD"=719233 "3x11"=860461 "711f"=1515701 "7hBD"=1667473 "JlDf"=4274021 "fHhD"=8919793 "hBfD"=9330073 "x11f"=12747701 "x1f7"=12750067 66: "31hD"=869695 "5JrT"=1523771 "7111"=2016895 "hDN5"=12420479 72: "27b&"=785515 78: "7111"=3328027 "711B"=3328037 "71B*"=3328877 "B111"=5226235 "&111"=31801147 "&11B"=31801157 84: "51@T"=2977577 "5Bxl"=3046139 "H5JB"=10112855 "HxBh"=10493239 "J15N"=11268875 "N1JH"=13640861 "N1<&"=13645951 90: "2hrl"=1811117 "2rDl"=1888517 "2rVl"=1890137 "3N&z"=2379391 "3lhB"=2571581 "3lhf"=2571611 "57DB"=3702881 "5DBx"=3751349 "7BTN"=5194733 "7zBT"=5598119 "B1JN"=8028833 "B7TN"=8078333 "DlBx"=9858749 "H1@b"=12408607 "NB*x"=16862549 "NH&D"=16910743 "T7Bz"=21198751 96: "2b7<"=2111215 "3x<J"=3205555 "7z*B"=6762155 "Tx,5"=26208101 "l11B"=41591915 "z11B"=53978219 Last fiddled with by sweety439 on 2021-01-05 at 16:07 |
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2021-01-05, 13:19
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#112 |
Jun 2003
23×607 Posts |
You've been asked to not quote entire posts unnecessarily.
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2021-01-05, 15:49
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#113 |
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Nov 2016
54048 Posts |
I want to make the reader know this post is a response of post #108
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2021-01-09, 17:08
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#114 |
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Nov 2016
1011000001002 Posts |
Update status files in base 36
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2021-01-09, 18:58
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#115 | |
"Serge"
Mar 2008
Phi(4,2^7658614+1)/2
22×2,333 Posts |
Quote:
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2021-01-09, 19:06
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#116 | |
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6809 > 6502
"""""""""""""""""""
Aug 2003
101×103 Posts
22×3×11×71 Posts |
Quote:
Oh wait, that is the norm... 
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2021-02-15, 16:05
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#118 |
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Nov 2016
22×3×5×47 Posts |
P81993SZ is minimal (probable) prime base 36, its formula is (5*36^81995+821)/7, now base 36 has only one unsolved family: O{L}Z (see https://github.com/RaymondDevillers/.../master/left36), and this family is searched to 100000 digits with no (probable) prime found.
Last fiddled with by sweety439 on 2021-02-21 at 15:53 |
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2021-02-20, 21:45
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#119 | |
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Nov 2016
282010 Posts |
Quote:
Code:
Minimal set of primes in decimal:
{2, 3, 5, 7, 11, 19, 41, 61, 89, 409, 449, 499, 881, 991, 6469, 6949, 9001, 9049, 9649, 9949, 60649, 666649, 946669, 60000049, 66000049, 66600049}
This set is complete.
Minimal set of primes in dozenal:
{2, 3, 5, 7, B, 11, 61, 81, 91, 401, A41, 4441, A0A1, AAAA1, 44AAA1, AAA0001, AA000001}
This set is complete.
Minimal set of composites in decimal:
{4, 6, 8, 9, 10, 12, 15, 20, 21, 22, 25, 27, 30, 32, 33, 35, 50, 51, 52, 55, 57, 70, 72, 75, 77, 111, 117, 171, 371, 711, 713, 731}
This set is complete.
Minimal set of composites in dozenal:
{4, 6, 8, 9, A, 10, 12, 13, 20, 21, 22, 23, 2B, 30, 32, 33, 50, 52, 53, 55, 70, 71, 72, 73, 77, 7B, B0, B1, B2, B3, BB, 115, 151, 15B, 257, 275, 311, 317, 31B, 351, B57, B75, 1111, 1117, 111B, 5111}
This set is complete.
Minimal set of prime powers (1 is not counted as prime power) in decimal:
{2, 3, 4, 5, 7, 8, 9, 11, 16, 61}
It is easy to prove that this set is complete, since combinations of only 0 and 6 cannot be prime power, since such numbers must be divisible by 6.
Minimal set of prime powers (1 is not counted as prime power) in dozenal:
{2, 3, 4, 5, 7, 8, 9, B, 11, 61, A1}
Since combinations of only 0, 6, and A is always even number, and if an even number is prime power, then this number must be power of 2, but except of 1 and 2, all powers of 2 ends with 4 or 8, which is a contradiction, thus this set is complete.
Minimal set of powers of 2 in decimal:
{1, 2, 4, 8, 65536}
This set is conjectured to be complete, but not proven, references: https://oeis.org/A071071, https://oeis.org/A071071/a071071.pdf
Minimal set of powers of 2 in dozenal:
{1, 2, 4, 8}
In dozenal such set is very easy to proven, since except 1 and 2, all powers of 2 end with either 4 or 8.
Minimal set of powers of 3 in decimal:
{1, 3, 9, 27}
This set is conjectured to be complete, but not proven.
Minimal set of powers of 3 in dozenal:
{1, 3, 9}
In dozenal such set is very easy to proven, since except 1, all powers of 3 end with either 3 or 9.
Minimal set of squares (0 is not counted as square) in decimal:
{1, 4, 9, 25, 36, 576, 676, 7056, 80656, 665856, 2027776, 2802276, 22282727076, 77770707876, 78807087076, 7888885568656, 8782782707776, 72822772707876, 555006880085056, 782280288087076, 827702888070276, 888288787822276, 2282820800707876, 7880082008070276, 80077778877070276, 88778000807227876, 782828878078078276, 872727072820287876, 2707700770820007076, 7078287780880770276, 7808287827720727876, 8008002202002207876, 27282772777702807876, 70880800720008787876, 72887222220777087876, 80028077888770207876, 80880700827207270276, 87078270070088278276, 88002002000028027076, 2882278278888228807876, 8770777780888228887076, 77700027222828822007876, 702087807788807888287876, 788708087882007280808827876, 880070008077808877000002276, 888000227087070707880827076, 888077027227228277087787076, 888588886555505085888555556, 7770000800780088788282227776, 7782727788888878708800870276, 5000060065066660656065066555556, 8070008800822880080708800087876, 80787870808888808272077777227076, 800008088070820870870077778827876, 822822722220080888878078820887876, ...}
This set is currently not known, and might be extremely difficult to found, although it is known that no repdigits (numbers whose all digits are same) are squares, references: https://oeis.org/A130448, http://recursed.blogspot.com/2006/12/prime-game.html
Minimal set of squares (0 is not counted as square) in dozenal:
{1, 4, 9, 30}
Interestingly, in dozenal such set is very easy to proven, since squares are end with 0, 1, 4, 9 in dozenal, thus a non-single digit minimal square in dozenal must end with 0, and in dozenal if a square ends with 0, then it must end with either 00 or 30, but 30 is already a square, and if x00 is square (when x represents any string), then x is also square, thus x00 cannot be minimal square, and this set is complete.
Minimal set of cubes (0 is not counted as cube) in decimal:
{1, 8, 27, 64, 343, 729, 3375, 4096, 35937, 39304, 46656, 50653, 79507, 97336, 300763, 405224, 456533, 474552, 493039, 636056, 704969, 3307949, 4330747, 5545233, 5639752, 5735339, 6539203, 9663597, 23393656, 23639903, 29503629, 37933056, 40353607, 45499293, 50243409, 54439939, 57066625, 57960603, 70444997, 70957944, 73560059, 76765625, 95443993, 202262003, 236029032, 350402625, 377933067, 379503424, 445943744, 454756609, 537367797, 549353259, 563559976, 567663552, 773620632, 907039232, ...}
This set is currently not known, and might be extremely difficult to found.
Minimal set of cubes (0 is not counted as cube) in dozenal:
{1, 8, 23, 54, A5, 247, 509, 6B4, 92B, 3460, 3B77, 705B, 36A60, 500BB, 94AB4, 2270B4, 29BB75, 329599, 3407B4, 479B09, 4B0B2B, 55B553, 9A79B3, B32299, 33B345B, 3993969, 435AA60, 46A44BB, 536722B, 67B97A7, 6992669, 73340B3, 7904599, 9396553, 9A69A39, A929760, 26626460, 2766795B, 2B7B9A60, 33497AB4, ...}
This set is currently not known, and might be extremely difficult to found.
Minimal set of perfect powers (0 and 1 are not counted as perfect powers) in decimal:
{4, 8, 9, 16, 25, 27, 32, 36, 100, 121, 512, 576, 676, 1331, 2601, 3375, 6561, 7056, 7776, 22201, 50653, 62001, 63001, 505521, 657721, 753571, 5000211, 5067001, 5177717, 5755201, 7557001, ...}
This set is not known, may not be complete.
Minimal set of perfect powers (0 and 1 are not counted as perfect powers) in dozenal:
{4, 8, 9, 21, 23, 30, A1, A5, 100, 160, 1331, 1761, 1B53, 3B77, 6761, 705B, 152A7, 16661, 16B61, 500BB, 51161, 57361, 6B161, 76361, 111101, 15315B, 161561, 55B553, 561661, 601701, 761501, B01101, B33761, 12B702B, 536722B, 5665561, 5B3B761, 6067B01, 611355B, 6615701, 7006061, 7710517, 7B70661, B055661, B56B561, B63BB61, B651661, 10BB1B61, 11355561, 135B375B, 1511B501, 1570A72B, 157B16B3, ...}
This set is not known, may not be complete.
Minimal set of powerful numbers (0 and 1 are not counted as powerful numbers) in decimal:
{4, 8, 9, 16, 25, 27, 32, 36, 72, 100, 121, 200, 500, 512, 576, 675, 676, 1152, 1331, 2601, 3375, 6561, 7056, 7776, 15552, 22201, 50653, 60552, 61731, 63001, 77175, 202612, 357075, 505521, 570375, 665523, 735075, 753571, 766656, 1037575, 3501153, 5177717, 6171373, 7555707, 15135133, 15150375, 15151531, ...}
This set is not known, may not be complete.
Minimal set of powerful numbers (0 and 1 are not counted as powerful numbers) in dozenal:
{4, 8, 9, 21, 23, 30, 60, A1, A5, 100, 200, 1331, 1761, 1B53, 3665, 3B77, 6761, 705B, 152A7, 156B3, 16661, 16B61, 500BB, 51161, 57361, 5B0B5, 5B7A7, 6B161, 73653, 76361, AA20B, 10B1B3, 111101, 11AB27, 15315B, 161561, 16672B, 317565, 3B63B3, 515B05, 516727, 53B525, 55B553, 561661, 5ABB7B, 6B1155, 751311, 7B7177, 7BA337, B01101, B33761, 1161553, 1275A07, 12B702B, 1572707, 1B22257, 22B6A2B, 3363353, 351567B, 50A76B3, 536722B, 5665561, 56B32BB, 577B165, 5B3B761, 611355B, 6157535, 6167AB3, 6551353, 75771B3, 7710517, 7B70661, ...}
This set is not known, may not be complete.
Minimal set of squarefree numbers in decimal:
{1, 2, 3, 5, 6, 7, 89, 94, 409, 449, 498, 499, 998}
This set is complete.
Minimal set of squarefree numbers in dozenal:
{1, 2, 3, 5, 6, 7, A, B, 49, 89}
This set is complete.
Minimal set of semiprimes in decimal:
{4, 6, 9, 10, 15, 21, 22, 25, 33, 35, 38, 51, 55, 57, 58, 77, 82, 85, 87, 111, 118, 123, 178, 183, 203, 237, 278, 301, 302, 327, 371, 502, 703, 713, 718, 723, 731, 753, 781, 803, 813, 818, 831, 1137, 1317, 3007, 3117, 8801, 8881, 28883, 50003, 80081, 888883, ...}
This set may not be complete.
Minimal set of semiprimes in dozenal:
{4, 6, 9, A, 12, 13, 21, 22, 2B, 32, 33, 52, 55, 71, 72, 73, 77, 7B, B1, B2, BB, 101, 10B, 115, 151, 15B, 187, 203, 207, 235, 257, 275, 287, 305, 311, 317, 351, 381, 385, 387, 501, 503, 50B, 537, 53B, 581, 802, 805, 807, 811, 815, 831, 837, 83B, 857, 875, 885, 887, B05, B07, B57, B75, B85, B87, 1007, 1057, 1075, 1085, 1117, 111B, 1181, 1507, 1881, 188B, 18B5, 3007, 300B, 301B, 318B, 31B5, 3507, 380B, 388B, 5087, 5107, 5111, 518B, 588B, 800B, 801B, 851B, 8801, 8803, 880B, 881B, 8882, 8B35, B003, 10005, 17005, 20085, 30001, 70005, 80001, 80881, 8088B, 88881, 1111111, ...}
This set may not be complete.
Minimal set of triangular numbers (0 is not counted as triangular number) in decimal:
{1, 3, 6, 28, 45, 55, 78, 820, 990, 2775, 7750, 9870, 24090, 25200, 40470, 49770, 57970, 70500, 97020, 292995, 299925, 422740, 442270, 588070, 702705, 749700, 870540, 2474200, 4744740, 5727420, 7279020, 7799275, 8588440, 20740020, 27524490, 27792240, 40002040, 52270200, 54920440, ...}
This set is currently not known, and might be extremely difficult to found.
Minimal set of triangular numbers (0 is not counted as triangular number) in dozenal:
{1, 3, 6, A, 24, 47, 77, 89, B4, 550, 584, 5B9, 820, B59, 2270, 2780, 4250, 4854, 4940, 5299, 5579, 7440, 7554, 9020, 98B0, 25880, 27050, 2B900, 42529, 59070, 87BB0, 97484, 200599, 202279, 209979, 299B80, 44B0B9, 494559, 500079, 594954, 595929, 74B280, 7B88B0, 884554, 902759, 9050B0, 920950, 92BB29, B079B9, 228B880, 25BB070, 2729599, 2BBB570, 4209280, 4225559, 422B2B0, 44BB220, 4544929, 4942929, 7090529, 7259900, 74B9229, 9292000, 9570529, 9574954, 979BBB0, B02B020, B70B280, B8B58B0, 20020850, 202B0800, 20759929, 25992980, 29B9B799, 2BB20850, ...}
This set is currently not known, and might be extremely difficult to found.
Minimal set of Pronic numbers (0 is not counted as Pronic number) in decimal:
{2, 6, 30, 90, 110, 870, 5550, 5700, 7140, 8010, 15500, 15750, 48180, 50400, 50850, 57840, 144780, 147840, 475410, 504810, 757770, 845480, 884540, 1477440, 4517750, 4754580, 7185080, 7450170, 10077450, 10500840, 14588580, 17770440, 17854850, 40710780, 41480040, 41544470, 45501770, 47755010, 54471780, ...}
This set is currently not known, and might be extremely difficult to found.
Minimal set of Pronic numbers (0 is not counted as Pronic number) in dozenal:
{2, 6, 10, 18, 48, B0, 3A0, 530, 578, 740, 890, 930, AA0, 3370, 37A8, 3900, 4050, 73A8, 7878, 7A30, 84A0, 8B78, 30730, 33330, 59A70, 5BAA8, 789A8, 83AA8, 97A80, 99B78, A7030, BA7A8, 370300, 39B9A8, 455090, 47A770, 495550, 590590, 5A9470, 70B9A8, 759570, 8850A8, 8A8580, 958540, 9705A0, 977790, 9A5AA8, 9A7880, A3BB78, A5A3A8, 3008878, 3033B78, 3373B78, 3780880, 4504040, 4700390, 4A94700, 55A0580, 5797770, 5844750, 5A09700, 7370950, 7385000, 77B59A8, 7998AA8, 8555880, 8808340, 9940A70, A53B9A8, A954970, ...}
This set is currently not known, and might be extremely difficult to found.
Minimal set of factorial numbers in decimal:
{1, 2, 6, 5040}
This set is conjectured to be complete, but not proven.
Minimal set of factorial numbers in dozenal:
{1, 2, 6, A0, 500}
This set is conjectured to be complete, but not proven.
Minimal set of non-single-digit primes in decimal:
{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}
This set is complete, reference: https://mersenneforum.org/attachment...9&d=1613720197
Minimal set of non-single-digit primes in dozenal:
{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}
This set is complete, reference: https://mersenneforum.org/attachment...9&d=1613720197
Minimal set of "lesser of twin primes" in decimal:
{3, 5, 11, 17, 29, 41, 71, 149, 227, 281, 809, 821, 827, 881, 2087, 2687, 4049, 4649, 4787, 4799, 4967, 6089, 6449, 6689, 6779, 6869, 6947, 7487, 7877, 7949, 8087, 8969, 8999, 9677, 9767, 10889, 19889, 20021, 20477, 20747, 26261, 27407, 40427, 40697, 40847, 42407, 44027, 44087, 44267, 44699, 46679, 46769, 47777, 48407, 48479, 48647, 48677, 48779, 48869, 48989, 49277, 49409, 60647, 60887, 64877, 66749, 67409, 67427, 68447, 68489, 68879, 68897, 70979, 70997, 72467, 72647, 74099, 74609, 76079, 76649, 77267, 77477, 77489, 77687, 78887, 79697, 79997, 80447, 80747, 80777, 81899, 84869, 84977, 88469, 88607, 88799, 89897, 94007, 94847, 94949, 96221, 97499, 97649, 97787, 97847, 98807, 98867, 98897, 99707, 99989, 168899, 202061, 240047, 244667, 246707, 247067, 247607, 248867, 266447, 266477, 267647, 267677, 276047, 277787, 278807, 284747, 288647, 400067, 400247, 400679, 402767, 407789, 408689, 409709, 409889, 426707, 427067, 428807, 442007, 442487, 444287, 447449, 448997, 449987, 460079, 460709, 460979, 464747, 467477, 467627, 468887, 469877, 470207, 474497, 474707, 476027, 476477, 479027, 480047, 484487, 487469, 487889, 488687, 490247, 490769, 496889, 498689, 602477, 604697, 604727, 606077, 607667, 624047, 624467, 624707, 626621, 640007, 640247, 640667, 642077, 644867, 646979, 648887, 649079, 649769, 664667, 664847, 666647, 667697, 668867, 669287, 670049, 670097, 670727, 674699, 676007, 678407, 678647, 679277, 679907, 684767, 689867, 696887, 697727, 699287, 700079, 700277, 704027, 704447, 704777, 706049, 707027, 707669, 708047, 708479, 708989, 709607, 728867, 744407, 746477, 746747, 747449, 747497, 747869, 747977, 762407, 764627, 764969, 767747, 767867, 770447, 770927, 777977, 778697, 779747, 780047, 790967, 796709, 796799, 796847, 798647, 844769, 848789, 848849, 849047, 864047, 864077, 867677, 868487, 868997, 869777, 869807, 870047, 874889, 876077, 876479, 876749, 877907, 878987, 879707, 879797, 880067, 884489, 884789, 886967, 886979, 887669, 887987, 888689, 888779, 888869, 889877, 889907, 894407, 894449, 896447, 897497, 897707, 897779, 898067, 900089, 902261, 904067, 904679, 904997, 906749, 907469, 907997, 908489, 908669, 908849, 908879, 909089, 909287, 909899, 920201, 944687, 944897, 946079, 946487, 946667, 948797, 949889, 960497, 962867, 972407, 974747, 977609, 986189, 986849, 988649, 989477, 990287, 990797, 990887, 992021, 994067, 994247, 994709, 996407, 996647, 997889, 998687, 1886699, 2007077, 2007767, 2020001, 2044067, 2046047, 2060447, 2062001, 2066201, 2404067, 2640707, 2666747, 2706677, 2744447, 2747447, 2766677, 2767067, 2778647, 2866607, 2878847, 2880467, 2887007, 2888747, 4000079, 4007207, 4044767, 4046477, 4047077, 4066877, 4070447, 4072007, 4074647, 4074767, 4076627, 4077947, 4079477, 4089887, 4244477, 4247447, 4404677, 4406747, 4408889, 4409609, 4440899, 4444469, 4446677, 4447607, 4447907, 4449077, 4449407, 4449479, 4449749, 4449899, 4462877, 4470047, 4470467, 4472747, 4474079, 4484897, 4490747, 4490777, 4494989, 4496069, 4497047, 4626467, 4628747, 4640477, 4648067, 4648487, 4672667, 4676447, 4700627, 4726067, 4740647, 4746647, 4747667, 4760069, 4777469, 4800707, 4876607, 4888067, 4888997, 4900997, 4907009, 4944479, 4949447, 4977689, 4987007, 4989947, 4994789, 6000479, 6002201, 6004067, 6009977, 6020447, 6040679, 6044627, 6047609, 6062807, 6074447, 6087077, 6087407, 6220001, 6220061, 6244487, 6267047, 6270767, 6404777, 6444677, 6447767, 6464207, 6466247, 6466277, 6466487, 6472007, 6474407, 6490907, 6600497, 6600677, 6602201, 6604097, 6609221, 6622661, 6628877, 6642707, 6646427, 6660677, 6664277, 6664769, 6674747, 6676667, 6692261, 6697277, 6698687, 6700607, 6704669, 6720407, 6740477, 6777047, 6777467, 6848867, 6876767, 6899777, 6908777, 6927887, 6970277, 6990749, 7004909, 7007069, 7028447, 7044449, 7044977, 7047749, 7060787, 7074047, 7078889, 7079069, 7087889, 7090427, 7096889, 7097807, 7098899, 7206767, 7400207, 7402007, 7404767, 7407077, 7426667, 7427447, 7440047, 7440089, 7440467, 7440689, 7444427, 7444727, 7447079, 7448489, 7460777, 7497407, 7499027, 7499207, 7600067, 7600469, 7600727, 7604669, 7606967, 7607207, 7607627, 7628807, 7662077, 7666247, 7666277, 7666667, 7670807, 7670987, 7674047, 7676609, 7679099, 7684007, 7684667, 7687499, 7690607, 7692887, 7696679, 7697969, 7699607, 7699889, 7707767, 7708007, 7708499, 7708847, 7724447, 7727777, 7746047, 7749947, 7760609, 7760999, 7767497, 7770779, 7770899, 7772447, 7776749, 7777499, 7790087, 7790897, 7796069, 7797899, 7804007, 7804607, 7860467, 7864067, 7864697, 7868849, 7884047, 7884647, 7898789, 7907069, 7920707, 7940627, 7976609, 7977797, 7978979, 7989407, 7989467, 7990667, 7990709, 7998047, 8004677, 8040707, 8440667, 8447849, 8448047, 8467007, 8467889, 8604767, 8644607, 8646707, 8647697, 8666477, 8666807, 8687477, 8704607, 8760467, 8767067, 8770007, 8776697, 8778899, 8784647, 8788667, 8788907, 8790077, 8806667, 8844047, 8844497, 8844767, 8864879, 8868887, 8874497, 8877497, 8877677, 8879747, 8886467, 8888489, 8888849, 8890667, 8898749, 8944487, 8947889, 8964887, 8987789, 8988449, 8998487, 9000779, 9004907, 9007049, 9024077, 9026447, 9028007, 9028067, 9044669, 9047849, 9070727, 9078449, 9079769, 9084077, 9084707, 9087899, 9088787, 9090749, 9096047, 9099887, 9204407, 9226601, 9400889, 9406247, 9426047, 9428777, 9440447, 9440477, 9444887, 9447077, 9447479, 9448067, 9469709, 9494207, 9497069, 9499079, 9499097, 9640487, 9642047, 9642887, 9649097, 9662447, 9664427, 9689987, 9699887, 9704489, 9707699, 9708869, 9708899, 9744869, 9770207, 9770909, 9777869, 9778889, 9779969, 9788489, 9794207, 9798869, 9804077, 9846467, 9846647, 9847889, 9866669, 9869789, 9870407, 9876899, 9888479, 9888887, 9900467, 9904889, 9907769, 9909407, 9909749, 9916889, 9946427, 9948749, 9977447, 9977909, 9981689, 9984647, 9992867, 9994097, 9999047, ...}
This set is currently not known, and might be extremely difficult to found.
Minimal set of "lesser of twin primes" in dozenal:
{3, 5, B}
Interestingly, in dozenal this set is very easy to proven to be complete, since except 3, all such numbers end with 5 or B.
Minimal set of "greater of twin primes" in decimal:
{5, 7, 13, 19, 31, 43, 61, 181, 229, 241, 283, 349, 421, 811, 823, 829, 883, 1021, 2089, 2689, 4801, 4969, 6949, 8089, 8221, 8389, 8821, 9001, 10141, 14011, 20023, 20233, 22111, 23203, 26263, 28099, 29023, 29389, 29881, 32323, 33289, 40429, 40699, 40849, 42409, 44029, 44089, 44269, 48409, 48481, 48649, 48991, 49411, 60649, 60889, 62989, 68449, 68899, 80449, 80491, 88609, 88801, 89899, 91141, 92221, 93889, 94009, 94111, 94441, 94849, 96223, 98809, 98869, 98899, 99991, 111121, 201121, 201211, 202063, 203323, 209203, 209623, 209821, 220021, 223063, 233923, 239233, 240049, 244669, 248869, 249499, 266449, 284899, 288649, 288991, 289021, 292081, 293263, 294649, 294949, 322633, 332203, 336223, 360289, 363889, 386989, 388699, 392263, 392809, 400069, 400249, 402949, 404011, 409891, 411001, 428809, 429889, 440941, 442009, 442489, 444001, 444289, 448999, 449989, 468889, 480049, 484489, 488689, 490249, 491041, 499141, 604699, 622333, 623263, 623869, 624049, 624469, 626623, 640009, 640249, 640669, 644869, 648889, 662899, 664669, 664849, 666649, 668869, 669289, 689869, 692389, 696889, 699289, 808441, 809401, 840841, 840991, 849049, 864049, 868489, 868999, 869809, 880069, 880909, 884491, 886969, 889699, 889909, 894409, 896449, 898069, 902263, 904069, 904999, 908491, 908881, 909289, 920203, 920281, 922081, 922333, 929011, 929809, 930289, 944689, 944899, 946489, 946669, 949891, 960499, 962869, 990289, 990889, 992023, 994069, 994249, 996409, 996649, 998689, 1100041, 1144141, 1144441, 1212121, 1444111, 2008081, 2020003, 2032663, 2044069, 2046049, 2049409, 2060449, 2062003, 2066203, 2090281, 2092021, 2092663, 2092801, 2099221, 2120101, 2203303, 2203633, 2260633, 2266633, 2326663, 2332663, 2333323, 2333869, 2389969, 2396923, 2399809, 2404069, 2446099, 2488909, 2499949, 2623333, 2630323, 2630923, 2669203, 2696923, 2699623, 2800981, 2849809, 2866609, 2880469, 2894449, 2896969, 2909281, 2920363, 2949889, 2966923, 2986969, 2992333, 2998201, 2998999, 2999233, 2999263, 3008809, 3062203, 3088669, 3090223, 3090289, 3092389, 3220603, 3220663, 3222223, 3269923, 3286669, 3289969, 3296233, 3298909, 3326623, 3329233, 3338899, 3386899, 3388069, 3388999, 3620923, 3623899, 3628969, 3866869, 3888889, 3889009, 3922033, 3926233, 3926623, 3928669, 3966889, 3992203, 4000081, 4009141, 4044499, 4049401, 4089889, 4140001, 4141441, 4200949, 4204999, 4299499, 4441111, 4441441, 4449409, 4449481, 4449901, 4484899, 4494991, 4626469, 4642999, 4648069, 4648489, 4649209, 4809949, 4888069, 4888999, 4890499, 4900999, 4949449, 4989949, 4990441, 4991011, 6002203, 6002389, 6004069, 6020449, 6020923, 6026233, 6029923, 6044629, 6062809, 6063289, 6209923, 6220003, 6220063, 6220633, 6238909, 6244489, 6260323, 6264499, 6269233, 6309223, 6320263, 6329203, 6329233, 6362203, 6362623, 6369889, 6448909, 6449899, 6464209, 6466249, 6466489, 6480499, 6490909, 6600499, 6602203, 6604099, 6609223, 6622663, 6629923, 6646429, 6662233, 6692263, 6698689, 6848869, 6866989, 6929203, 6992233, 8099281, 8208001, 8440669, 8448049, 8449099, 8464699, 8644609, 8666809, 8680699, 8800999, 8806669, 8844049, 8844499, 8868889, 8886469, 8888491, 8890669, 8899441, 8944489, 8964889, 8998489, 9004909, 9023869, 9023899, 9026449, 9028009, 9028069, 9029203, 9089281, 9096049, 9099889, 9204409, 9223003, 9223603, 9226603, 9236023, 9236923, 9238969, 9286999, 9294499, 9296923, 9298699, 9299203, 9299449, 9323989, 9326623, 9328999, 9329869, 9329989, 9392923, 9406249, 9426049, 9426499, 9440449, 9444889, 9448069, 9494209, 9499081, 9499099, 9602389, 9640489, 9642049, 9642889, 9649099, 9662449, 9664429, 9689989, 9699889, 9804481, 9844099, 9846469, 9846649, 9888481, 9888889, 9889081, 9896989, 9900469, 9902911, 9904891, 9909409, 9923899, 9923989, 9929203, 9929323, 9946429, 9962389, 9980281, 9984649, 9991441, 9992869, 9994099, 9998281, 9999049, 10100011, 11104411, 11110111, 11110441, 11111101, 11141401, 11210011, ...}
This set is currently not known, and might be extremely difficult to found, e.g. there is an unsolved family {1} (the repunit numbers), since there are no known repunit primes which are greater of twin primes, and the search limit is >=10^6, since the only repunit primes with length <= 10^6 are length {2, 19, 23, 317, 1031, 49081, 86453, 109297, 270343}, and none of the corresponding repunit primes are greater of twin primes (even there is no known prime n except 3 such that (10^n-1)/9-2 is prime, see https://oeis.org/A293272 and https://stdkmd.net/nrr/prime/primecount.txt).
Minimal set of "greater of twin primes" in dozenal:
{5, 7, 11, 61, 91, 221, 2B1, 301, 421, 801, A41, B21, B31, 2481, 2A31, 3241, 3321, 3331, 34A1, 3821, 3A31, 4031, 4041, 4401, 4A31, 4BB1, 83B1, 8441, 8A21, 8A81, A331, B0B1, BB41, 20041, 32831, 344B1, 38B81, 3AAA1, 43341, 43481, 44481, 448A1, 448B1, 48431, 4BA01, 84881, 88331, 88381, 8B481, A00A1, A0AB1, A2081, A3A21, AAAA1, AB8B1, B4001, B4841, B8841, BA081, BBA81, 238431, 2388A1, 23A881, 23AA81, 244331, 2833A1, 2888A1, 2AA881, 338841, 338BA1, 344381, 348881, 400001, 4008A1, 40A8B1, 433881, 4343B1, 438AB1, 44AAA1, 44B841, 483381, 488AA1, 488B41, 48B881, 4AB081, 884AB1, 8AABA1, 8B4BA1, 8BB881, 8BBAB1, A3BA81, A88A31, A8AAB1, AA0881, AA23A1, AA8821, AAB8A1, ABA8A1, ABAAB1, ABAB01, ABBBA1, B44441, B88BA1, B8AAA1, BB0AA1, BBA0A1, BBBBB1, 2003381, 200A8A1, 20338A1, 20444A1, 2083341, 20883A1, 2843AA1, 2A88881, 388AAB1, 388BBA1, 3A88BA1, 3BB8881, 40BA881, 433A8A1, 43A88B1, 444B4A1, 4A88AB1, 4AA88A1, 4ABAA81, 4B000A1, ...}
This set is currently not known, and might be extremely difficult to found.
Minimal set of "average of twin primes" in decimal:
{4, 6, 12, 18, 30, 72, 150, 228, 270, 282, 522, 570, 810, 822, 828, 858, 882, 2088, 2550, 2592, 3252, 3258, 3528, 3558, 3582, 3852, 5010, 5100, 5280, 5502, 5520, 5850, 5880, 7590, 7758, 7878, 7950, 8088, 8388, 8838, 8970, 9000, 10710, 11070, 11700, 11970, 17190, 17790, 17910, 20022, 20232, 20358, 20508, 22092, 22110, 23292, 23538, 25032, 25578, 25800, 25998, 27738, 28098, 29022, 29388, 29880, 32322, 33288, 33588, 35532, 35592, 35838, 35898, 37338, 50052, 50550, 50592, 50892, 53592, 53898, 55050, 55332, 55338, 55932, 57558, 58392, 58788, 58908, 59052, 59208, 59358, 70980, 70998, 77550, 78888, 79398, 79998, 80778, 85092, 85200, 85332, 85932, 88590, 88800, 89520, 89898, 90528, 92220, 92958, 93888, 95088, 95190, 95802, 95958, 95988, 97170, 97500, 97578, 97788, 98808, 98898, 99258, 99528, 99708, 99990, 107070, 109170, 117990, 119100, 170100, 197970, 202290, 203322, 203352, 205398, 207798, 208500, 208590, 209202, 209580, 220020, 229590, 233922, 235788, 238878, 239232, 253788, 255588, 257988, 277788, 280338, 280590, 288990, 290532, 291900, 295038, 295080, 295200, 295878, 299358, 327798, 328788, 357738, 358878, 500238, 500808, 502500, 503382, 503778, 503928, 505278, 508020, 511110, 525000, 525378, 529050, 529578, 535938, 538332, 539838, 550008, 550938, 551910, 553278, 553758, 573738, 577398, 577938, 580032, 583338, 589290, 595038, 595578, 595950, 598932, 700080, 701010, 708990, 710910, 717090, 719010, 733938, 753588, 759558, 759798, 770838, 773778, 773988, 777978, 778050, 778080, 787770, 790170, 791970, 793788, 795798, 800520, 805032, 805500, 807870, 837378, 839352, 859050, 877398, 877908, 877938, 878988, 879798, 880908, 880950, 887988, 888780, 888870, 889878, 889908, 895050, 895902, 899850, 900552, 900588, 901170, 902598, 907398, 907998, 908850, 908880, 909288, 919110, 920202, 922332, 925080, 929010, 929808, 933552, 939738, 955038, 955938, 955992, 958932, 980592, 983532, 985992, 989250, 990288, 990798, 990888, 992022, 995052, 995550, 995592, 997110, 997890, 998070, 1171110, 1709970, 1909110, 2000352, 2000520, 2005020, 2008050, 2020002, 2025900, 2029020, 2029500, 2037378, 2057598, 2057778, 2075538, 2089050, 2090352, 2092590, 2099220, 2200590, 2220552, 2222250, 2222502, 2225052, 2239332, 2250090, 2290032, 2293392, 2299950, 2333322, 2333532, 2333952, 2399598, 2500890, 2502000, 2573358, 2588898, 2599110, 2800950, 2839938, 2895090, 2907978, 2909520, 2922552, 2922990, 2929392, 2980038, 2980950, 2989038, 2990190, 2992332, 2998938, 2998998, 2999232, 3222222, 3238398, 3283338, 3289878, 3329232, 3335952, 3338898, 3339978, 3357798, 3373788, 3388998, 3539778, 3593352, 3777378, 3777888, 3837738, 3879738, 3888888, 3897798, 3923838, 3937398, 3995778, 5000928, 5003838, 5003982, 5005908, 5009088, 5009580, 5009958, 5009988, 5023398, 5025078, 5029008, 5033838, 5052558, 5053578, 5055978, 5057838, 5073378, 5079588, 5080938, 5090250, 5090778, 5098032, 5099988, 5200050, 5203338, 5255358, 5258778, 5273838, 5299338, 5333328, 5339238, 5379378, 5393328, 5399832, 5509080, 5539578, 5539998, 5551110, 5553798, 5555058, 5555190, 5555508, 5559738, 5575398, 5579838, 5580978, 5593788, 5595588, 5759838, 5800020, 5803938, 5835552, 5889888, 5900382, 5908338, 5909088, 5909820, 5923998, 5939778, 5939832, 5952798, 5952978, 5957388, 5959398, 5977338, 5989020, 5990250, 5990820, 5993328, 5995080, 5995398, 5998038, 7007070, 7035798, 7050078, 7059738, 7071900, 7071990, 7077738, 7078398, 7078890, 7079070, 7093578, 7101000, 7110990, 7119900, 7190970, 7199910, 7500798, 7503738, 7505358, 7507938, 7550988, 7553598, 7559988, 7700388, 7703388, 7708008, 7708500, 7770780, 7770900, 7777500, 7790088, 7790898, 7797900, 7890990, 7977798, 7978338, 7978980, 7990710, 7991100, 7997388, 7999710, 8000052, 8000790, 8007090, 8008770, 8009592, 8033592, 8099250, 8099550, 8550990, 8590290, 8595552, 8700870, 8703378, 8705550, 8708070, 8755500, 8770008, 8775000, 8778900, 8780700, 8788908, 8790078, 8800998, 8850000, 8878998, 8880798, 8888850, 8909502, 8993532, 8999778, 9019710, 9022590, 9023898, 9025110, 9029202, 9035778, 9039378, 9055590, 9059382, 9070338, 9079338, 9079770, 9085290, 9088788, 9089052, 9099888, 9101790, 9197010, 9200952, 9229950, 9257838, 9273378, 9275838, 9279888, 9287988, 9289038, 9289338, 9290250, 9295020, 9297888, 9299202, 9299502, 9322932, 9322992, 9323988, 9328998, 9329988, 9353778, 9359352, 9387798, 9392922, 9503952, 9520038, 9529908, 9533388, 9533598, 9537588, 9539238, 9550578, 9550998, 9555738, 9558552, 9582900, 9708870, 9711900, 9770910, 9777870, 9779970, 9795558, 9797190, 9798870, 9837978, 9888888, 9902910, 9905058, 9907770, 9909750, 9909870, 9923898, 9923988, 9929202, 9929322, 9950538, 9950778, 9950820, 9957888, 9958938, 9959778, 9977910, 9980250, 9985500, 9985902, 9991770, 9992532, 9995598, 9998700, 10007970, 10091910, 10100010, 11110110, 11111100, 11199900, ...}
This set is currently not known, and might be extremely difficult to found.
Minimal set of "average of twin primes" in dozenal:
{4, 6, 10, 50, 90, 220, 2B0, 300, 720, 770, 800, B20, B30, B70, 2070, 2A30, 3320, 3330, 3820, 3A30, 7B00, 83B0, 8730, 8A20, 8A80, A330, A700, B0B0, 23730, 28870, 32830, 32A70, 38870, 38B80, 3A7B0, 3AAA0, 70380, 738A0, 73BA0, 7A080, 7AAA0, 7ABB0, 7BAA0, 87BA0, 88330, 88380, A00A0, A07B0, A0AB0, A2080, A3A20, A73B0, A7A30, A7BB0, AA7A0, AA7B0, AAAA0, AB8B0, BA080, BBA80, 238780, 2388A0, 23A880, 23AA80, 2833A0, 2888A0, 2AA780, 2AA880, 338BA0, 37A8A0, 3A2870, 700BA0, 7088A0, 733BB0, 788AB0, 78B880, 7A3B80, 878880, 8827A0, 8AABA0, 8BB880, 8BBAB0, A28370, A378B0, A37BA0, A3A780, A3BA80, A88870, A88A30, A8AAB0, AA0270, AA0880, AA23A0, AA3780, AA8820, AAB8A0, ABA8A0, ABAAB0, ABAB00, ABBBA0, B88BA0, B8AAA0, BB0AA0, BBA0A0, BBBBB0, 2003380, 200A8A0, 20338A0, 20883A0, 2A88880, 3237AA0, 337AB80, 378ABA0, 37AA8B0, 388AAB0, 388BBA0, 3A788A0, 3A7A880, 3A88BA0, 3BB8880, 7000A00, 7088880, 70B8BA0, 70BBB80, 78883A0, 7A88B80, 7BB88A0, 83278A0, 83287A0, 838AA70, 8882AA0, 8883AA0, A008370, A008BB0, A032A80, A038AA0, A03B8A0, A0832A0, A083880, A088370, A0A0380, A2AA870, A3287A0, A3ABBA0, AA00B80, AAA0380, AAABB80, AB0A000, ABBB080, B000A80, B888880, BAAB0A0, BB88B80, BB8BB80, ...}
This set is currently not known, and might be extremely difficult to found.
Minimal set of Mersenne primes in decimal:
{3, 7, 8191}
This set is conjectured to be complete, but not proven.
Minimal set of Mersenne primes in dozenal:
{3, 7}
This set is complete, since except 3, all Mersenne primes end with 7.
Minimal set of Fermat primes in decimal:
{3, 5, 17}
This set is conjectured to be complete, but not proven.
Minimal set of Fermat primes in dozenal:
{3, 5}
This set is complete, since except 3, all Fermat primes end with 5.
Minimal set of "primes plus 1" in decimal:
{3, 4, 6, 8, 12, 20, 72, 90, 110, 150, 252, 500, 510, 522, 570, 710, 770, 992, 1070, 1700, 2222, 5052, 5502, 5592, 7550, 9222, 9552, 555555555552, 5555555555555555555555555555555555555555555555555555555555555555555550}
I think that this set is complete.
Minimal set of "primes plus 1" in dozenal:
{3, 4, 6, 8, 10, 12, 20, 50, 52, 70, 90, 92, B0, 222, 272, 2A2, 2B2, 722, 772, 7A2, A00, AA0, B22, B72, A0A2, 7BBB2, A07B2, A7BB2, AA7B2, AAAA2, AABB2, BAAA2, BBBB2, ABAAB2, ABBAB2, ABBBA2, BABAB2, BBAAB2, BBBAA2}
I think that this set is complete.
Minimal set of "primes plus 2" in decimal:
{4, 5, 7, 9, 13, 21, 31, 33, 61, 63, 81, 111, 283, 823, 883, 2223, 20023, 20203, 22000003}
This set is complete, since any remain number must end with 3 (if end with 1, only from 1{0} can be in front of it, but such numbers minus 2 are divisible by 9 and cannot be prime), and the digits in front of 3 can only be 2{0} or 8{0}, but such numbers minus 2 are divisible by 3 and cannot be prime.
Minimal set of "primes plus 2" in dozenal:
{4, 5, 7, 9, 11, 13, 21, 33, 61, 63, 83, B1, 223, 2A3, 2B3, 301, 801, A01, AA1, B23, 2003, 3A81, 8A81, A0A3, A203, A381, B003, B0B3, BB03, 38881, 88881, AAAA3, AABB3, BAA03, BAAA3, BBBB3, AA00B3, ABAAB3, ABBAB3, ABBBA3, BABAB3, BBAAB3, BBBAA3}
I think that this set is complete.
Minimal set of "primes plus 3" in decimal:
{5, 6, 8, 10, 14, 20, 22, 32, 34, 40, 44, 70, 74, 92, 112, 412, 712, 772, 994, 9004}
This set is complete since for all numbers end with 0, 2, 4, all remain numbers are divisible by 3.
Minimal set of "primes plus 3" in dozenal:
{5, 6, 8, A, 12, 14, 22, 34, 42, 72, 92, 94, B2, 244, 274, 2B4, 302, 332, 404, 474, 4B4, 704, 774, B74, 2004, 4444, 7444, 7B44, B004, B0B4, BB04, BB44, 7BBB4, BBBB4}
This set is complete, since any remain number must end with 4, and any possible digits combo in front of it are already in the set.
Minimal set of "primes plus 4" in decimal:
{6, 7, 9, 11, 15, 21, 23, 33, 35, 41, 45, 51, 83, 143, 255, 285, 443, 503, 525, 801, 825, 881, 885, 2225, 5055, 5085, 5505, 5585, 5805, 5855, 8505, 20025, 20205, 80055, 22000005, 80555555, 555555555555}
I think that this set is complete.
Minimal set of "primes plus 4" in dozenal:
{6, 7, 9, B, 13, 15, 23, 35, 43, 53, 55, 85, 225, 245, 2A5, 303, 333, 405, 425, 803, 833, A03, A45, AA3, 2005, 3A83, 4445, 8A83, A0A5, A205, A383, 38883, 88883, AAAA5, 44AAA5, AAA0005, AA000005}
I think that this set is complete.
Minimal set of "primes minus 1" in decimal:
{1, 2, 4, 6, 30, 58, 70, 78, 88, 990, 5050, 5500, 5590, 8500, 8950, 9000, 9550, 9850, 80050, 80555550, 555555555550}
I think that this set is complete.
Minimal set of "primes minus 1" in dozenal:
{1, 2, 4, 6, A, 30, 50, 80, 90, 700, 770, B70, B000, B0B0, BB00, 7BBB0, BBBB0}
This set is complete, since any remain number must end with 0 and only digits 0, 7, and B can be in front of it, but 700, 770, B000, and BBBB0 are already in the set.
Minimal set of "primes minus 2" in decimal (0 is not counted):
{1, 3, 5, 9, 27, 77, 87, 407, 447, 6467, 60647, 666647, 60000047, 66000047, 66600047}
I think that this set is complete.
Minimal set of "primes minus 2" in dozenal (0 is not counted):
{1, 3, 5, 9, B}
This set is very easy to proven to be complete, since any such number ends with 1, 3, 5, 9, or B.
Minimal set of "primes minus 3" in decimal (0 is not counted):
{2, 4, 8, 10, 16, 50, 56, 70, 76}
This set is complete, since any remain number must end with 0 or 6 and only digits 0, 3, 6, 9 can be in front of it, but combinations of only 0, 3, 6, 9 cannot be such numbers, since all such numbers are divisible by 3.
Minimal set of "primes minus 3" in dozenal (0 is not counted):
{2, 4, 8, A}
This set is very easy to proven to be complete, since any such number ends with 2, 4, 8, or A.
Minimal set of "primes minus 4" in decimal:
{1, 3, 7, 9, 25, 55, 85, 405, 445, 6465, 60645, 666645, 60000045, 66000045, 66600045}
I think that this set is complete.
Minimal set of "primes minus 4" in dozenal:
{1, 3, 7, 9}
This set is very easy to proven to be complete, since any such number ends with 1, 3, 7, or 9.
Minimal set of "primes == 1 mod 4" in decimal:
{5, 13, 17, 29, 37, 41, 61, 73, 89, 97, 101, 109, 149, 181, 233, 277, 281, 349, 409, 433, 449, 677, 701, 709, 769, 821, 877, 881, 1669, 2221, 3001, 3121, 3169, 3221, 3301, 3833, 4969, 4993, 6469, 6833, 6949, 7121, 7477, 7949, 9001, 9049, 9221, 9649, 9833, 9901, 9949, 11969, 19121, 20021, 20201, 21121, 23021, 23201, 43669, 44777, 47777, 60493, 60649, 66749, 80833, 90121, 91121, 91921, 91969, 94693, 111121, 112121, 119921, 199921, 220301, 466369, 470077, 666493, 666649, 772721, 777221, 777781, 779981, 799921, 800333, 803333, 806033, 833033, 833633, 860333, 863633, 901169, 946369, 946669, 999169, 1111169, 1999969, 4007077, 4044077, 4400477, 4666693, 8000033, 8000633, 8006633, 8600633, 8660033, 8830033, 8863333, 8866633, 22000001, 40400077, 44040077, 60000049, 66000049, 66600049, 79999981, 80666633, 83333333, 86606633, 86666633, 88600033, 88883033, 88886033, 400000477, 400444477, 444000077, 444044477, 836666333, 866663333, 888803633, 888806333, 888880633, 888886333, 8888800033, 8888888033, 88888883333, 440444444477, 7777777777921, 8888888888333, 40000000000777, 44444444400077, 40444444444444477, 44444444444444477, 88888888888888633, 999999999999999121, 8888888888888888888888888888888888888888888888888888888888888888888888888888833}
This set is complete, references: https://oeis.org/A111055, https://github.com/curtisbright/mepn...minimal.10.txt
Minimal set of "primes == 1 mod 4" in dozenal:
{5, 11, 31, 61, 81, 91, 221, 241, 271, 2A1, 2B1, 401, 421, 471, 4B1, 701, 721, 771, 7A1, A41, B21, B71, 2001, 4441, 7441, 7B41, A0A1, A201, B001, B0B1, BB01, BB41, 7BBB1, A07B1, A7BB1, AA7B1, AAAA1, AABB1, B04A1, BAA01, BAAA1, BBBB1, 44AAA1, AA00B1, ABAAB1, ABBAB1, ABBBA1, BABAB1, BBAAB1, BBBAA1, AAA0001, AA000001}
This set is complete, reference: https://oeis.org/A111057
Minimal set of "primes == 3 mod 4" in decimal:
{3, 7, 11, 19, 59, 251, 491, 499, 691, 991, 2099, 2699, 2999, 4051, 4451, 4651, 5051, 5651, 5851, 6299, 6451, 6551, 6899, 8291, 8699, 8951, 8999, 9551, 9851, 22091, 22291, 66851, 80051, 80651, 84551, 85451, 86851, 88651, 92899, 98299, 98899, 200891, 208891, 228299, 282299, 545551, 608851, 686051, 822299, 828899, 848851, 866051, 880091, 885551, 888091, 888451, 902299, 909299, 909899, 2000291, 2888299, 2888891, 8000099, 8000891, 8000899, 8028299, 8808299, 8808551, 8880551, 8888851, 9000451, 9000899, 9908099, 9980099, 9990899, 9998099, 9999299, 60000851, 60008651, 60086651, 60866651, 68666651, 80088299, 80555551, 80888299, 88808099, 88808899, 88880899, 90000299, 90080099, 222222899, 800888899, 808802899, 808880099, 808888099, 888800299, 888822899, 992222299, 2222288899, 8808888899, 8888800099, 8888888299, 8888888891, 48555555551, 555555555551, 999999999899, 88888888888099, 2228888888888899, 9222222222222299, 2288888888888888888888899, 888888888888888888888888888888888888888888899, 86666666666666666666666666666666666666666666666651, 21915199}
This set is complete, references: https://oeis.org/A111056, https://github.com/curtisbright/mepn...minimal.10.txt
Minimal set of "primes == 3 mod 4" in dozenal:
{3, 7, B}
This set is very easy to proven to be complete, since except 3, all primes == 3 mod 4 end with 7 or B.
Minimal set of "primes == 1 mod 3" in decimal:
{7, 13, 19, 31, 43, 61, 151, 181, 211, 223, 229, 241, 283, 349, 409, 421, 499, 523, 541, 811, 823, 829, 853, 859, 883, 991, 1021, 1201, 2053, 2089, 2221, 2251, 2281, 2389, 2503, 2521, 2539, 2551, 2593, 2659, 2689, 2851, 2953, 3253, 3259, 3529, 3559, 3583, 3889, 4051, 4111, 4441, 4549, 4591, 4801, 4951, 5011, 5059, 5101, 5209, 5281, 5449, 5503, 5521, 5563, 5569, 5581, 5653, 5659, 5683, 5689, 5821, 5839, 5851, 5869, 5881, 5953, 6469, 6529, 6553, 6949, 8089, 8221, 8389, 8521, 8581, 8689, 8821, 8941, 9001, 9049, 9511, 9649, 9949, 10111, 10141, 14011, 14401, 20359, 20509, 20599, 20959, 23059, 23509, 23563, 23599, 24469, 24889, 25609, 25633, 25801, 25849, 25969, 25981, 25999, 26449, 28069, 28081, 28099, 28201, 28669, 28909, 28921, 29059, 29569, 29581, 29599, 29881, 29959, 29989, 30553, 32869, 33289, 33589, 35053, 35089, 35353, 35533, 35809, 35899, 42589, 42649, 44101, 44269, 44449, 44851, 45259, 45289, 45589, 46489, 48481, 48649, 48889, 49081, 49411, 50053, 50221, 50329, 50383, 50551, 50833, 50893, 50929, 50989, 52021, 52051, 52201, 52489, 52501, 52951, 52999, 53089, 53269, 53299, 53353, 53359, 53593, 53629, 53899, 53959, 54559, 55009, 55051, 55249, 55333, 55339, 55399, 55501, 55849, 55933, 56269, 56299, 56629, 56929, 58099, 58363, 58603, 58909, 58963, 59029, 59083, 59221, 59359, 59509, 59629, 59809, 59833, 59863, 59929, 60259, 60289, 60589, 60649, 60889, 62533, 62563, 62653, 62869, 62989, 63589, 64489, 64849, 65089, 65599, 65809, 65899, 65983, 66889, 68449, 68899, 69259, 80251, 80449, 80491, 82051, 84481, 84649, 88069, 88609, 88801, 88951, 88969, 89449, 89809, 89899, 89989, 90121, 90289, 90481, 90529, 90583, 90841, 91141, 91411, 92353, 92569, 92581, 92809, 92821, 92899, 92959, 93553, 94849, 95083, 95089, 95383, 95539, 95629, 95803, 95929, 95959, 95989, 96259, 96289, 96589, 98041, 98251, 98809, 98869, 98899, 99259, 99289, 99529, 99559, 100411, 111121, 114001, 200569, 200881, 205081, 205549, 209449, 209821, 233353, 236653, 255049, 255589, 256699, 259009, 259459, 263533, 265333, 266353, 280009, 286009, 286999, 288049, 288649, 289489, 289999, 290821, 295459, 298021, 298999, 300589, 302989, 306589, 309289, 329899, 353389, 359389, 404011, 408841, 410401, 424849, 442489, 444289, 444529, 444589, 446569, 449011, 450001, 484489, 491041, 500029, 500083, 500299, 500389, 500629, 500809, 502669, 502699, 503389, 504289, 508009, 508489, 511111, 525949, 530533, 530983, 533389, 533809, 533893, 533989, 535099, 535999, 538093, 538303, 538333, 539389, 550489, 552589, 555589, 559099, 559459, 559549, 559939, 560029, 580033, 580093, 580303, 580633, 580663, 586633, 588949, 589903, 589933, 589993, 590251, 590389, 590899, 592849, 594889, 595549, 598489, 598903, 598999, 599383, 599803, 599899, 599959, 599983, 604249, 605509, 606559, 609253, 620569, 623353, 625489, 625699, 625909, 629509, 645889, 650863, 653083, 653893, 654889, 655489, 658303, 658633, 658663, 660559, 662059, 662353, 662449, 662899, 664459, 665359, 665803, 666559, 666649, 669289, 684889, 688669, 689869, 695389, 695509, 696253, 699253, 800281, 805501, 808081, 808441, 825001, 840841, 844489, 848851, 880909, 880981, 884491, 884881, 886999, 888451, 888469, 889081, 889489, 889909, 890551, 890881, 895051, 898669, 900253, 900259, 900553, 900589, 901111, 901441, 902563, 902599, 902653, 905053, 905551, 905599, 908851, 908881, 909253, 909889, 911011, 911101, 914041, 925663, 926533, 928849, 929869, 941041, 942889, 944551, 944659, 944689, 946669, 950029, 950251, 950269, 952669, 953053, 954259, 955993, 958333, 958849, 958933, 966583, 969253, 969889, 980851, 982801, 988051, 988489, 988501, 988849, 990589, 990889, 993589, 995053, 995833, 995983, 996253, 998989, 999553, 1100041, 1100101, 2004559, 2005459, 2080021, 2500009, 2500081, 2500669, 2535553, 2536663, 2555353, 2588809, 2588899, 2595559, 2656663, 2665363, 2880901, 2888449, 2950009, 2955559, 2955889, 3000289, 3002899, 3653989, 3662809, 3665989, 3909589, 4000081, 4004881, 4014001, 4140001, 4446259, 4455001, 4455559, 4484869, 4642459, 4644259, 4884469, 5000251, 5008063, 5009803, 5028949, 5080003, 5255959, 5266969, 5295559, 5298889, 5300803, 5300863, 5308663, 5330803, 5333329, 5333929, 5338033, 5338633, 5380003, 5380009, 5389003, 5393329, 5528899, 5528989, 5529889, 5551111, 5555509, 5555929, 5555983, 5559259, 5559529, 5559889, 5595559, 5598889, 5599999, 5830333, 5888899, 5889889, 5938003, 5939893, 5952559, 5952889, 5955259, 5955529, 5958889, 5995999, 6004429, 6004459, 6005383, 6005863, 6020449, 6058033, 6058333, 6059959, 6060583, 6066583, 6090559, 6095359, 6095833, 6200059, 6233959, 6250009, 6250099, 6256669, 6288889, 6293359, 6366289, 6442459, 6444259, 6508003, 6533383, 6533803, 6533833, 6539989, 6599389, 6623959, 6624589, 6625459, 6625669, 6653389, 6653833, 6653989, 6655549, 6662809, 6662959, 6663289, 6664429, 6665383, 6666589, 6695863, 6880009, 6900559, 6999589, 8000401, 8005051, 8020801, 8048881, 8090851, 8092801, 8208001, 8288881, 8488981, 8505001, 8550001, 8800009, 8800501, 8800999, 8808841, 8840401, 8844001, 8850001, 8884849, 8885551, 8886649, 8888449, 8888491, 8888851, 8888989, 8889889, 9005509, 9025333, 9058999, 9060559, 9095899, 9099589, 9250009, 9250909, 9253333, 9266563, 9284449, 9424669, 9444889, 9525589, 9550999, 9558889, 9559909, 9559999, 9600559, 9605359, 9606253, 9844501, 9844669, 9845551, 9884551, 9888481, 9888841, 9888889, 9926563, 9953389, 9953533, 9953809, 9958999, 9965863, 9992533, 9996583, 9999889, ...}
This set is very large, may not be complete.
Minimal set of "primes == 1 mod 3" in dozenal:
{7, 11, 31, 51, 61, 81, 91, 221, 241, 2A1, 2B1, 401, 421, 4B1, A41, B21, 2001, 4441, A0A1, A201, B001, B0B1, BB01, BB41, AAAA1, AABB1, B04A1, BAA01, BAAA1, BBBB1, 44AAA1, AA00B1, ABAAB1, ABBAB1, ABBBA1, BABAB1, BBAAB1, BBBAA1, AAA0001, AA000001}
I think that this set is complete.
Minimal set of "primes == 2 mod 3" in decimal:
{2, 5, 11, 17, 41, 47, 71, 83, 89, 149, 443, 449, 677, 743, 773, 797, 881, 887, 977, 1433, 3767, 3779, 6143, 7079, 7307, 7349, 7499, 7607, 7649, 7877, 7949, 9749, 10343, 10463, 13043, 13463, 16943, 19403, 19463, 30707, 31643, 33377, 33749, 36749, 37049, 37337, 37409, 66749, 67049, 67409, 70067, 70667, 73379, 73637, 73679, 73709, 74609, 76367, 76379, 76667, 76679, 77069, 77687, 77699, 77867, 77969, 77999, 80777, 87767, 91463, 91943, 100043, 100403, 100943, 104003, 109943, 133403, 136343, 136403, 139343, 139943, 140603, 140663, 146063, 146603, 160403, 163403, 163643, 164663, 166043, 166403, 166643, 193943, 196043, 196643, 199343, 300749, 301403, 301463, 301943, 303143, 310043, 313343, 314003, 314063, 314603, 319343, 331043, 331943, 338777, 374669, 387077, 387707, 391403, 393143, 603749, 674669, 704009, 704069, 706337, 767909, 770909, 900143, 901403, 901643, 903143, 910643, 991043, 991343, 991643, 1339643, 1460003, 1909043, 1933643, 1990643, 1999043, 3000077, 3000377, 3007787, 3199943, 3301343, 3309143, 3331463, 3339143, 3377807, 3390143, 3391343, 3707777, 3770807, 3778007, 3901043, 3999143, 7000337, 7007087, 7033787, 7070087, 7333367, 7337777, 7400069, 7633337, 7676609, 7700807, 7707767, 7708007, 7766009, 7777409, 7777667, 7777709, 7777769, 7780007, 8707007, 8770007, 9136643, 9140003, 9301043, 9333143, 9909143, 9930143, ...}
This set is very large, may not be complete.
Minimal set of "primes == 2 mod 3" in dozenal:
{2, 5, B}
This set is very easy to proven to be complete, since except 2, all primes == 2 mod 3 end with 5 or B.
Minimal set of non-repdigit primes in decimal:
{13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 151, 181, 211, 227, 251, 257, 277, 281, 349, 409, 449, 499, 521, 557, 577, 587, 727, 757, 787, 811, 821, 827, 857, 877, 881, 887, 911, 991, 2087, 2221, 5011, 5051, 5081, 5501, 5581, 5801, 5851, 6469, 6949, 8501, 9001, 9049, 9221, 9551, 9649, 9851, 9949, 20021, 20201, 50207, 55511, 60649, 80051, 111121, 511111, 666649, 946669, 5200007, 22000001, 60000049, 66000049, 66600049, 80555551, 555555555551, 5000000000000000000000000000027}
I think that this set is complete.
Minimal set of non-repdigit primes in dozenal:
{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, 141, 221, 241, 2A1, 2B1, 2BB, 401, 421, 447, 471, 497, 565, 655, 665, 701, 70B, 711, 721, 747, 771, 77B, 797, 7A1, 7BB, 907, 90B, 9BB, A11, A41, B11, B21, B2B, 1011, 1021, 1101, 11A1, 1201, 1211, 1A01, 2001, 200B, 2011, 202B, 2111, 222B, 229B, 292B, 299B, 4111, 41A1, 4441, 4707, 4777, 6A05, 6AA5, 729B, 7441, 7B41, 929B, 9777, 992B, 9947, 997B, 9997, A0A1, A201, A605, A6A5, AA65, B001, B0B1, BB01, BB41, 100A1, 600A5, 7999B, 9999B, AAAA1, B04A1, B0B9B, BAA01, BAAA1, BB09B, BBBB1, 1AAA21, 44AAA1, A00065, BBBAA1, AAA0001, B00099B, AA000001, BBBBBB99B, B0000000000000000000000000009B, 400000000000000000000000000000000000000077}
This set may not be complete, as there is an unsolved family 1{0}1, and this family is already searched to length 2^24 with no prime found, see http://www.prothsearch.com/GFN12.html
Minimal set of palindromic primes in decimal:
{2, 3, 5, 7, 11, 919, 94049, 94649, 94849, 94949, 96469, 98689, 9809089, 9888889, 9889889, 9908099, 9980899, 9989899, 900808009, 906686609, 906989609, 908000809, 908444809, 908808809, 909848909, 960898069, 968999869, 988000889, 989040989, 996686699, 996989699, 999686999, 90689098609, 90899999809, 90999899909, 96099899069, 96600800669, 96609890669, 98000000089, 98844444889, 9009004009009, 9099094909909, 9600098900069, 9668000008669, 9699998999969, 9844444444489, 9899900099989, 9900004000099, 9900994990099, 900006898600009, 900904444409009, 966666989666669, 966668909866669, 966699989996669, 999090040090999, 999904444409999, 90000006860000009, 90000008480000009, 90000089998000009, 90999444444499909, 96000060806000069, 99900944444900999, 99990009490009999, 99999884448899999, 9000090994990900009, 9000094444444900009, 9666666080806666669, 9666666668666666669, 9909999994999999099, 9999444444444449999, 9999909994999099999, 9999990994990999999, 900000000080000000009, 900999994444499999009, 90000000009490000000009, 90909444444444444490909, 98999999444444499999989, 9904444444444444444444099, 999999999844444448999999999, 90944444444444444444444444909, 99999999999944444999999999999, 99999999999999499999999999999, 9999999999990004000999999999999, 900000000999999949999999000000009, 989999999999998444899999999999989, 9000000999999999994999999999990000009, ..., 9943401999, ...}
This set may not be complete, the reference https://oeis.org/A114835/b114835.txt does not include the large 34023-digit prime in this set, which was found by me.
Minimal set of palindromic primes in dozenal:
{2, 3, 5, 7, B, 11}
In dozenal such set is very easy to proven, since except 2 and 3, all primes end with 1, 5, 7, B, and palindromic primes must have substring 5, 7, B, or 11, except 2 and 3.
Minimal set of non-palindromic primes in decimal:
{13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 211, 227, 251, 257, 277, 281, 349, 409, 449, 499, 521, 557, 577, 587, 811, 821, 827, 857, 877, 881, 887, 911, 991, 1021, 1051, 1151, 1181, 1201, 1511, 1801, 2087, 2221, 5011, 5051, 5081, 5101, 5501, 5581, 5801, 5851, 6469, 6949, 7027, 7057, 7207, 7507, 7727, 7757, 8501, 9001, 9049, 9221, 9551, 9649, 9851, 9949, 10111, 20021, 20201, 50207, 51551, 55511, 60649, 78007, 80051, 111121, 150001, 185551, 511111, 666649, 700087, 777787, 946669, 1000081, 1100101, 5200007, 7700807, 11111101, 22000001, 60000049, 66000049, 66600049, 80555551, 555555555551, 5000000000000000000000000000027}
I think that this set is complete.
Minimal set of non-palindromic primes in dozenal:
{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, 655, 665, 701, 70B, 711, 721, 771, 77B, 7A1, 7BB, 907, 90B, 9BB, A11, A41, B11, B21, 1011, 1021, 1041, 1101, 11A1, 1201, 1211, 1A01, 2001, 200B, 2011, 202B, 2111, 222B, 229B, 292B, 299B, 4111, 41A1, 4441, 4707, 4777, 5565, 56A5, 6A05, 6AA5, 7097, 729B, 7441, 7477, 7747, 7797, 7B41, 929B, 9777, 992B, 9947, 997B, 9997, A0A1, A201, A565, A605, A6A5, AA65, B001, B0B1, BB01, BB2B, BB41, 100A1, 14AA1, 600A5, 70047, 79977, 7999B, 9999B, AAAA1, B002B, B022B, B04A1, B0B9B, BAA01, BAAA1, BB09B, BBBB1, 111141, 114141, 1AAA21, 44AAA1, 704007, A00065, BBBAA1, 1114411, 5000065, 5000A65, 500A065, 5060005, 7400007, AAA0001, B00099B, AA000001, BBBBBB99B, 56000000005, B0000000000000000000000000009B, 400000000000000000000000000000000000000077}
I think that this set is complete.
Last fiddled with by sweety439 on 2021-02-24 at 18:08 |
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2021-02-21, 11:34
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#120 | |
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Nov 2016
22×3×5×47 Posts |
Quote:
More references: https://primes.utm.edu/glossary/page...t=MinimalPrime http://www.bitman.name/math/article/730 https://cs.uwaterloo.ca/~cbright/reports/mepn.pdf https://cs.uwaterloo.ca/~cbright/tal...mal-slides.pdf https://www.primepuzzles.net/puzzles/puzz_178.htm https://arxiv.org/pdf/1607.01548.pdf https://scholar.colorado.edu/downloads/hh63sw661 http://www.cs.uwaterloo.ca/~shallit/Papers/minimal5.pdf http://www.bitman.name/math/table/497 https://github.com/curtisbright/mepn-data https://github.com/RaymondDevillers/primes Last fiddled with by sweety439 on 2021-02-22 at 21:35 |
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2021-02-21, 11:44
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#121 | |
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Nov 2016
22×3×5×47 Posts |
Quote:
Update sieve files. |
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