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Old 2021-01-05, 12:51   #111
sweety439
 
Nov 2016

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Quote:
Originally Posted by sweety439 View Post
Up to base 84 are checked:
Up to base 96 are checked:

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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Old 2021-01-05, 13:19   #112
axn
 
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You've been asked to not quote entire posts unnecessarily.
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Old 2021-01-05, 15:49   #113
sweety439
 
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I want to make the reader know this post is a response of post #108
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Old 2021-01-09, 17:08   #114
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Update status files in base 36
Attached Files
File Type: txt base 36 OLZ family status.txt (78.6 KB, 16 views)
File Type: txt base 36 PSZ family status.txt (101.0 KB, 16 views)
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Old 2021-01-09, 18:58   #115
Batalov
 
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Quote:
Originally Posted by sweety439 View Post
I want to make the reader know this post is a response of post #108
What reader? You think that you have readers? Wow!
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Old 2021-01-09, 19:06   #116
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Quote:
Originally Posted by Batalov View Post
What reader? You think that you have readers? Wow!
Don't you love a response without any real content?


Oh wait, that is the norm...
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Old 2021-02-15, 16:02   #117
sweety439
 
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9943401999 is minimal palindromic (probable) prime, its formula is (895*10^34021+491)/9, the b-file for A114835 (minimal palindromic prime) is not complete.

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Old 2021-02-15, 16:05   #118
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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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Old 2021-02-20, 21:45   #119
sweety439
 
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Quote:
Originally Posted by sweety439 View Post
Many hard problems:

* Minimal set of strings of primes in bases 2<=b<=256
* Minimal set of strings of primes with >=2 digits in bases 2<=b<=256
* Minimal set of strings of composites in bases 2<=b<=256
* Minimal set of strings of composites with >=2 digits in bases 2<=b<=256
* Minimal set of strings of squares in bases 2<=b<=256
* Minimal set of strings of squares with >=2 digits in bases 2<=b<=256
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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Old 2021-02-21, 11:34   #120
sweety439
 
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Quote:
Originally Posted by sweety439 View Post
Minimal set of powers of 2 in decimal:

{1, 2, 4, 8, 65536}

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, ...}

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, ...}

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, ...}

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, ...}

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}

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, ...}

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, ...}

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, ...}

Minimal set of emirps in decimal:

{13, 17, 31, 37, 71, 73, 79, 97, 149, 199, 359, 389, 941, 953, 983, 991, 1009, 1021, 1061, 1069, 1091, 1109, 1151, 1181, 1201, 1229, 1259, 1511, 1559, 1601, 1619, 1669, 1811, 1901, 3023, 3049, 3083, 3203, 3299, 3343, 3433, 3463, 3469, 3583, 3643, 3803, 3853, 3929, 7027, 7057, 7207, 7457, 7507, 7547, 7577, 7687, 7757, 7867, 9001, 9011, 9029, 9161, 9209, 9221, 9293, 9349, 9403, 9439, 9521, 9551, 9601, 9643, 9661, 9923, 10159, 10859, 10889, 11159, 11161, 11621, 12119, 12241, 12611, 12641, 12689, 12809, 12841, 14081, 14221, 14251, 14551, 14621, 14821, 15241, 15289, 15461, 15541, 15661, 16111, 16451, 16481, 16651, 16829, 18041, 18089, 18169, 18269, 18461, 18691, 18859, 19681, 30029, 32009, 32233, 32353, 32369, 32563, 32633, 32693, 32933, 33029, 33223, 33329, 33623, 33863, 33923, 35323, 35363, 36209, 36269, 36353, 36523, 36833, 39623, 70667, 74077, 76607, 77047, 90023, 90059, 90089, 90263, 90499, 90821, 90989, 91121, 91129, 92003, 92033, 92119, 92189, 92333, 92369, 92459, 92489, 92639, 92861, 92899, 92959, 93629, 94559, 94889, 95009, 95101, 95111, 95429, 95549, 95801, 95881, 95929, 96181, 96263, 96281, 96289, 96323, 96329, 98009, 98081, 98129, 98251, 98269, 98299, 98429, 98621, 98801, 98849, 98909, 98999, 99289, 99409, 99829, 99989, 100411, 101119, 101141, 104801, 108401, 108881, 111119, 111211, 111829, 111869, 112111, 114001, 114041, 116911, 119611, 121421, 121921, 124121, 125551, 125651, 125821, 126551, 126851, 128521, 128551, 129121, 129281, 140411, 141101, 144481, 144541, 145441, 145861, 146161, 146581, 152981, 155521, 155581, 155621, 155821, 155851, 156521, 158551, 158621, 158981, 161641, 162881, 168541, 182921, 184441, 185291, 185551, 185641, 185681, 185819, 186581, 186889, 188261, 188609, 188801, 189251, 189851, 192581, 302609, 302629, 305603, 305633, 306329, 306503, 322229, 322249, 322649, 323333, 324293, 328883, 329663, 330053, 333253, 333323, 333563, 335633, 335653, 336503, 336533, 338383, 340453, 340909, 340999, 344053, 344293, 344453, 344843, 348443, 349399, 349493, 350033, 350443, 350663, 352333, 354043, 354443, 356533, 362293, 362339, 362449, 363683, 364909, 365333, 366053, 366239, 366293, 366409, 366923, 383683, 383833, 386363, 386383, 388823, 392263, 392423, 392443, 392663, 394943, 394993, 399493, 700067, 704447, 704747, 707767, 724747, 725587, 725827, 727487, 727877, 728527, 728747, 740087, 744407, 746267, 746747, 747407, 747427, 747647, 747827, 748877, 755267, 760007, 762557, 762647, 766477, 767707, 774667, 777787, 777877, 778727, 778777, 778847, 780047, 780887, 784727, 785527, 787777, 788087, 904459, 904663, 906203, 906881, 906949, 908549, 908959, 909043, 909463, 909599, 911101, 911111, 915869, 918581, 918829, 921629, 922223, 922499, 923603, 924299, 926129, 926203, 928111, 928159, 928289, 928469, 928819, 928859, 932663, 933263, 940469, 942223, 942569, 944263, 945089, 945809, 946223, 946459, 946949, 948659, 949609, 949649, 950569, 951089, 951829, 951859, 952859, 954409, 954649, 954869, 956569, 956689, 956849, 958159, 958259, 958829, 959489, 959689, 959809, 959869, 960889, 964049, 964829, 965059, 965189, 965249, 965659, 965969, 968111, 968459, 968519, 968959, 969569, 980159, 980549, 981569, 982829, 984959, 986659, 986959, 988069, 988681, 992429, 993943, 994229, 995699, 995909, 996599, 999043, ...}
Although some such minimal sets are not known, and might be extremely difficult to found (these sets may be more difficult to found in larger bases, such as bases 17 through 64), the basic theorem of minimal sets says that the minimal set is always finite. However, determining the number of elements in these minimal sets is very difficult.

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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Old 2021-02-21, 11:44   #121
sweety439
 
Nov 2016

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Quote:
Originally Posted by sweety439 View Post
Base 40:

S{Q}d (86*40^n+37)/3: currently at n=87437, no (probable) prime found
Reserve this family to n=100K

Update sieve files.
Attached Files
File Type: txt k.txt (12 Bytes, 11 views)
File Type: txt sr_40.txt (1.3 KB, 13 views)
File Type: log srsieve.log (159 Bytes, 13 views)
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