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Old 2021-01-03, 10:41   #67
sweety439
 
Nov 2016

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Originally Posted by sweety439 View Post
This file is the smallest prime (not include x or y themselves) in given simple family x{y} or {x}y (where x,y are base b digits) in given base 2<=b<=24, where gcd(x,y) = 1, gcd(y,b) = 1 (searched up to 5000 base b digits, 0 if no such prime found (include the case such that x{y} or {x}y proven composite by all or partial algebra factors)

format of file:

b,x,{y}: smallest prime of the form x{y} in base b
b,{x},y: smallest prime of the form {x}y in base b

such primes are generalized near-repdigit primes base b

already excluded families x{y} and {x}y with NUMERICAL covering set (e.g. {1}3, {1}4, 3{1}, 4{1} in base 5)

Such primes are ALWAYS minimal prime (start with 2 digits) in base b, except when the repeating digit (i.e. y in x{y}, or x in {x}y) is 1 and base b has generalized repunit primes (i.e. all digits are 1) smaller than the prime (in base b = 9, 25, 32, 49, 64, 81, 121, 125, 144, 169, 216, 225, 243, 289, 324, 343, 361, 441, 484, 529, 625, ..., no generalized repunit primes exist, thus in these bases b, such primes are always minimal primes (start with 2 digits) in base b)

Also,

* in base 35, all such primes with <= 313 digits are minimal primes (start with 2 digits)
* in base 39, all such primes with <= 349 digits are minimal primes (start with 2 digits)
* in base 47, all such primes with <= 127 digits are minimal primes (start with 2 digits)
* in base 51, all such primes with <= 4229 digits are minimal primes (start with 2 digits)
* in base 91, all such primes with <= 4421 digits are minimal primes (start with 2 digits)
* in base 92, all such primes with <= 439 digits are minimal primes (start with 2 digits)
* in base 124, all such primes with <= 599 digits are minimal primes (start with 2 digits)
* in base 135, all such primes with <= 1171 digits are minimal primes (start with 2 digits)
* in base 139, all such primes with <= 163 digits are minimal primes (start with 2 digits)
* in base 142, all such primes with <= 1231 digits are minimal primes (start with 2 digits)
* in base 152, all such primes with <= 270217 digits are minimal primes (start with 2 digits)
* in base 171, all such primes with <= 181 digits are minimal primes (start with 2 digits)
* in base 174, all such primes with <= 3251 digits are minimal primes (start with 2 digits)
* in base 182, all such primes with <= 167 digits are minimal primes (start with 2 digits)
* in base 183, all such primes with <= 223 digits are minimal primes (start with 2 digits)
* in base 184, all such primes with <= 16703 digits are minimal primes (start with 2 digits)
* in base 185, all such primes with <= 66337 (at least) digits are minimal primes (start with 2 digits)
* in base 199, all such primes with <= 577 digits are minimal primes (start with 2 digits)
* in base 200, all such primes with <= 17807 digits are minimal primes (start with 2 digits)
* in base 201, all such primes with <= 271 digits are minimal primes (start with 2 digits)
Family x{d}y (where d is base-b digit, x.y are base-b strings (may be empty)) in base b means family xddd...dddy in base b, i.e. means this set {xy, xdy, xddy, xdddy, xddddy, xdddddy, xddddddy, ...} in base b
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Old 2021-01-03, 11:07   #68
sweety439
 
Nov 2016

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Originally Posted by sweety439 View Post
search the simple families x{0}y with gcd(x,y) = 1, gcd(y,b) = 1, gcd(x+y,b-1) = 1

Such primes are ALWAYS minimal prime (start with 2 digits) in base b, except when the base (b) is prime, and x = 1 (while 10 is prime and a subsequence of the prime, but with LaurV's suggestion, the prime 10 (i.e. the prime = base) is also not counted just as the primes < base, all such primes (i.e. all smallest primes of the form x{0}y) is ALWAYS minimal prime (start with b+1) in base b)
the formula of family x{0}y in base b is very easy, it is just x*b^n+y

since any number ending with the digit 0 is divisible by the base (b) and thus cannot be prime >b (our set is the primes >b in base b), and any number cannot begin with the digit 0, therefore, all digits 0 in the numbers in the minimal set of our set (in any base b>=2) are middle digits (i.e. neither the left-most digit nor the right-most digit).
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Old 2021-01-03, 11:10   #69
sweety439
 
Nov 2016

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Quote:
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Minimal set of prime-strings (> base, as LaurV's suggestion) in bases 2 to 12 (only bases 2 to 8 are proved to be complete)
start searching bases 13 to 16

the minimal primes (start with b+1) are

Code:
13: {14, 16, 1A, 23, 25, 2B, 32, 34, 38, 41, 47, 49, 52, 56, 58, 61, 65, 6B, 76, 7A, 7C, 83, 85, 89, 9A, A1, A7, A9, B6, B8, C1, C7, CB, 10C, 119, 11B, 122, 133, 155, 157, 173, 179, 17B, 188, 197, 1B1, 1B5, 1CC, 209, 212, 218, 229, 272, 274, 281, 287, 292, 296, 298, 29C, 2C9, 311, 313, 331, 33B, 355, 371, 373, 379, 397, 3A6, 3AA, 3B3, 3B9, 3BB, 3CA, 43C, 445, 44B, 45A, 463, 4A3, 4A5, 4B2, 4B4, 4BA, 50C, 511, 515, 533, 54A, 551, 559, 571, 575, 57B, 595, 599, 5B3, 5B9, 5CC, 607, 629, 63A, 643, 674, 704, 715, 724, 728, 731, 737, 739, 742, 751, 75B, 773, 775, 779, 782, 784, 791, 793, 797, 7B1, 812, 818, 874, 878, 8AB, 8B4, 902, 919, 922, 926, 92C, 937, 93B, 946, 95B, 962, 968, 971, 977, 979, 982, 98C, 9B3, 9B5, A03, A3C, A45, A4B, A54, AA3, AAB, B02, B0C, B11, B15, B17, B24, B33, B39, B42, B57, B59, B71, B93, B9B, BA4, BAA, BB1, BB9, BC2, BCC, C29, C43, C98, CA3, 1013, 1031, 1037, 105B, 1075, 10B7, 10BB, 1105, 1112, 1121, 1127, 113C, 1172, 1187, 1208, 1211, 1277, 12C8, 1307, 1309, 131C, 139C, 151C, 1721, 1727, 1787, 1901, 1909, 1912, 1918, 193C, 1981, 198B, 199C, 19B2, 19C3, 1B29, 1BB2, 1BBC, 1C28, 1C39, 2021, 2078, 2117, 2201, 2221, 2267, 2278, 2627, 2678, 2711, 2771, 2788, 3037, 3053, 306A, 3077, 3091, 309B, 30AC, 3305, 353C, 35AB, 35BA, 35BC, 3677, 3905, 390B, 39C5, 3A0C, 3AB5, 3B5C, 3C35, 3C59, 3C95, 403A, 40AB, 4333, 435B, 4403, 44C3, 4535, 4544, 454C, 45B5, 45BB, 480B, 4B35, 4B5B, 4C36, 5057, 5077, 509B, 50A4, 5107, 5305, 530B, 539C, 53AB, 53C9, 5444, 5455, 54C4, 5503, 5545, 55AB, 5774, 5794, 590B, 594B, 5974, 59B4, 5A4C, 5A53, 5AA4, 5AB5, 5ABB, 5ACA, 5B4B, 5B5A, 5BA5, 5CA4, 6227, 6278, 6667, 6698, 6733, 6872, 6928, 6944, 694C, 6973, 6986, 6997, 69C8, 6AC3, 6C92, 6C94, 7019, 7057, 70B5, 7103, 710B, 7118, 7127, 7129, 7172, 7178, 7192, 7211, 7217, 7219, 7271, 7303, 7408, 7433, 7444, 7505, 7507, 7574, 770B, 7774, 7778, 7787, 7871, 7877, 7888, 794B, 7994, 79B4, 7B43, 7B74, 7B94, 7BB2, 8027, 8072, 8081, 80BA, 8171, 8207, 821C, 848B, 8687, 8711, 8722, 87BB, 8867, 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D00BA9, D05EDD, D077D7, D10CCB, D22207, D4000B, D4040D, D4044D, D40CCB, D70077, D7D00D, D90009, D900BB, DB00BB, DB4441, DD400D, DDD109, DDD1A9, DDD919, DDD941, DED00D, E00D4D, E00EEB, E0AAE1, E0AE41, E0AEA1, E0B44D, E0BCCD, E0BEBB, E0D0DD, E0E441, E4048D, E4448D, E800CD, E8200D, EA0E41, EAA0E1, EBB00B, ECCCAB, EDDDDD, EEBE0B, F00263, F0056F, F00A45, F02C63, F03F23, F05405, F060AF, F08585, F0A4A5, F0F2C3, F0F323, F2CCC3, F33203, F33C23, F5F66F, F5FF6F, F68CCF, F6AA8F, F888AF, FA0F45, FAA045, FAA545, FAFC69, FC0AAF, FC66AF, FCCCAF, FCFFAF, FF0323, FF056F, FF3203, FF7903, FFA045, FFA4A5, FFAA45, FFC0AF, FFF4A5, FFF575, FFFA45, FFFCAF, 10A009B, 20000D1, 2CCC663, 30A00FF, 30CCCAF, 30FA00F, 30FCCAF, 3333C23, 333C2C3, 33C3AAF, 33FCAAF, 33FFFAF, 3A0A00F, 3AAAA0F, 3AF000F, 3AFAAAF, 3C0CA0F, 3CCC3AF, 3CFF323, 3F33F23, 3FAA00F, 3FF3323, 4004441, 400DDD1, 400E00D, 400ED0D, 404404D, 404448D, 404E4DD, 440EDDD, 4440EDD, 44444ED, 4444E4D, 44DDDDD, 4A000A5, 4CCCCAB, 4D0CCCB, 4E4404D, 4E4444D, 4E4DDDD, 5000021, 5004221, 5006AAF, 500FF6F, 5042201, 508CCCD, 5400005, 5400AA5, 5555405, 5808007, 5AA4005, 5C0008D, 5CCC8CD, 5D4444D, 5EEEEEB, 5F40005, 5F554A5, 5F6AAAF, 60000AF, 60006A9, 600866F, 6008AAF, 600AA8F, 600F6A9, 606608F, 606686F, 608666F, 60AA08F, 60AAA8F, 66000AF, 66666A9, 6666AF9, 6866A8F, 6AAAAAF, 70070D7, 70077DD, 700DDDD, 707077D, 707D007, 70D00DD, 770077D, 770400D, 770740D, 7777775, 77777B7, 77777DD, 7777ACB, 77B88E7, 77DD00D, 77DDDDD, 7D0D00D, 7DD0D07, 7DDD00D, 800002D, 8000CED, 80C0E0D, 80CECCD, 840400D, 844000D, 844E00D, 868688F, 880444D, 884404D, 887D007, 8888801, 8888881, 8888E07, 8888F77, 8888FE7, 88A8AFF, 88AAAFF, 88FAFFF, 8A8AAAF, 8A8AAFF, 8AAA8FF, 8C00ECD, 8C8444D, 8E4400D, 8FCCCCF, 900BBAB, 90CC4AB, 9908AA1, 99E0E01, 9B00801, 9B6CCC9, A000FF9, A006069, A00A8FF, A01CCCB, A05F545, A0BEEEB, A0E4AA1, AA0008F, AA08FFF, AA40AA5, AA8FFFF, AAAA405, AE04AA1, AE44441, AE4AAA1, AECCCCB, AF40005, AFA5A45, AFFFC69, B000BAB, B000EBB, B0D0007, B222227, B6CCCC9, B8880A1, BA000EB, BA0BEEB, BAEEEEB, BB000CD, BB00C0D, BB0B00D, BC6CC69, BC6CCC9, BCCCC69, BCCCCED, C0000A9, C00068F, C000CFD, C000E2D, C000FAF, C004D4D, C00E20D, C00E8CD, C00F68F, C033A0F, C0802CD, C086AAF, C0A00AF, C0AFFFF, C0C086F, C0C0F8F, C0CA00F, C0CC08F, C0D044D, C0F0AFF, C0FF023, C0FFFAF, C33FA0F, C33FAAF, C3CA00F, C3FFCAF, C8002CD, C8200CD, CCC668F, CCCAA8F, CCCC0A9, CCCC3AF, CCCCCA9, CCCDC4B, CE0008D, CE2000D, CE8CCCD, CF000AF, CFF0AAF, CFFF0AF, D0000EB, D0005EB, D000775, D000EDD, D007077, D00DDD9, D00ED0D, D0AAA45, D0AAAA5, D0EDDDD, D19000B, D4404ED, D4440ED, D5BBBBB, DCCCC4B, DD00DD9, DD07077, DD0DD09, DD0DDD9, DD99999, DDD0D09, DDDD0D9, DDDD9E1, DDDDD09, DDDDD99, DE0DDDD, DEEEEEB, E00001B, E0004A1, E000CAB, E00A041, E00BB0B, E00BBBB, E00C80D, E00CCCB, E044DDD, E0AA4A1, E0AAA41, E0BBB0B, E0D444D, E40444D, E4DDD4D, E88CCCD, E8C000D, E8CCCCD, EA04441, EA0A4A1, EBB000D, EBCCCCD, ED0D00D, EEAAA01, EEBBBBB, EEE000B, F0002C3, F002CC3, F003323, F005545, F00F4A5, F033323, F0400A5, F0A5545, F333323, F333F23, F6660AF, F733333, FA00009, FA004A5, FAAAA45, FC6668F, FCC668F, FD00AA5, FEE7777, FF0F263, FF26003, FF3F323, FF5F887, FFAFF45, FFFF263, FFFF379, 2CCCCC63, 30CCA00F, 33333319, 3333FCAF, 3333FFAF, 33FFA00F, 3C00CCAF, 3C00FCAF, 3CF3FF23, 40000441, 40000CAB, 4000DAA1, 400440DD, 400ACCCB, 400CCCAB, 400E44DD, 4040D00D, 404400DD, 40444EDD, 4044D00D, 40ACCCCB, 40DDDDDD, 440000D1, 44000DDD, 4400DD0D, 44E400DD, 4A00004B, 4A0AAAA5, 5000C08D, 52000CCD, 555400A5, 55540A05, 58800007, 58888087, 5A540005, 5C00020D, 5F5400A5, 5F888887, 60006AAF, 600093AF, 600AAAAF, 608CCCCF, 6600686F, 6606866F, 6688AAAF, 7000077D, 70000D5D, 7000707B, 7000707D, 7000740D, 70500D0D, 7070040D, 707007DD, 7070777B, 7077744D, 7077777B, 77007D0D, 7700B44D, 7707000B, 7707D00D, 7770700D, 7770777B, 7777740D, 7777770B, 7777777D, 77777CAB, 7777B887, 778888E7, 788888E7, 79333333, 7ACCCCCB, 7D0000DD, 7D00D0DD, 7DD00D0D, 7DDDDDA9, 80000081, 80000087, 8000E0CD, 80400E4D, 80A0AAA1, 80EC000D, 84000E4D, 8404444D, 84400E4D, 868AAAAF, 86AAAA8F, 8884044D, 88FFFE77, 8C44444D, 8CCCCAAF, 8E40004D, 900000BB, 90000B0B, 90100009, 90800AA1, 93333AAF, 94AAAAA1, 980000A1, 998AAAA1, A00000F9, A0000EEB, A0005A45, A0055545, A00AAA45, A0666669, A0AAA045, A0AAAA45, A0AAE4A1, A0B44441, A4A00005, A6066669, A8AAFFFF, AA055545, AA0AA045, AAA00A45, AAAAA045, B00000AB, B000EEEB, B00EEE0B, B0900081, B0BBBBAB, B7777787, B9000081, B9008001, B9800001, BA00000B, BBBB0ABB, BCCCCCC9, ...}

Last fiddled with by sweety439 on 2021-01-04 at 14:10
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Old 2021-01-03, 11:13   #70
sweety439
 
Nov 2016

B0416 Posts
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Largest minimal primes in simple families for bases 13 to 16 (written in decimal):

x{y} and {x}y:

Code:
13, {1}, 1: 30941
13, 1, {1}: 30941
13, {1}, 2: 2381
13, 1, {2}: 197
13, {1}, 3: 883708283
13, 1, {3}: 211
13, {1}, 4: 17
13, 1, {4}: 17
13, 1, {5}: 239
13, {1}, 6: 19
13, 1, {6}: 19
13, 1, {7}: 253217502498750291800692183145337720992638880271493569431738157631027569095215561
13, 1, {8}: 281
13, {1}, 9: 191
13, 1, {9}: 27130132404659193376721686434661
13, {1}, 10: 23
13, 1, {10}: 23
13, {1}, 11: 193
13, 1, {11}: 820195757799727198696695842441476208994963187388611974376331352215246160014077762227387500472080168786083449145277773838858800795991958180632280974279961571375401216690067428169654353503039823288371270608465421195831630752643523277310127185899536482776399744897654945387104284638368882957170938237278749541598968947323011480936838588602269948473325034154837529102013688233548651627077006312693219100274314806851388327828512220445130387062692274032398838117351850000439212156970074507805886649022038706852334408911410638474707605620484783396663735375992701354072765197190488304749330337810593696686818871540035682031309739770552183078238960961
13, {1}, 12: 36898271981403391525359432679065451
13, 1, {12}: 337
13, {2}, 1: 4759
13, 2, {1}: 106637277112689077
13, {2}, 3: 29
13, 2, {3}: 29
13, {2}, 5: 31
13, 2, {5}: 31
13, {2}, 7: 804473
13, 2, {7}: 959173
13, {2}, 9: 373
13, {2}, 11: 37
13, 2, {11}: 37
13, {3}, 1: 547
13, 3, {1}: 521
13, {3}, 2: 41
13, 3, {2}: 41
13, {3}, 4: 43
13, 3, {4}: 43
13, {3}, 5: 6220138738168647434831423806501836269264860388724755065149565951442236175936796719067011700334477855412396475124991691344796424350196661821800919998163958711515136934394634097179800502201458093822871
13, 3, {5}: 577
13, 3, {7}: 2923035083
13, {3}, 8: 47
13, 3, {8}: 47
13, 3, {10}: 647
13, {3}, 11: 557
13, 3, {11}: 661
13, {4}, 1: 53
13, 4, {1}: 53
13, {4}, 3: 3534833123
13, 4, {3}: 9337
13, {4}, 5: 733
13, 4, {5}: 13799574804865291194219202692403868309075434457702185936718792737
13, {4}, 7: 59
13, 4, {7}: 59
13, {4}, 9: 61
13, 4, {9}: 61
13, {4}, 11: 739
13, {5}, 1: 911
13, 5, {1}: 859
13, {5}, 2: 67
13, 5, {2}: 67
13, {5}, 3: 4418541403
13, 5, {3}: 887
13, {5}, 4: 277256920492991599
13, 5, {4}: 11717
13, {5}, 6: 71
13, 5, {6}: 71
13, 5, {7}: 89921716241132417850870043495861784708189815671814466610282151602592361843582839405888101361717803180497433838124247307746249298923864405720133765156792108299126338169033221693415179426566058950783399084612596039386427021973466021161486411164221010228501523
13, {5}, 8: 73
13, 5, {8}: 73
13, {5}, 9: 919
13, 5, {9}: 971
13, {5}, 11: 3604339966408890811
13, 5, {11}: 815662743439
13, 5, {12}: 1013
13, {6}, 1: 79
13, 6, {1}: 79
13, {6}, 5: 83
13, 6, {5}: 83
13, {6}, 7: 14281
13, 6, {7}: 9624308578305020228623
13, {6}, 11: 89
13, 6, {11}: 89
13, {7}, 1: 178099219309623994113347865115830245881192672331136086644940932390281437773862542264058288467257753659801600961075130547842788841314191170557163588785806820352498775469685611756153126667905685532469070927112200362751893732722602014850726541568809237898479136618304580608401918281234930143064257920309170965969499129532757126377921069913907908471671716574775646406870993578835592914767961359736945068963155952491960373644449418976551172435322847621930721542144058540845006834135041368564051814099612330378729974668784588333951075548113502723356005589020090498660438660159816508080021058414799091444088419142837713138761057222122289170719363548408846549381700350017033339161308029642370902736513421829569951497819557491865511689513661142885392815700489495652470272195143512237819660601060635160718950091398065703223634277265560584842616957317643843802700518533058037207116004784352250055837483109124201522964355475650386943932211405298597905457035869654221665584889762003382262425915627119116388998518127939798007098237014217655662706251901683109346257926396141021838296111310372968225022079707942822795129744698997404408233889930805048836987175297692891091686536077480054995854699722419639970465336199946812908942288140516904083829095941075082576747429233198452931491612278651981884049653724609828879600848950258125404977404144280808141884961849927596265042459349498828074205132275149771178481317527349922743964367067517723323072236797216719438348669059261865669008875549670954883841127641347209610157306553559175511522826280497259599838188113180381333755455900349307408532955497746124837076056334168406604339828775363723268449080391071951364995984584279858031164184210011834081
13, {7}, 2: 2815633
13, 7, {2}: 425554642597531069476088349319265344520247822549860785329030452393239592223053102610453
13, {7}, 3: 1277
13, {7}, 4: 16657
13, 7, {4}: 16111
13, {7}, 5: 1279
13, {7}, 6: 97
13, 7, {6}: 97
13, {7}, 8: 16661
13, 7, {8}: 16843
13, {7}, 9: 1283
13, {7}, 10: 101
13, 7, {10}: 101
13, {7}, 11: 176677145512151
13, {7}, 12: 103
13, 7, {12}: 103
13, {8}, 1: 201916737728161
13, {8}, 3: 107
13, 8, {3}: 107
13, {8}, 5: 109
13, 8, {5}: 109
13, {8}, 7: 948090435833789898349839159224250448489822963231314463923791
13, 8, {7}: 91021952951
13, {8}, 9: 113
13, 8, {9}: 113
13, {8}, 11: 247531
13, 8, {11}: 107838136118779143544110144382426092596204365203552428434248370827269303781694390904131147239057991725836898042412261207750257957583868509005708271654322505648777710176276577432900281939636108415883809919452408705622432187985384594384558596142925993309255985009535307478523156581759235779167162869019703196846751228017411378948853640401871368029723240618590933942573810568367308325289
13, {9}, 1: 121215212228974701436931483878589891140303089832220953724825443452928225193998713337930372911067805791799869147809411855365238746332705480023760792811129961633638311977064834817213347105220337863208732908350674718836123175940417328579685338580206495293657407565505774954042639105332034212418635645417015477944123471063765451261909023510740139021
13, 9, {1}: 259429
13, 9, {2}: 1549
13, {9}, 4: 68799997644951462493799712001
13, 9, {4}: 36748846266526697
13, {9}, 5: 1513040167438129868806971467209988783047172742217267483154894329901717663980542315652349848778853894915555170807951322572378209558278635157172717513628242832984817118678991631720815452108000099210400379825153731084262244319493812890311789045794884092831356123242061758082732806604188005520228674939217665328388067373699901394102699076617987792004875548394303529748128202045356868637958421168851264492251414451671881645020719873271254395994874212298538928180953385418967331680881078358158544926445810098124494008083153487609262254200180462767265649895085196812223539172699394835278304012594889076972978898251549233575133572026086685197059747614441040152909798221820479642718133459387770777298554261474133030690753241864544153004060794064644652707512761735881885543957256884192628158805686561809594061924393932922840678322622784462425947093214038759553887929649916953155720669786551358215226363554462047120571340688059948977555505902980499655568680185582793237970604923030974220265795047325241015419809657491893019473031136829858500145730181190585716816328018616913990223894736777230358689553647254936178082288866348094694739653443883605310239766413123460690534793812785934073248428969751425870653389046775244015718236495145673518071665801891709920766181983586792103960637565600979310820525883807031028492116501924047338205921075081927116926896242518609770952106202341273961329572020549726342482699063714395183214771739428693004365223506346305628182421466017950778081664584660392184816346242270287808532367340974202505243
13, 9, {5}: 0
13, 9, {7}: 1619
13, {9}, 8: 21419
13, 9, {8}: 1332632087873
13, {9}, 10: 127
13, 9, {10}: 127
13, {9}, 11: 193272315039175487326233511816066059384236614936125602856112933108936878519237472904660314716980770567675038998906449425413153983914911198004481338508457524880312210951611198276223531290887108690292738026664067061295245572474703602610246125803179560118335836826468952746508374948156130103443176084512499609881640980577720176544209084145250756004293999763606271755552801884151163797064193306051490695131926580100826963694188422295198532753628423986864538180785129561486984798531474623258138617820724652921878726441978127950818375993469697424536456843865475449486163418975049457428208240361723655765920087962062440804142883445511641266703241468837936119086839611677856638014115541263266704520590558533279312304037176827720788243643056423797345536794811687465498057714283585292329995513313040502867333497441355894682583294242422882099383092926007095933296099723170595507590307152252179729407846066017402908790460912572771982724554085389116205103789017519828339606017138742770709364271683876516827264126734705389614550068455954555083870819276074206224473000405522699421186643568300931
13, 9, {11}: 188465890767567927768109
13, {10}, 1: 131
13, 10, {1}: 131
13, {10}, 3: 1823
13, {10}, 7: 137
13, 10, {7}: 137
13, {10}, 9: 139
13, 10, {9}: 139
13, {10}, 11: 1831
13, 10, {11}: 19564417634903
13, {11}, 1: 2003
13, 11, {1}: 1873
13, {11}, 2: 26171
13, 11, {2}: 24533
13, {11}, 3: 9720791083
13, 11, {3}: 1901
13, 11, {4}: 9244948171
13, {11}, 5: 16947116540528994433003374971133476766755585467759746042637787
13, 11, {5}: 55106069
13, {11}, 6: 149
13, 11, {6}: 149
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x{0}y:

Code:
13, 1, 4: 17
13, 1, 6: 19
13, 1, 10: 23
13, 1, 12: 181
13, 2, 3: 29
13, 2, 5: 31
13, 2, 9: 347
13, 2, 11: 37
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14, 1, 9: 23
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15, 13, 14: 2939
15, 14, 1: 211
15, 14, 11: 10631261
15, 14, 13: 223
16, 1, 1: 17
16, 1, 3: 19
16, 1, 7: 23
16, 1, 13: 29
16, 1, 15: 31
16, 2, 5: 37
16, 2, 9: 41
16, 2, 11: 43
16, 2, 15: 47
16, 3, 1: 769
16, 3, 5: 53
16, 3, 11: 59
16, 3, 13: 61
16, 4, 3: 67
16, 4, 7: 71
16, 4, 9: 73
16, 4, 13: 274877906957
16, 4, 15: 79
16, 5, 3: 83
16, 5, 9: 89
16, 5, 11: 1291
16, 6, 1: 97
16, 6, 5: 101
16, 6, 7: 103
16, 6, 11: 107
16, 6, 13: 109
16, 7, 1: 113
16, 7, 9: 1801
16, 7, 15: 127
16, 8, 3: 131
16, 8, 5: 2053
16, 8, 9: 137
16, 8, 11: 139
16, 8, 15: 2063
16, 9, 5: 149
16, 9, 7: 151
16, 9, 13: 157
16, 10, 1: 40961
16, 10, 3: 163
16, 10, 7: 167
16, 10, 9: 11529215046068469769
16, 10, 13: 173
16, 11, 3: 179
16, 11, 5: 181
16, 11, 15: 191
16, 12, 1: 193
16, 12, 5: 197
16, 12, 7: 199
16, 12, 11: 3083
16, 13, 1: 3329
16, 13, 3: 211
16, 13, 9: 55834574857
16, 13, 15: 223
16, 14, 3: 227
16, 14, 5: 229
16, 14, 9: 233
16, 14, 15: 239
16, 15, 1: 241
16, 15, 7: 3847
16, 15, 11: 251
16, 15, 13: 3853

Last fiddled with by sweety439 on 2021-01-03 at 15:43
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Old 2021-01-03, 15:49   #71
sweety439
 
Nov 2016

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

x{y} and {x}y:

Code:
13, {1}, 1: 30941
13, 1, {1}: 30941
13, {1}, 2: 2381
13, 1, {2}: 197
13, {1}, 3: 883708283
13, 1, {3}: 211
13, {1}, 4: 17
13, 1, {4}: 17
13, 1, {5}: 239
13, {1}, 6: 19
13, 1, {6}: 19
13, 1, {7}: 253217502498750291800692183145337720992638880271493569431738157631027569095215561
13, 1, {8}: 281
13, {1}, 9: 191
13, 1, {9}: 27130132404659193376721686434661
13, {1}, 10: 23
13, 1, {10}: 23
13, {1}, 11: 193
13, 1, {11}: 820195757799727198696695842441476208994963187388611974376331352215246160014077762227387500472080168786083449145277773838858800795991958180632280974279961571375401216690067428169654353503039823288371270608465421195831630752643523277310127185899536482776399744897654945387104284638368882957170938237278749541598968947323011480936838588602269948473325034154837529102013688233548651627077006312693219100274314806851388327828512220445130387062692274032398838117351850000439212156970074507805886649022038706852334408911410638474707605620484783396663735375992701354072765197190488304749330337810593696686818871540035682031309739770552183078238960961
13, {1}, 12: 36898271981403391525359432679065451
13, 1, {12}: 337
13, {2}, 1: 4759
13, 2, {1}: 106637277112689077
13, {2}, 3: 29
13, 2, {3}: 29
13, {2}, 5: 31
13, 2, {5}: 31
13, {2}, 7: 804473
13, 2, {7}: 959173
13, {2}, 9: 373
13, {2}, 11: 37
13, 2, {11}: 37
13, {3}, 1: 547
13, 3, {1}: 521
13, {3}, 2: 41
13, 3, {2}: 41
13, {3}, 4: 43
13, 3, {4}: 43
13, {3}, 5: 6220138738168647434831423806501836269264860388724755065149565951442236175936796719067011700334477855412396475124991691344796424350196661821800919998163958711515136934394634097179800502201458093822871
13, 3, {5}: 577
13, 3, {7}: 2923035083
13, {3}, 8: 47
13, 3, {8}: 47
13, 3, {10}: 647
13, {3}, 11: 557
13, 3, {11}: 661
13, {4}, 1: 53
13, 4, {1}: 53
13, {4}, 3: 3534833123
13, 4, {3}: 9337
13, {4}, 5: 733
13, 4, {5}: 13799574804865291194219202692403868309075434457702185936718792737
13, {4}, 7: 59
13, 4, {7}: 59
13, {4}, 9: 61
13, 4, {9}: 61
13, {4}, 11: 739
13, {5}, 1: 911
13, 5, {1}: 859
13, {5}, 2: 67
13, 5, {2}: 67
13, {5}, 3: 4418541403
13, 5, {3}: 887
13, {5}, 4: 277256920492991599
13, 5, {4}: 11717
13, {5}, 6: 71
13, 5, {6}: 71
13, 5, {7}: 89921716241132417850870043495861784708189815671814466610282151602592361843582839405888101361717803180497433838124247307746249298923864405720133765156792108299126338169033221693415179426566058950783399084612596039386427021973466021161486411164221010228501523
13, {5}, 8: 73
13, 5, {8}: 73
13, {5}, 9: 919
13, 5, {9}: 971
13, {5}, 11: 3604339966408890811
13, 5, {11}: 815662743439
13, 5, {12}: 1013
13, {6}, 1: 79
13, 6, {1}: 79
13, {6}, 5: 83
13, 6, {5}: 83
13, {6}, 7: 14281
13, 6, {7}: 9624308578305020228623
13, {6}, 11: 89
13, 6, {11}: 89
13, {7}, 1: 178099219309623994113347865115830245881192672331136086644940932390281437773862542264058288467257753659801600961075130547842788841314191170557163588785806820352498775469685611756153126667905685532469070927112200362751893732722602014850726541568809237898479136618304580608401918281234930143064257920309170965969499129532757126377921069913907908471671716574775646406870993578835592914767961359736945068963155952491960373644449418976551172435322847621930721542144058540845006834135041368564051814099612330378729974668784588333951075548113502723356005589020090498660438660159816508080021058414799091444088419142837713138761057222122289170719363548408846549381700350017033339161308029642370902736513421829569951497819557491865511689513661142885392815700489495652470272195143512237819660601060635160718950091398065703223634277265560584842616957317643843802700518533058037207116004784352250055837483109124201522964355475650386943932211405298597905457035869654221665584889762003382262425915627119116388998518127939798007098237014217655662706251901683109346257926396141021838296111310372968225022079707942822795129744698997404408233889930805048836987175297692891091686536077480054995854699722419639970465336199946812908942288140516904083829095941075082576747429233198452931491612278651981884049653724609828879600848950258125404977404144280808141884961849927596265042459349498828074205132275149771178481317527349922743964367067517723323072236797216719438348669059261865669008875549670954883841127641347209610157306553559175511522826280497259599838188113180381333755455900349307408532955497746124837076056334168406604339828775363723268449080391071951364995984584279858031164184210011834081
13, {7}, 2: 2815633
13, 7, {2}: 425554642597531069476088349319265344520247822549860785329030452393239592223053102610453
13, {7}, 3: 1277
13, {7}, 4: 16657
13, 7, {4}: 16111
13, {7}, 5: 1279
13, {7}, 6: 97
13, 7, {6}: 97
13, {7}, 8: 16661
13, 7, {8}: 16843
13, {7}, 9: 1283
13, {7}, 10: 101
13, 7, {10}: 101
13, {7}, 11: 176677145512151
13, {7}, 12: 103
13, 7, {12}: 103
13, {8}, 1: 201916737728161
13, {8}, 3: 107
13, 8, {3}: 107
13, {8}, 5: 109
13, 8, {5}: 109
13, {8}, 7: 948090435833789898349839159224250448489822963231314463923791
13, 8, {7}: 91021952951
13, {8}, 9: 113
13, 8, {9}: 113
13, {8}, 11: 247531
13, 8, {11}: 107838136118779143544110144382426092596204365203552428434248370827269303781694390904131147239057991725836898042412261207750257957583868509005708271654322505648777710176276577432900281939636108415883809919452408705622432187985384594384558596142925993309255985009535307478523156581759235779167162869019703196846751228017411378948853640401871368029723240618590933942573810568367308325289
13, {9}, 1: 121215212228974701436931483878589891140303089832220953724825443452928225193998713337930372911067805791799869147809411855365238746332705480023760792811129961633638311977064834817213347105220337863208732908350674718836123175940417328579685338580206495293657407565505774954042639105332034212418635645417015477944123471063765451261909023510740139021
13, 9, {1}: 259429
13, 9, {2}: 1549
13, {9}, 4: 68799997644951462493799712001
13, 9, {4}: 36748846266526697
13, {9}, 5: 1513040167438129868806971467209988783047172742217267483154894329901717663980542315652349848778853894915555170807951322572378209558278635157172717513628242832984817118678991631720815452108000099210400379825153731084262244319493812890311789045794884092831356123242061758082732806604188005520228674939217665328388067373699901394102699076617987792004875548394303529748128202045356868637958421168851264492251414451671881645020719873271254395994874212298538928180953385418967331680881078358158544926445810098124494008083153487609262254200180462767265649895085196812223539172699394835278304012594889076972978898251549233575133572026086685197059747614441040152909798221820479642718133459387770777298554261474133030690753241864544153004060794064644652707512761735881885543957256884192628158805686561809594061924393932922840678322622784462425947093214038759553887929649916953155720669786551358215226363554462047120571340688059948977555505902980499655568680185582793237970604923030974220265795047325241015419809657491893019473031136829858500145730181190585716816328018616913990223894736777230358689553647254936178082288866348094694739653443883605310239766413123460690534793812785934073248428969751425870653389046775244015718236495145673518071665801891709920766181983586792103960637565600979310820525883807031028492116501924047338205921075081927116926896242518609770952106202341273961329572020549726342482699063714395183214771739428693004365223506346305628182421466017950778081664584660392184816346242270287808532367340974202505243
13, 9, {5}: 0
13, 9, {7}: 1619
13, {9}, 8: 21419
13, 9, {8}: 1332632087873
13, {9}, 10: 127
13, 9, {10}: 127
13, {9}, 11: 193272315039175487326233511816066059384236614936125602856112933108936878519237472904660314716980770567675038998906449425413153983914911198004481338508457524880312210951611198276223531290887108690292738026664067061295245572474703602610246125803179560118335836826468952746508374948156130103443176084512499609881640980577720176544209084145250756004293999763606271755552801884151163797064193306051490695131926580100826963694188422295198532753628423986864538180785129561486984798531474623258138617820724652921878726441978127950818375993469697424536456843865475449486163418975049457428208240361723655765920087962062440804142883445511641266703241468837936119086839611677856638014115541263266704520590558533279312304037176827720788243643056423797345536794811687465498057714283585292329995513313040502867333497441355894682583294242422882099383092926007095933296099723170595507590307152252179729407846066017402908790460912572771982724554085389116205103789017519828339606017138742770709364271683876516827264126734705389614550068455954555083870819276074206224473000405522699421186643568300931
13, 9, {11}: 188465890767567927768109
13, {10}, 1: 131
13, 10, {1}: 131
13, {10}, 3: 1823
13, {10}, 7: 137
13, 10, {7}: 137
13, {10}, 9: 139
13, 10, {9}: 139
13, {10}, 11: 1831
13, 10, {11}: 19564417634903
13, {11}, 1: 2003
13, 11, {1}: 1873
13, {11}, 2: 26171
13, 11, {2}: 24533
13, {11}, 3: 9720791083
13, 11, {3}: 1901
13, 11, {4}: 9244948171
13, {11}, 5: 16947116540528994433003374971133476766755585467759746042637787
13, 11, {5}: 55106069
13, {11}, 6: 149
13, 11, {6}: 149
13, 11, {7}: 275614952732329653238232466846964959764342234851112847239174811330605042147473869409955112827912038180459570611709057122713557738754852977301670028971129467515066229407333223877783672306282366369543871115968877415280318537960852589377566524955025517265527862915865085457853330890387002896522575722038083366786743993408179910457825505003824935898016650965949690327645173471015392053739386753124739151576511051033085127452855493526018723218218416151702223054562562122092583257452732969841804887384398911206575041813220337355043752278695720428137
13, {11}, 8: 151
13, 11, {8}: 151
13, {11}, 9: 2011
13, 11, {9}: 335591
13, {11}, 10: 52950113757237678592993805854491324369268684216704693554936717423836128996957000831399984295437037597224946915093310665683439
13, 11, {10}: 1999
13, {11}, 12: 16836900297891418080414469547118518955584357920776290786511507224819852347973193037600665289070901330976115445902783343792856149076064327963454445124840887022352433623214149015015943271257627167012185236811023315748308075343126054090560004563875124190448995227748073744916159908957819701603274854998000296763254125672206384758348891742961717040363229489213108521955314350073857925001010097317113705164622416602981584525394558649693204742511309000575073486313783914987497483013408328355077527202814535784777000148396721007194688339582681878366906510944731328876064735814127172451578146421749559114747412555063799277435883965467381
13, 11, {12}: 2027
13, {12}, 1: 157
13, 12, {1}: 157
13, {12}, 5: 1792160394029
13, {12}, 7: 163
13, 12, {7}: 163
13, {12}, 11: 167
13, 12, {11}: 167
14, {1}, 1: 211
14, 1, {1}: 211
14, {1}, 3: 17
14, 1, {3}: 17
14, {1}, 5: 19
14, 1, {5}: 19
14, {1}, 9: 23
14, 1, {9}: 23
14, {1}, 11: 41381
14, 1, {11}: 70921
14, {1}, 13: 223
14, 1, {13}: 76831
14, {2}, 1: 29
14, 2, {1}: 29
14, {2}, 3: 31
14, 2, {3}: 31
14, 2, {5}: 467
14, {2}, 9: 37
14, 2, {9}: 37
14, {2}, 11: 431
14, 2, {11}: 557
14, {2}, 13: 41
14, 2, {13}: 41
14, {3}, 1: 43
14, 3, {1}: 43
14, {3}, 5: 47
14, 3, {5}: 47
14, {3}, 11: 53
14, 3, {11}: 53
14, {3}, 13: 643
14, {4}, 1: 32434921
14, 4, {1}: 156619
14, {4}, 3: 59
14, 4, {3}: 59
14, {4}, 5: 61
14, 4, {5}: 61
14, 4, {9}: 919
14, {4}, 11: 67
14, 4, {11}: 67
14, {4}, 13: 853
14, 4, {13}: 0
14, {5}, 1: 71
14, 5, {1}: 71
14, {5}, 3: 73
14, 5, {3}: 73
14, {5}, 9: 79
14, 5, {9}: 79
14, {5}, 11: 1061
14, {5}, 13: 83
14, 5, {13}: 83
14, {6}, 1: 48652381
14, {6}, 5: 89
14, 6, {5}: 89
14, {6}, 11: 248231
14, {6}, 13: 97
14, 6, {13}: 97
14, {7}, 1: 1471
14, 7, {1}: 271867
14, {7}, 3: 101
14, 7, {3}: 101
14, {7}, 5: 103
14, 7, {5}: 103
14, {7}, 9: 107
14, 7, {9}: 107
14, {7}, 11: 109
14, 7, {11}: 109
14, {7}, 13: 1483
14, 7, {13}: 1567
14, {8}, 1: 113
14, 8, {1}: 113
14, 8, {3}: 1613
14, 8, {5}: 1527891201751406184274498849901197440218663
14, {8}, 9: 14893189141836674105809869073123801
14, 8, {9}: 333923
14, {8}, 11: 3178964943473909010162782853260586588933938914164889347525573201287387822569505763054988368562457051
14, 8, {11}: 1733
14, {8}, 13: 1693
14, 8, {13}: 0
14, {9}, 1: 127
14, 9, {1}: 127
14, {9}, 5: 131
14, 9, {5}: 131
14, {9}, 11: 137
14, 9, {11}: 137
14, {9}, 13: 139
14, 9, {13}: 139
14, {10}, 1: 81087301
14, {10}, 3: 450545636966997425132717095409966070465562893779498582938637002140743
14, 10, {3}: 5502353
14, {10}, 9: 149
14, 10, {9}: 149
14, {10}, 11: 151
14, 10, {11}: 151
14, {10}, 13: 2113
14, {11}, 1: 2311
14, 11, {1}: 0
14, {11}, 3: 157
14, 11, {3}: 157
14, 11, {5}: 437351
14, {11}, 9: 163
14, 11, {9}: 163
14, {11}, 13: 167
14, 11, {13}: 167
14, {12}, 1: 2521
14, 12, {1}: 1273070779
14, {12}, 5: 173
14, 12, {5}: 173
14, {12}, 11: 179
14, 12, {11}: 179
14, {12}, 13: 181
14, 12, {13}: 181
14, {13}, 1: 2731
14, 13, {1}: 19298779963
14, 13, {3}: 2593
14, {13}, 5: 0
14, 13, {5}: 276540164647
14, {13}, 9: 191
14, 13, {9}: 191
14, {13}, 11: 193
14, 13, {11}: 193
15, {1}, 1: 241
15, 1, {1}: 241
15, {1}, 2: 17
15, 1, {2}: 17
15, {1}, 4: 19
15, 1, {4}: 19
15, {1}, 7: 303629285816214089001622765197550636782710041318620954247
15, 1, {7}: 337
15, {1}, 8: 23
15, 1, {8}: 23
15, {1}, 11: 251
15, 1, {11}: 401
15, {1}, 13: 12204253
15, 1, {13}: 433
15, {1}, 14: 29
15, 1, {14}: 29
15, {2}, 1: 31
15, 2, {1}: 31
15, {2}, 7: 37
15, 2, {7}: 37
15, {2}, 11: 41
15, 2, {11}: 41
15, {2}, 13: 43
15, 2, {13}: 43
15, {3}, 1: 417041800362721
15, 3, {1}: 691
15, {3}, 2: 47
15, 3, {2}: 47
15, {3}, 4: 2440849
15, 3, {4}: 739
15, {3}, 7: 727
15, 3, {7}: 787
15, {3}, 8: 53
15, 3, {8}: 53
15, {3}, 11: 162731
15, 3, {11}: 491182564871651
15, {3}, 13: 733
15, 3, {13}: 883
15, {3}, 14: 59
15, 3, {14}: 59
15, {4}, 1: 61
15, 4, {1}: 61
15, {4}, 7: 67
15, 4, {7}: 67
15, {4}, 11: 71
15, 4, {11}: 71
15, {4}, 13: 73
15, 4, {13}: 73
15, {5}, 1: 1201
15, 5, {1}: 2924441266741
15, {5}, 2: 18077
15, 5, {2}: 3905357
15, {5}, 4: 79
15, 5, {4}: 79
15, {5}, 7: 13729771207
15, 5, {7}: 1237
15, {5}, 8: 83
15, 5, {8}: 83
15, {5}, 11: 271211
15, 5, {11}: 1301
15, {5}, 13: 1213
15, 5, {13}: 15194280133
15, {5}, 14: 89
15, 5, {14}: 89
15, {6}, 1: 4881691
15, 6, {1}: 1037360491
15, {6}, 7: 97
15, 6, {7}: 97
15, {6}, 11: 101
15, 6, {11}: 101
15, {6}, 13: 103
15, 6, {13}: 103
15, {7}, 1: 379681
15, 7, {1}: 80547991
15, {7}, 2: 107
15, 7, {2}: 107
15, {7}, 4: 109
15, 7, {4}: 109
15, {7}, 8: 113
15, 7, {8}: 113
15, {7}, 11: 218946945190429691
15, 7, {11}: 2588949882704871041434151
15, {7}, 13: 1693
15, 7, {13}: 1783
15, {8}, 1: 28921
15, 8, {1}: 27241
15, {8}, 7: 127
15, 8, {7}: 127
15, {8}, 11: 131
15, 8, {11}: 131
15, {8}, 13: 1933
15, 8, {13}: 30133
15, {9}, 1: 2161
15, 9, {1}: 103329241
15, {9}, 2: 137
15, 9, {2}: 137
15, {9}, 4: 139
15, 9, {4}: 139
15, {9}, 7: 24713588167
15, 9, {7}: 2137
15, {9}, 8: 1647572543
15, 9, {8}: 2153
15, {9}, 11: 488171
15, 9, {11}: 495401
15, {9}, 13: 5560557338173
15, 9, {13}: 502633
15, {9}, 14: 149
15, 9, {14}: 149
15, {10}, 1: 151
15, 10, {1}: 151
15, {10}, 7: 157
15, 10, {7}: 157
15, {10}, 11: 2411
15, 10, {11}: 8190401
15, {10}, 13: 163
15, 10, {13}: 163
15, {11}, 1: 881786829315764563424246641
15, 11, {1}: 560491
15, {11}, 2: 167
15, 11, {2}: 167
15, {11}, 4: 39769
15, 11, {4}: 2539
15, {11}, 7: 2647
15, 11, {7}: 130992187
15, {11}, 8: 173
15, 11, {8}: 173
15, {11}, 13: 596653
15, 11, {13}: 2683
15, {11}, 14: 179
15, 11, {14}: 179
15, {12}, 1: 181
15, 12, {1}: 181
15, {12}, 7: 2887
15, 12, {7}: 42187
15, {12}, 11: 191
15, 12, {11}: 191
15, {12}, 13: 193
15, 12, {13}: 193
15, {13}, 1: 3121
15, 13, {1}: 661741
15, {13}, 2: 197
15, 13, {2}: 197
15, {13}, 4: 199
15, 13, {4}: 199
15, {13}, 7: 705127
15, 13, {7}: 3037
15, {13}, 8: 10577003
15, 13, {8}: 2318805803
15, {13}, 11: 91488544954572405131
15, 13, {11}: 157027901
15, {13}, 14: 535461077009
15, 13, {14}: 408700964355468749
15, {14}, 1: 211
15, 14, {1}: 211
15, {14}, 11: 3371
15, 14, {11}: 42527645637007506364690405981881277901
15, {14}, 13: 223
15, 14, {13}: 223
16, {1}, 1: 17
16, 1, {1}: 17
16, {1}, 3: 19
16, 1, {3}: 19
16, {1}, 5: 277
16, 1, {5}: 0
16, {1}, 7: 23
16, 1, {7}: 23
16, {1}, 9: 281
16, 1, {9}: 409
16, {1}, 11: 283
16, 1, {11}: 443
16, {1}, 13: 29
16, 1, {13}: 29
16, {1}, 15: 31
16, 1, {15}: 31
16, {2}, 1: 8737
16, 2, {1}: 581714951868689
16, {2}, 3: 547
16, 2, {3}: 563
16, {2}, 5: 37
16, 2, {5}: 37
16, {2}, 7: 725935716098002055388532495854438851111
16, 2, {7}: 631
16, {2}, 9: 41
16, 2, {9}: 41
16, {2}, 11: 43
16, 2, {11}: 43
16, {2}, 13: 557
16, 2, {13}: 733
16, {2}, 15: 47
16, 2, {15}: 47
16, {3}, 1: 253530120045645880299340641073
16, 3, {1}: 13171233041
16, {3}, 5: 53
16, 3, {5}: 53
16, {3}, 7: 823
16, 3, {7}: 887
16, {3}, 11: 59
16, 3, {11}: 59
16, {3}, 13: 61
16, 3, {13}: 61
16, {4}, 1: 0
16, 4, {1}: 16657
16, {4}, 3: 67
16, 4, {3}: 67
16, {4}, 5: 1093
16, 4, {5}: 1109
16, {4}, 7: 71
16, 4, {7}: 71
16, {4}, 9: 73
16, 4, {9}: 73
16, {4}, 11: 17483
16, 4, {11}: 19387
16, 4, {13}: 444540081354816304286954136617869418478679481821
16, {4}, 15: 79
16, 4, {15}: 79
16, {5}, 1: 1361
16, 5, {1}: 1297
16, {5}, 3: 83
16, 5, {3}: 83
16, {5}, 7: 1367
16, 5, {7}: 1399
16, {5}, 9: 89
16, 5, {9}: 89
16, {5}, 11: 21851
16, 5, {11}: 1613789866474427
16, {5}, 13: 1373
16, 5, {13}: 24029
16, {6}, 1: 97
16, 6, {1}: 97
16, {6}, 5: 101
16, 6, {5}: 101
16, {6}, 7: 103
16, 6, {7}: 103
16, {6}, 11: 107
16, 6, {11}: 107
16, {6}, 13: 109
16, 6, {13}: 109
16, {7}, 1: 113
16, 7, {1}: 113
16, {7}, 3: 1907
16, 7, {3}: 0
16, {7}, 5: 125269877
16, 7, {5}: 1877
16, {7}, 9: 1913
16, 7, {9}: 498073
16, {7}, 11: 32069089147
16, 7, {11}: 1979
16, {7}, 13: 2004318077
16, 7, {13}: 9972184721795404625107398548957
16, {7}, 15: 127
16, 7, {15}: 127
16, {8}, 1: 143165569
16, 8, {1}: 0
16, {8}, 3: 131
16, 8, {3}: 131
16, {8}, 5: 34949
16, 8, {5}: 0
16, {8}, 7: 56166555556563832905556281431290897236744050880292859335632521351
16, 8, {7}: 34679
16, {8}, 9: 137
16, 8, {9}: 137
16, {8}, 11: 139
16, 8, {11}: 139
16, {8}, 13: 8947853
16, 8, {13}: 2269
16, 8, {15}: 0
16, {9}, 1: 39313
16, 9, {1}: 594193
16, {9}, 5: 149
16, 9, {5}: 149
16, {9}, 7: 151
16, 9, {7}: 151
16, {9}, 11: 2459
16, 9, {11}: 637883
16, {9}, 13: 157
16, 9, {13}: 157
16, {10}, 1: 733007751841
16, 10, {1}: 41233
16, {10}, 3: 163
16, 10, {3}: 163
16, {10}, 7: 167
16, 10, {7}: 167
16, {10}, 9: 2729
16, 10, {9}: 2713
16, {10}, 11: 2731
16, 10, {11}: 43963
16, {10}, 13: 173
16, 10, {13}: 173
16, {11}, 1: 48049
16, 11, {1}: 2833
16, {11}, 3: 179
16, 11, {3}: 179
16, {11}, 5: 181
16, 11, {5}: 181
16, {11}, 7: 2999
16, 11, {7}: 49248958327
16, {11}, 9: 3001
16, 11, {9}: 2969
16, {11}, 13: 12303293
16, 11, {13}: 3037
16, {11}, 15: 191
16, 11, {15}: 191
16, {12}, 1: 193
16, 12, {1}: 193
16, {12}, 5: 197
16, 12, {5}: 197
16, {12}, 7: 199
16, 12, {7}: 199
16, {12}, 11: 0
16, 12, {11}: 3259
16, {12}, 13: 0
16, 12, {13}: 843229
16, {13}, 1: 3722304977
16, 13, {1}: 0
16, {13}, 3: 211
16, 13, {3}: 211
16, {13}, 5: 3541
16, 13, {5}: 3413
16, {13}, 7: 908759
16, 13, {7}: 882551
16, {13}, 9: 999198637325934041
16, 13, {9}: 73749768669482915691491069321318626688914012237296060805206525525363591813836272035774910527919776180923677912554968891935394987986240496179621997893655532569315694783635803112700208508303413378891202384198997794213422176304573334413545606463639516918296257466344350258969657713796137622531892391975484473872559979575727003547753581022912486703477573912049826765132053211177341761946288632815391744689614375401028459100583268869723038988660204614984245471691470023113466364417874725337512714447532250846778586077760659205293618044147237229216306717637040861705334444181470200752974579322509544738704990857820454867202261704090678797538558326245584064671252468247095559023662993480878895077936090038599163027885118597295012047000583187251486272120581780591332114804425847265975339536610666721934463637123044596968540088909735294287236975077640291184075261370631154339959438296960077041200837369288569872632621900878776171600339056308989379011831334774153700978537796879728062464221403872190754080933322090903482454554035388405527848042146114638163297543714301665763439499883062039897590145382317298449300948509392928786633391596392577283746404680586571591203059252060135897745743650628381225179713605144028836751431506430002554673660083008864161778283325595456594614681
16, {13}, 11: 3547
16, 13, {11}: 0
16, {13}, 15: 223
16, 13, {15}: 223
16, {14}, 1: 61153
16, {14}, 3: 227
16, 14, {3}: 227
16, {14}, 5: 229
16, 14, {5}: 229
16, {14}, 9: 233
16, 14, {9}: 233
16, {14}, 11: 1300876803247619683256250232571154421182187
16, 14, {11}: 58304019973926508829195794288364830930948296694792337729075131089632305865113112154282586276837027138881157744159537913596428450753911485229682880296019616116874769754520009659
16, {14}, 13: 3821
16, 14, {13}: 15588829
16, {14}, 15: 239
16, 14, {15}: 239
16, {15}, 1: 241
16, 15, {1}: 241
16, {15}, 7: 0
16, 15, {7}: 66428827511
16, {15}, 11: 251
16, 15, {11}: 251
16, {15}, 13: 4093
16, 15, {13}: 1039837
x{0}y:

Code:
13, 1, 4: 17
13, 1, 6: 19
13, 1, 10: 23
13, 1, 12: 181
13, 2, 3: 29
13, 2, 5: 31
13, 2, 9: 347
13, 2, 11: 37
13, 3, 2: 41
13, 3, 4: 43
13, 3, 8: 47
13, 3, 10: 14480437
13, 4, 1: 53
13, 4, 3: 54020737582614507942458917440610823901767221634062888289700852810815079724323258965596943883469013562594127941397398999016481603427622205011908900266071076096475248216593088654359952926621741299090558666311191216978954704152791
13, 4, 7: 59
13, 4, 9: 61
13, 5, 2: 67
13, 5, 6: 71
13, 5, 8: 73
13, 5, 12: 857
13, 6, 1: 79
13, 6, 5: 83
13, 6, 7: 1021
13, 6, 11: 89
13, 7, 4: 1187
13, 7, 6: 97
13, 7, 10: 101
13, 7, 12: 103
13, 8, 3: 107
13, 8, 5: 109
13, 8, 9: 113
13, 8, 11: 38614483
13, 9, 2: 1523
13, 9, 4: 19777
13, 9, 8: 43441289
13, 9, 10: 127
13, 10, 1: 131
13, 10, 3: 1693
13, 10, 7: 137
13, 10, 9: 139
13, 11, 2: 1861
13, 11, 6: 149
13, 11, 8: 151
13, 11, 12: 1871
13, 12, 1: 157
13, 12, 5: 2729251996728070131798006327033140931418231688924554627711654775633915597843291014832282995312758790354592964227251689466987151209150175544113
13, 12, 7: 163
13, 12, 11: 167
14, 1, 1: 197
14, 1, 3: 17
14, 1, 5: 19
14, 1, 9: 23
14, 1, 11: 0
14, 1, 13: 2177953337809371149
14, 2, 1: 29
14, 2, 3: 31
14, 2, 5: 397
14, 2, 9: 37
14, 2, 13: 41
14, 3, 1: 43
14, 3, 5: 47
14, 3, 11: 53
14, 3, 13: 601
14, 4, 1: 0
14, 4, 3: 59
14, 4, 5: 61
14, 4, 11: 67
14, 4, 13: 797
14, 5, 1: 71
14, 5, 3: 73
14, 5, 9: 79
14, 5, 11: 991
14, 5, 13: 83
14, 6, 1: 45177217
14, 6, 5: 89
14, 6, 11: 1187
14, 6, 13: 97
14, 7, 1: 1373
14, 7, 3: 101
14, 7, 5: 103
14, 7, 9: 107
14, 7, 11: 109
14, 7, 13: 0
14, 8, 1: 113
14, 8, 3: 1571
14, 8, 9: 21961
14, 8, 11: 1579
14, 8, 13: 0
14, 9, 1: 127
14, 9, 5: 131
14, 9, 11: 137
14, 9, 13: 139
14, 10, 1: 75295361
14, 10, 9: 149
14, 10, 11: 151
14, 10, 13: 1973
14, 11, 1: 0
14, 11, 3: 157
14, 11, 5: 2161
14, 11, 9: 163
14, 11, 13: 167
14, 12, 5: 173
14, 12, 11: 179
14, 12, 13: 181
14, 13, 1: 2549
14, 13, 3: 2551
14, 13, 5: 35677
14, 13, 9: 191
14, 13, 11: 193
15, 1, 2: 17
15, 1, 4: 19
15, 1, 8: 23
15, 1, 14: 29
15, 2, 1: 31
15, 2, 7: 37
15, 2, 11: 41
15, 2, 13: 43
15, 3, 2: 47
15, 3, 8: 53
15, 3, 14: 59
15, 4, 1: 61
15, 4, 7: 67
15, 4, 11: 71
15, 4, 13: 73
15, 5, 4: 79
15, 5, 8: 83
15, 5, 14: 89
15, 6, 7: 97
15, 6, 11: 101
15, 6, 13: 103
15, 7, 2: 107
15, 7, 4: 109
15, 7, 8: 113
15, 8, 1: 1801
15, 8, 7: 127
15, 8, 11: 131
15, 9, 2: 137
15, 9, 4: 139
15, 9, 8: 1537734383
15, 9, 14: 149
15, 10, 1: 151
15, 10, 7: 157
15, 10, 13: 163
15, 11, 2: 167
15, 11, 4: 8353129
15, 11, 8: 173
15, 11, 14: 179
15, 12, 1: 181
15, 12, 7: 2707
15, 12, 11: 191
15, 12, 13: 193
15, 13, 2: 197
15, 13, 4: 199
15, 13, 14: 2939
15, 14, 1: 211
15, 14, 11: 10631261
15, 14, 13: 223
16, 1, 1: 17
16, 1, 3: 19
16, 1, 7: 23
16, 1, 13: 29
16, 1, 15: 31
16, 2, 5: 37
16, 2, 9: 41
16, 2, 11: 43
16, 2, 15: 47
16, 3, 1: 769
16, 3, 5: 53
16, 3, 11: 59
16, 3, 13: 61
16, 4, 3: 67
16, 4, 7: 71
16, 4, 9: 73
16, 4, 13: 274877906957
16, 4, 15: 79
16, 5, 3: 83
16, 5, 9: 89
16, 5, 11: 1291
16, 6, 1: 97
16, 6, 5: 101
16, 6, 7: 103
16, 6, 11: 107
16, 6, 13: 109
16, 7, 1: 113
16, 7, 9: 1801
16, 7, 15: 127
16, 8, 3: 131
16, 8, 5: 2053
16, 8, 9: 137
16, 8, 11: 139
16, 8, 15: 2063
16, 9, 5: 149
16, 9, 7: 151
16, 9, 13: 157
16, 10, 1: 40961
16, 10, 3: 163
16, 10, 7: 167
16, 10, 9: 11529215046068469769
16, 10, 13: 173
16, 11, 3: 179
16, 11, 5: 181
16, 11, 15: 191
16, 12, 1: 193
16, 12, 5: 197
16, 12, 7: 199
16, 12, 11: 3083
16, 13, 1: 3329
16, 13, 3: 211
16, 13, 9: 55834574857
16, 13, 15: 223
16, 14, 3: 227
16, 14, 5: 229
16, 14, 9: 233
16, 14, 15: 239
16, 15, 1: 241
16, 15, 7: 3847
16, 15, 11: 251
16, 15, 13: 3853
Forms are not listed here means they have a NUMERICAL covering set (including trivial 1-cover) (only for families x{y} and {x}y, for families x{0}y, only families with trivial 1-cover (the trivial factor must divides either b or b-1) are not listed)

Forms with 0 means either they make a full covering set with (all or partial) ALGEBRAIC factors or they are not ruled out as only contain composites but have no primes or PRPs with length <= 5000

Last fiddled with by sweety439 on 2021-01-06 at 12:52
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Old 2021-01-03, 15:51   #72
sweety439
 
Nov 2016

22×3×5×47 Posts
Default

All of the primes are minimal primes (start with b+1) except the case which the repeating digit (i.e. x in {x}y, or y in x{y}) is 1 (the x{0}y case is always minimal primes (start with b+1), but if x = 1 and the base is prime, then the corresponding primes are not minimal primes (start with b))

Last fiddled with by sweety439 on 2021-01-03 at 15:52
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Old 2021-01-04, 04:16   #73
sweety439
 
Nov 2016

22·3·5·47 Posts
Default

Quote:
Originally Posted by sweety439 View Post
start searching bases 13 to 16

the minimal primes (start with b+1) are
newest data:

Code:
13: {14, 16, 1A, 23, 25, 2B, 32, 34, 38, 41, 47, 49, 52, 56, 58, 61, 65, 6B, 76, 7A, 7C, 83, 85, 89, 9A, A1, A7, A9, B6, B8, C1, C7, CB, 10C, 119, 11B, 122, 133, 155, 157, 173, 179, 17B, 188, 197, 1B1, 1B5, 1CC, 209, 212, 218, 229, 272, 274, 281, 287, 292, 296, 298, 29C, 2C9, 311, 313, 331, 33B, 355, 371, 373, 379, 397, 3A6, 3AA, 3B3, 3B9, 3BB, 3CA, 43C, 445, 44B, 45A, 463, 4A3, 4A5, 4B2, 4B4, 4BA, 50C, 511, 515, 533, 54A, 551, 559, 571, 575, 57B, 595, 599, 5B3, 5B9, 5CC, 607, 629, 63A, 643, 674, 704, 715, 724, 728, 731, 737, 739, 742, 751, 75B, 773, 775, 779, 782, 784, 791, 793, 797, 7B1, 812, 818, 874, 878, 8AB, 8B4, 902, 919, 922, 926, 92C, 937, 93B, 946, 95B, 962, 968, 971, 977, 979, 982, 98C, 9B3, 9B5, A03, A3C, A45, A4B, A54, AA3, AAB, B02, B0C, B11, B15, B17, B24, B33, B39, B42, B57, B59, B71, B93, B9B, BA4, BAA, BB1, BB9, BC2, BCC, C29, C43, C98, CA3, 1013, 1031, 1037, 105B, 1075, 10B7, 10BB, 1105, 1112, 1121, 1127, 113C, 1172, 1187, 1208, 1211, 1277, 12C8, 1307, 1309, 131C, 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F88888F7, FE777777, FF33F2C3, FF3FFF23, FF588887, FFFF02C3, FFFF5F6F, FFFFF887, FFFFFF79, 10CCCCCAB, 266666603, ...}

Last fiddled with by sweety439 on 2021-01-04 at 14:00
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Old 2021-01-04, 08:19   #74
LaurV
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Please stop posting all these endless lists of numbers that nobody read nor needs. If you are really interested in studying these minimal sets, make a web page, put them in zip files, provide links, etc. The way you do it, "text mode", and with additional quote to quote to quote to quote, which makes the posts double size, you are only cluttering the forum, and some people believe that you are doing it intentionally (i.e. trolling), which means you are coming very close to a ban (subject already in discussion for a while on moderator threads!).

At least, wtf man? if you like your text mode so much, can't you at least use "code" tags? Scrolling down 20 pages of numbers (even more for mobile browsers) is a pain in the butt. Really.

Last fiddled with by LaurV on 2021-01-04 at 08:21
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Old 2021-01-04, 13:51   #75
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Quote:
Originally Posted by LaurV View Post
Please stop posting all these endless lists of numbers that nobody read nor needs. If you are really interested in studying these minimal sets, make a web page, put them in zip files, provide links, etc. The way you do it, "text mode", and with additional quote to quote to quote to quote, which makes the posts double size, you are only cluttering the forum, and some people believe that you are doing it intentionally (i.e. trolling), which means you are coming very close to a ban (subject already in discussion for a while on moderator threads!).

At least, wtf man? if you like your text mode so much, can't you at least use "code" tags? Scrolling down 20 pages of numbers (even more for mobile browsers) is a pain in the butt. Really.
OK, I stopped and going to make a web page about these primes.

Should I delete these posts?

Last fiddled with by sweety439 on 2021-01-04 at 13:52
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Old 2021-01-04, 13:57   #76
LaurV
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Quote:
Originally Posted by sweety439 View Post
Should I delete these posts?
Putting the long text in "code tags" is enough, don't need to delete. You can however delete the quotes to former posts in each (which just duplicate the text). Anyhow, the issue is not the disk space (just some text), but the difficulty in reading through such long strings of numbers.
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Old 2021-01-04, 14:02   #77
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Quote:
Originally Posted by LaurV View Post
Putting the long text in "code tags" is enough, don't need to delete. You can however delete the quotes to former posts in each (which just duplicate the text). Anyhow, the issue is not the disk space (just some text), but the difficulty in reading through such long strings of numbers.
Is this OK? I put all my data for bases 13~16 with codes, must I also put data for bases <=12?

(base 13~16 data is much longer)

Last fiddled with by sweety439 on 2021-01-04 at 14:03
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