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 2021-01-06, 08:54 #89 sweety439   Nov 2016 22×3×5×47 Posts Now, we proved the set of minimal primes (start with b+1, which is equivalent to start with b, if b is composite) of base b=10: Code: 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
 2021-01-06, 08:56 #90 sweety439   Nov 2016 54048 Posts There are totally 77 minimal primes (start with 2 digits) in base 10, there are 75 such primes in base 8
2021-01-06, 10:36   #91
sweety439

Nov 2016

22·3·5·47 Posts

Quote:
 Originally Posted by sweety439 Proof of base 4:
Proof of base 6:

The possible (first digit,last digit) for an element with >=3 digits in the minimal set of the strings for primes with at least two digits are:

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

* Case (1,1):

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

* Case (1,5):

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

* Case (2,1):

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

* Case (2,5):

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

* Case (3,1):

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

* Case (3,5):

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

* Case (4,1):

** Since 45, 11, 21, 31, 51 are primes, we only need to consider the family 4{0,4}1 (since any digits 1, 2, 3, 5 between them will produce smaller primes)

*** Since 4401 and 4441 are primes, we only need to consider the families 4{0}1 and 4{0}41 (since any digits combo 40 and 44 between them will produce smaller primes)

**** All numbers of the form 4{0}1 are divisible by 5, thus cannot be prime.

**** The smallest prime of the form 4{0}41 is 40041

* Case (4,5):

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

* Case (5,1):

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

* Case (5,5):

** Since 51, 15, 25, 35, 45 are primes, we only need to consider the family 5{0,5}5 (since any digits 1, 2, 3, 4 between them will produce smaller primes)

*** All numbers of the form 5{0,5}5 are divisible by 5, thus cannot be prime.

 2021-01-06, 12:48 #92 sweety439   Nov 2016 22×3×5×47 Posts Currently status for bases 13 to 16: 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, 88B2, 88BA, 8B22, 8B2A, 8BAC, 9004, 9017, 9031, 9053, 9055, 9073, 9091, 90BB, 90C8, 9107, 9118, 913C, 9181, 91C3, 9284, 935C, 93C5, 9424, 9428, 9448, 9509, 959C, 96C4, 9703, 9743, 9745, 974B, 97B2, 9811, 981B, 987B, 98B1, 991C, 9967, 9998, 9B12, 9B74, 9B92, 9BBC, 9C55, 9C86, 9CC4, A0BA, A306, A436, A535, A5B5, A636, A6C3, A80B, AB04, AB22, AB35, AB3B, AB4C, AB55, ABAC, ABB5, AC36, ACA5, B044, B04A, B0B7, B129, B1B2, B219, B222, B291, B299, B2CA, B35A, B3A5, B404, B44C, B45B, B4B3, B501, B51C, B55A, B5A5, B5AB, B5C3, B707, B792, B794, B905, B912, B9C5, BA5B, BAB3, BB03, BB45, BB72, BBA5, BBB2, BC44, BC53, BC95, BC99, C30A, C36A, C395, C454, C535, C553, C593, C944, C953, C964, CC94, 10015, 10051, 10099, 10118, 10291, 10712, 10772, 10811, 10877, 10921, 10B92, 11111, 11135, 11171, 111C8, 11531, 11C03, 13001, 13177, 13777, 13915, 13951, 13991, 159BB, 17018, 17102, 17111, 17117, 17171, 17177, 17708, 17711, 17801, 18071, 18101, 18271, 18B27, 19003, 19153, 19315, 19351, 19591, 19913, 19951, 1C099, 20171, 20177, 20207, 20227, 20777, 21011, 21077, 2111C, 21707, 22207, 30017, 300B5, 301C9, 3033A, 303A3, 303C5, 3050B, 305C9, 3095C, 30B05, 31007, 3159B, 31999, 31C09, 3330A, 33353, 33593, 33757, 33C5C, 33CC5, 35003, 3591B, 39353, 39539, 39935, 39995, 3ACCC, 3C5C3, 3CC53, 40043, 40306, 405C4, 408BC, 40BBB, 40C54, 43066, 4366A, 4443A, 45055, 45505, 45554, 4555C, 455BC, 455C3, 45C04, 488BC, 4B03B, 4B0B5, 4B55C, 4BB0B, 4C003, 4C054, 4C5C4, 50053, 500B1, 5035A, 504B5, 5053A, 50554, 505B4, 50A35, 50B07, 50BBA, 5139B, 519BB, 51BB7, 535AC, 53A5C, 53AC5, 53BAC, 54004, 54035, 5403B, 545C3, 54B05, 54B5C, 54BBC, 54C53, 55357, 5535B, 553AC, 554BC, 55537, 55544, 5554C, 55577, 555A4, 555BB, 55A5C, 55B04, 55B55, 55B77, 55BB5, 55BC4, 55C54, 55C5A, 57403, 591BB, 59443, 59BB7, 5A044, 5AC04, 5AC35, 5B001, 5B007, 5B0AB, 5B0B4, 5B4C5, 5B544, 5B555, 5B5BB, 5B744, 5B777, 5BA0B, 5BB44, 5BB55, 5BBC4, 5BC54, 5C039, 5C35A, 5C53A, 60098, 60964, 60988, 60A63, 66094, 66377, 66692, 66694, 669C2, 669C4, 66A36, 67022, 67099, 67222, 67277, 67772, 68627, 69088, 690C4, 69808, 69994, 6A663, 7007B, 70181, 70222, 70277, 70772, 70808, 70B0B, 70B29, 71113, 71711, 71908, 71999, 7199B, 71BB7, 71BBB, 74035, 74305, 7430B, 74503, 75443, 75454, 75535, 77072, 77108, 77177, 77717, 77BBB, 78011, 79BBB, 7B007, 7B7B7, 7B7BB, 7BBB3, 7BBB7, 80117, 80221, 80771, 80777, 80807, 8084B, 80B7B, 80BBB, 81107, 8400B, 86267, 87107, 87277, 87727, 87B27, 88111, 88201, 88702, 88771, 8888B, 88B77, 88BBB, 8B1BC, 8B727, 90035, 90059, 90088, 90095, 9009B, 90101, 90103, 90305, 90488, 904CC, 90574, 90644, 9064C, 90806, 908B7, 9090B, 90994, 90B09, 90C35, 90C59, 90C64, 91111, 91135, 91315, 9180B, 92008, 92408, 92488, 93359, 93395, 944C2, 944CC, 94505, 9455C, 94804, 94888, 94C0C, 94C33, 94C4C, 95045, 95504, 95573, 955C4, 95C54, 96044, 97BBB, 98066, 98408, 98444, 98804, 98848, 99001, 99005, 9900B, 99074, 990BC, 99113, 99175, 99278, 99335, 99454, 994C3, 99517, 99593, 9984B, 99881, 99904, 99917, 99935, 99955, 99973, 999BB, 999C2, 999C4, 99B99, 9B00B, 9B04B, 9B0B4, 9B1BB, 9BB04, 9C059, 9C244, 9C404, 9C44C, 9C488, 9C503, 9C5C9, 9C644, 9C664, 9CC88, 9CCC2, A00B4, A05BB, A08B2, A08BC, A0BC4, A3336, A3633, A443A, A4443, A50BB, A55C5, A5AAC, A5BBA, A5C53, A5C55, AACC5, AB05B, AB0BB, AB40A, ABBBC, ABC4A, ACC5A, ACCC3, B0053, B0075, B010B, B0455, B0743, B0774, B0909, B0BB4, B2277, B2A2C, B3005, B351B, B37B5, B3A0B, B3ABC, B3B0A, B400A, B4035, B403B, B4053, B4305, B4BC5, B4C0A, B504B, B50BA, B530A, B5454, B54BC, B54C5, B5544, B55B5, B5B44, B5B4C, B5BB5, B7403, B7535, B77BB, B7955, B7B7B, B9207, B9504, B9999, BA055, BA305, BABC5, BAC35, BB054, BB05A, BB207, BB3B5, BB4C3, BB504, BB544, BB54C, BB5B5, BB753, BB7B7, BBABC, BBB04, BBB4C, BBB55, BBBAC, BC035, BC455, C0353, C0359, C03AC, C0904, C0959, C0A5A, C0CC5, C3059, C335C, C5A0A, C5A44, C5AAC, C6692, C69C2, C904C, C9305, C9905, C995C, C99C5, C9C04, C9C59, C9CC2, CA50A, CA5AC, CAA05, CAA5A, CC335, CC544, CC5AA, CC935, CC955, 100039, 100178, 100718, 100903, 101177, 101708, 101711, 101777, 102017, 102071, 103999, 107081, 107777, 108217, 109111, 109151, 110078, 110108, 110717, 111017, 111103, 1111C3, 111301, 111707, 113501, 115103, 117017, 117107, 117181, 117701, 120701, 13C999, 159103, 170717, 177002, 177707, 180002, 187001, 18C002, 19111C, 199903, 1B0007, 1BB077, 1BBB07, 1C0903, 1C8002, 1C9993, 200027, 207107, 217777, 219991, 220027, 222227, 270008, 271007, 277777, 290444, 300059, 300509, 303359, 303995, 309959, 30B50A, 3336AC, 333707, 33395C, 335707, 3360A3, 350009, 36660A, 3666AC, 370007, 377B07, 39001C, 399503, 3BC005, 400366, 400555, 400B3B, 400B53, 400BB5, 400CC3, 4030B5, 40B053, 40B30B, 40B505, 43600A, 450004, 4A088B, 4B0503, 4B5C05, 4BBBB5, 4BC505, 500039, 50045B, 50405B, 504B0B, 50555B, 5055B5, 505B0A, 509003, 50A50B, 50B045, 50B054, 539B01, 550054, 5500BA, 55040B, 553BC5, 5553C5, 55550B, 5555C3, 555C04, 55B00A, 55BB0B, 570007, 5A500B, 5A555B, 5AC505, 5B055B, 5B0B5B, 5B5B5C, 5B5BC5, 5BB05B, 5BBB0B, 5BBB54, 5BBBB4, 5BBC0A, 5BC405, 5C5A5A, 5CA5A5, 600694, 6060A3, 609992, 637777, 6606A3, 6660A3, 667727, 667808, 668777, 669664, 670088, 679988, 696064, 69C064, 6A6333, 700727, 700811, 700909, 70098B, 700B92, 701117, 701171, 701717, 707027, 707111, 707171, 707201, 707801, 70788B, 7080BB, 708101, 70881B, 70887B, 70B227, 710012, 710177, 711002, 711017, 711071, 717707, 718001, 718111, 720077, 722002, 727777, 74BB3B, 74BB53, 770102, 770171, 770801, 777112, 777202, 777727, 777772, 778801, 77B772, 780008, 78087B, 781001, 788B07, 79088B, 794555, 7B000B, 7B0535, 7B077B, 7B2777, 7B4BBB, 7BB4BB, 800021, 800717, 801077, 80BB07, 811117, 870077, 8777B7, 877B77, 880177, 88071B, 88077B, 8808BC, 887017, 88707B, 888227, 88877B, 8887B7, 888821, 888827, 888BB7, 8B001B, 8B00BB, 8BBB77, 8BBBB7, 900097, 900BC9, 901115, 903935, 904033, 90440C, 908008, 908866, 909359, 909C05, 90B944, 90C95C, 90CC95, 91008B, 91115C, 911503, 920888, 930335, 933503, 935903, 940033, 94040C, 940808, 94CCCC, 950005, 950744, 95555C, 9555C5, 95C003, 95C005, 96400C, 96440C, 96664C, 966664, 966994, 969094, 969964, 97008B, 97080B, 975554, 97800B, 97880B, 980006, 980864, 980B07, 984884, 986006, 986606, 986644, 988006, 988088, 988664, 988817, 988886, 988B0B, 98B007, 990115, 990151, 990694, 990B44, 990C5C, 991501, 993059, 99408B, 994555, 995404, 995435, 996694, 9978BB, 998087, 999097, 999103, 99944C, 999503, 9995C3, 999754, 999901, 99990B, 999B09, 99B4C4, 99C0C5, 99C539, 99CC05, 9B9444, 9B9909, 9C0484, 9C0808, 9C2888, 9C400C, 9C4CCC, 9C6994, 9C90C5, 9C9C5C, 9CC008, 9CC5C3, 9CC905, 9CCC08, A0055B, A005AC, A0088B, A00B2C, A00BBB, A0555C, A05CAA, A0A5AC, A0A5CA, A0AC05, A0AC5A, A0B50B, A0BB0B, A0BBB4, A0C5AC, A3660A, A5050B, A555AC, A5B00B, AA0C05, AAA05C, AAA0C5, AAC05C, AB4444, ABB00B, AC050A, AC333A, B0001B, B00099, B0030B, B004B5, B00A35, B00B54, B030BA, B05043, B0555B, B05B0A, B05B5B, B07B53, B09074, B09755, B09975, B09995, B0AB0B, B0B04B, B0B535, B0BB53, B4C055, B50003, B5003A, B500A3, B50504, B50B04, B53BC5, B54BBB, B550BB, B555BC, B55C55, B5B004, B5B0BB, B5B50B, B5B554, B5B55C, B5B5B4, B5BBB4, B5BBBC, B5BC0A, B5C045, B5C054, B70995, B70B3B, B74555, B74B55, B99921, B99945, BAC505, BB0555, BB077B, BB0B5B, BB0BB5, BB500A, BB53BC, BB53C5, BB5505, BB55BC, BB5BBA, BB5C0A, BB7BB4, BBB00A, BBB74B, BBBB54, BBBBAB, BC5054, BC5504, C00094, C00694, C009C4, C00C05, C03035, C050AA, C05309, C05404, C0544C, C05AC4, C05C39, C06092, C06694, C09035, C094CC, C09992, C09994, C09C4C, C09C95, C0CC3A, C0CC92, C33539, C35009, C4C555, C50309, C50AAA, C53009, C550A5, C555CA, C55A5A, C55CA5, C5AC55, C60094, C60694, C93335, C95405, C99094, CA05CA, CA0AC5, CA555C, CAC5CA, CC05A4, CC0AA5, CC0C05, CC3509, CC4555, CC5039, CC5554, CC555A, CC6092, CCC0C5, CCC353, CCC959, CCC9C2, 1000271, 1000802, 1000871, 1001771, 1001801, 1007078, 1008002, 1008107, 1008701, 1010117, 1027001, 1070771, 1077107, 1077701, 1080107, 1101077, 1110008, 1111078, 1115003, 1117777, 1170008, 1170101, 1700078, 1700777, 1800017, 1877017, 18B7772, 18BBB0B, 1999391, 1999931, 1BBBB3B, 2011001, 2107001, 2110001, 2700017, 2700707, 300000A, 3000019, 3000A33, 3003335, 3003395, 3009335, 300A05B, 3010009, 30A3333, 3335C09, 3339359, 3353777, 336A333, 3393959, 33AC333, 3537007, 3577777, 3636337, 3757777, 395C903, 3AC3333, 40003B5, 400B0B3, 400BBC3, 403B005, 405050B, 40B5555, 40BB555, 40CC555, 4436606, 4444306, 45C5555, 4BC5555, 4C55555, 4CC5004, 4CCC0C3, 500001B, 50003A5, 50005BA, 500B55B, 501000B, 505004B, 505B05B, 50B50B5, 50B550B, 50BB004, 5300009, 5400B0B, 54B000B, 5500BBB, 550B05B, 553000A, 5537777, 555054B, 55505BA, 5550B74, 5555054, 5555BAC, 5555C05, 555B005, 555C00A, 555CA55, 55AC005, 55AC555, 55B005B, 55CA0A5, 5A00004, 5AA5C05, 5B05B05, 5B50B05, 5B5C004, 5BBBBB5, 5BBBBCA, 5C00093, 5C003A5, 5C00A0A, 5C0A055, 5C505AA, 5C5555A, 6000692, 600A333, 606A333, 6363337, 6720002, 6906664, 7000112, 7000712, 7001201, 7001777, 7005553, 70088B7, 7009555, 7010771, 7070881, 7088107, 709800B, 70B9992, 7100021, 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C0668F, C0844D, C0A0FF, C0AFF9, C0C3AF, C0C68F, C0CAAF, C0CDED, C0D0ED, C0E80D, C0EC2D, C0EC8D, C0FA0F, C0FAAF, C2CC63, C30CAF, C333AF, C3CAAF, C3CCAF, C4048D, C40D4D, C4404D, C4408D, C4440D, C44DDD, C4ACCB, C4DCCB, C4DD4D, C6068F, C66AAF, C68AAF, C6AA8F, C8044D, C8440D, C8666F, CA00FF, CA0FFF, CAAAAF, CAAFFF, CAFF0F, CBE0CD, CC008F, CC0C8F, CC3CAF, CC4ACB, CC608F, CC66AF, CCBECD, CCC4AB, CCCA0F, CCCC8F, CCCE8D, CE0C8D, CF0F23, CF0FAF, CFAFFF, CFCAAF, CFFAFF, D0005D, 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, C000004D, C000086F, C0000AFF, C0000E8D, C0000FDD, C00033AF, C0003CAF, C000448D, C000AFFF, C000CF8F, C004444D, C00663AF, C00F00AF, C00FCCAF, C0FFCCAF, C844444D, CC3A000F, CCCCCBED, CCCCCE2D, CCCCD999, CCDCCC4B, CD44444D, CFAF000F, CFFFF023, D00400ED, D004404D, D00777A5, D00E00DD, D0444E0D, D40000ED, D444E00D, D7DDDDDD, DD00D007, DD0D0077, DD0D0707, DDD0040D, DDDDDD19, DDDDDDD1, E0000CCB, E0044441, E00A4AA1, E888820D, E8888CCD, E888C80D, E8AAAAA1, EB00C0CD, EBBC00CD, ECCCCCCB, F00006AF, F00040A5, F00066AF, F06666AF, F0F004A5, F33FFF23, F60006AF, F6AAA0AF, F88888F7, FE777777, FF33F2C3, FF3FFF23, FF588887, FFFF02C3, FFFF5F6F, FFFFF887, FFFFFF79, 10CCCCCAB, 266666603, 333333AAF, 333333F23, 3333FF2C3, 333CCCCAF, 333FFCCAF, 3A000000F, 3FA00000F, 40000048D, 4000004DD, 4000040D1, 40000ACCB, 4000400D1, 4040000DD, 404D0000D, 40A000005, 40E00444D, 40ED0000D, 444E000DD, 444ED000D, 48444444D, 4A0000005, 4AAAAAAA5, 500000C8D, 500000F8D, 50CCCCC8D, 50FFFFF6F, 5AAAAAA45, 5C020000D, 5E444444D, 666666AFF, 70000044D, 70000440D, 700007CCB, 700007D07, 70044000D, 70070007D, 77070007D, 77700040D, 77700070D, 77707044D, 77770000D, 77777777B, 777888887, 7D0DDDDDD, 7DD0000D7, 8008880A1, 800888A01, 800C000ED, 888800087, 88888AF8F, 888CCCCCD, 88CCCCCCD, 8AAAAAFFF, 8AAFFFFFF, 8CECCCCCD, 8CFFFFCFF, 8EC00000D, 900010009, 908A0AAA1, 9800AAAA1, 9B0CCCCC9, A00000669, A00005545, A0000A545, A000FFF45, A0AAAAA8F, A4000004B, A55540005, A5F554005, AA0A0AA45, AA0AAA8FF, AA4000005, AAA0AA8FF, AAAA0A8FF, AAAA0AA8F, ...}
 2021-01-06, 12:55 #93 sweety439   Nov 2016 22×3×5×47 Posts Proven minimal primes (start with b+1) set: (bases b = 2, 3, 4, 5, 6, 8, 10, 12) (base b = 7 is not proven, but I cannot find other such primes) Code: 2: {11} 3: {12, 21, 111} 4: {11, 13, 23, 31, 221} 5: {12, 21, 23, 32, 34, 43, 104, 111, 131, 133, 313, 401, 414, 3101, 10103, 14444, 30301, 33001, 33331, 44441, 300031, 100000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000013} 6: {11, 15, 21, 25, 31, 35, 45, 51, 4401, 4441, 40041} 7: {14, 16, 23, 25, 32, 41, 43, 52, 56, 61, 65, 113, 115, 131, 133, 155, 212, 221, 304, 313, 335, 344, 346, 364, 445, 515, 533, 535, 544, 551, 553, 1022, 1051, 1112, 1202, 1211, 1222, 2111, 3031, 3055, 3334, 3503, 3505, 3545, 4504, 4555, 5011, 5455, 5545, 5554, 6034, 6634, 11111, 11201, 30011, 30101, 31001, 31111, 33001, 33311, 35555, 40054, 100121, 150001, 300053, 351101, 531101, 1100021, 33333301, 5100000001, 33333333333333331} 8: {13, 15, 21, 23, 27, 35, 37, 45, 51, 53, 57, 65, 73, 75, 107, 111, 117, 141, 147, 161, 177, 225, 255, 301, 343, 361, 401, 407, 417, 431, 433, 463, 467, 471, 631, 643, 661, 667, 701, 711, 717, 747, 767, 3331, 3411, 4043, 4443, 4611, 5205, 6007, 6101, 6441, 6477, 6707, 6777, 7461, 7641, 47777, 60171, 60411, 60741, 444641, 500025, 505525, 3344441, 4444477, 5500525, 5550525, 55555025, 444444441, 744444441, 77774444441, 7777777777771, 555555555555525, 44444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444447} 10: {11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 227, 251, 257, 277, 281, 349, 409, 449, 499, 521, 557, 577, 587, 727, 757, 787, 821, 827, 857, 877, 881, 887, 991, 2087, 2221, 5051, 5081, 5501, 5581, 5801, 5851, 6469, 6949, 8501, 9001, 9049, 9221, 9551, 9649, 9851, 9949, 20021, 20201, 50207, 60649, 80051, 666649, 946669, 5200007, 22000001, 60000049, 66000049, 66600049, 80555551, 555555555551, 5000000000000000000000000000027} 12: {11, 15, 17, 1B, 25, 27, 31, 35, 37, 3B, 45, 4B, 51, 57, 5B, 61, 67, 6B, 75, 81, 85, 87, 8B, 91, 95, A7, AB, B5, B7, 221, 241, 2A1, 2B1, 2BB, 401, 421, 447, 471, 497, 565, 655, 665, 701, 70B, 721, 747, 771, 77B, 797, 7A1, 7BB, 907, 90B, 9BB, A41, B21, B2B, 2001, 200B, 202B, 222B, 229B, 292B, 299B, 4441, 4707, 4777, 6A05, 6AA5, 729B, 7441, 7B41, 929B, 9777, 992B, 9947, 997B, 9997, A0A1, A201, A605, A6A5, AA65, B001, B0B1, BB01, BB41, 600A5, 7999B, 9999B, AAAA1, B04A1, B0B9B, BAA01, BAAA1, BB09B, BBBB1, 44AAA1, A00065, BBBAA1, AAA0001, B00099B, AA000001, BBBBBB99B, B0000000000000000000000000009B, 400000000000000000000000000000000000000077} Last fiddled with by sweety439 on 2021-01-26 at 16:35
 2021-01-06, 12:57 #94 sweety439   Nov 2016 22×3×5×47 Posts Let L(b) be the minimal set of the strings for the primes >b in base b Code: b |L(b)| largest element in L(b) largest element in L(b) in base b written in decimal 2 1 11 3 3 3 111 13 4 5 221 41 5 22 109313 5^95+8 6 11 40041 5209 7 71 3161 (7^17-5)/2 8 75 42207 (4*8^221+17)/7 10 77 502827 5*10^30+27 12 106 403977 4*12^41+91 (base b = 7 is not proven, but I cannot find other such primes) Last fiddled with by sweety439 on 2021-01-26 at 16:36
 2021-01-06, 17:11 #95 sweety439   Nov 2016 B0416 Posts Last fiddled with by sweety439 on 2021-01-26 at 16:21
 2021-01-07, 04:34 #96 LaurV Romulan Interpreter     Jun 2011 Thailand 24A316 Posts Haha, quite ingenious putting them in fdb in a product. Hat off! You made me click on the links because I could not imagine how you can put a list of numbers in fdb under a single identifier. Pleasantly surprised. (this encouragement post should not be taken as a license to restart spamming the forum, I really do appreciate the cleverness of the solution, but that's all )
2021-01-07, 07:53   #97
sweety439

Nov 2016

22×3×5×47 Posts

Quote:
 Originally Posted by LaurV Haha, quite ingenious putting them in fdb in a product. Hat off! You made me click on the links because I could not imagine how you can put a list of numbers in fdb under a single identifier. Pleasantly surprised. (this encouragement post should not be taken as a license to restart spamming the forum, I really do appreciate the cleverness of the solution, but that's all )
Now I computed the set for bases 2, 3, 4, 5, 6, 8, 10 and proved that my sets are all complete (by hand), see post #93, in this post, you said that my set in post #1 for bases 7 and 8 are not complete, and then I tried to solved base 8, and found there are only 4 primes not in the list in post #1:

Code:
77774444441
7777777777771
555555555555525
44444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444447
For the fully proof of base 8, see post #62

For the fully proof of base 10, see posts #84~#88 in page 8

Now I try to proof base 7, 9, 12 (other bases are too difficult to prove by hand, but can still prove by computer program, but I have no such program that can compute the simple families), base 9 may be harder, since this base has many families which can be ruled out to only contain composites (e.g. {1}, 3{1}, 5{1}, 6{1}, 3{8}, {8}5 and even the non-simple family {1}6{1})

 2021-01-07, 14:32 #98 sweety439   Nov 2016 B0416 Posts some 2-digits number xy in base b such that the smallest prime of the form xy*b^n+1 is minimal prime (start with b+1) in base b: (they have the property that both x*b^n+1 and y*b^n+1 are ruled out to only contain composites) b=11: Code: 12: 1453 16: 3429742097 18: 199 20: 56204873696128495701224278067068097021 36: 397 38: 419 40: 8574355241 42: 463 48: 6922076933111963233808451075889 56: 617 58: 7019 60: 661 62: 683 78: 859 80: 881 82: 9923 86: 947 100: 12101 102: 1123 106: 187785467 108: 191328589 b=13: Code: 16: 35153 18: 32258887092667 22: 3719 24: 313 28: 4733 34: 443 40: 521 42: 547 46: 599 48: 48*13^6267+1 66: 859 70: 911 72: 937 76: 166973 94: 1223 96: 1249 100: 1301 102: 1327 106: 25476138101289768358558528047514833801370032844988650960096232747 112: 2609385533717873 118: 211474926496367 120: 120*13^1552+1 124: 1613 126: 276823 144: 1873 148: 25013 150: 1951 154: 2003
 2021-01-07, 15:13 #99 sweety439   Nov 2016 1011000001002 Posts some 3-digits number xyz in base b such that the smallest prime of the form xyz*b^n+1 is minimal prime (start with b+1) in base b: (they have the property that all of x*b^n+1, y*b^n+1, z*b^n+1, xy*b^n+1, xz*b^n+1, yz*b^n+1 are ruled out to only contain composites) base 11: Code: 158: 210299 168: 27056569 378: 4159 408: 49369 498: 5479 518: 689459 548: 6029 568: 68729 588: 6469 658: 1165687139 708: 7789 898: 103832913046615294450266950864953559038103755928866797157252835645367668431847219 928: 112289 1138: 137699 1148: 124382424230098512904021128149 base 13: Code: 184: 2393 190: 917093711 196: 2549 202: 75001187 262: 3407 268: 45293 274: 46307 280: 615161 352: 10053473 358: 292031598119 382: 4967 388: 852437 406: 5279 412: 152972717 430: 5591 436: 5669 460: 5981 466: 13309427 [no known prime for S13 k=484, 2B3{0}1 is unsolved family in base 13] 490: 82811 538: 1181987 544: 1195169 574: 97007 580: 7541 616: 8009 622: 8087 652: 531856430093 658: 90710887636643 874: 264712843161629123 880: 148721 886: 11519 892: 11597 952: 12377 958: 2104727 964: 162917 970: 12611 1198: 34216079 1204: 2645189 1210: 15731 1216: 15809 1276: 80067107693 1282: 36615203 1288: 2829737 1294: 16823 1366: 652758153339229918146262170195611899412691250234661073608281535607441327245782154274226163627511856910242680997040239850166408242104102157590110059393064877741749052590786638302327896695517321882076385248095689607714279245947320040491619053213124101066946141643999231501774298529691499984819614064572813909251081765622950846989229615735750095118621279499641063473790506905622553548920945551313745769942678628213020977681299198153663054881959698284804196609230966508441216040759006656791607 1372: 17837 1396: 18149 1402: 12127883118972635782067 1420: 18461 1426: 18539 1444: 18773 1450: 41413451 1474: 19163 1480: 448255157756534441 1498: 34900531513476539 1504: 19553 1552: 20177 1558: 263303 1588: 2321529421116208423755077 1594: 269387 1630: 21191 1636: 21269 1666: 47582627 1672: 21737 1888: 821828298004735653739505427655167983139612951149032024600411489000613133621406169905393832026983244445467247164419724522787738176502646863025015739869100207704101192365412789485747731430993843624803368161051869616583490868130229284719241207489969244826144259775702398961583808046895149627729346180500126934284621986296138708001440861667299169 1894: 24623 1900: 1549888369901 1906: 4187483 1966: 4319303 1972: 333269 1978: 56493659 1984: 25793

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