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2021-01-08, 20:21   #111
sweety439

Nov 2016

22×3×5×47 Posts

Quote:
 Originally Posted by sweety439 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
484*13^15198+1 (2B30151971) is prime!!!

This prime is likely the third-largest "base 13 minimal prime (start with base+1)" (there is a larger probable prime 8032017111, and there is an unsolved family in base 13: 9{5})

Last fiddled with by sweety439 on 2021-01-08 at 20:23

 2021-01-08, 20:37 #112 pinhodecarlos     "Carlos Pinho" Oct 2011 Milton Keynes, UK 114268 Posts Have you thought about writing a book about all of this exciting stuff?
 2021-01-08, 21:11 #113 VBCurtis     "Curtis" Feb 2005 Riverside, CA 3·1,579 Posts 14 posts on a single topic in 36 hours. Are you trying to get banned? It's working.
2021-01-09, 10:34   #114
sweety439

Nov 2016

B0416 Posts

Quote:
 Originally Posted by pinhodecarlos Have you thought about writing a book about all of this exciting stuff?
I have a pdf file for the proof (not complete, continue updating), I will complete the proof for bases 7, 9, 12
Attached Files
 minimal primes gt b in base b.pdf (238.6 KB, 41 views)

 2021-01-09, 10:38 #115 sweety439   Nov 2016 22×3×5×47 Posts It is conjectured that for all simple families x{y}z cannot be proved as only contain composites (for numbers > base) in one of these four ways: ** Periodic sequence p of prime divisors with p(n) | (xyyy...yyyz with n y's) ** Algebraic factors (e.g. difference-of-squares factorization, difference-of-cubes factorization, sum-of-cubes factorization, difference-of-5th-powers factorization, sum-of-5th-powers factorization, Aurifeuillian factorization of x^4+4*y^4, etc.) of x{y}z ** The combine of the above two ways (like the case of {B}9B in base 12) ** Reduced to (b^(r*n+s)+1)/gcd(b+1,2), and r*n+s can never be power of 2 (like the case of 8{0}1 in base 128) Then x{y}z contain primes (for numbers > base). Last fiddled with by sweety439 on 2021-04-05 at 16:12
 2021-01-09, 10:53 #116 sweety439   Nov 2016 22·3·5·47 Posts The simple families x{y}z (where x and z are strings of base b digits, y is base b digit) in base b are of the form (a*b^n+c)/gcd(a+c,b-1) (where a>=1, c != 0, gcd(a,c) = 1, gcd(b,c) = 1), this number has algebra factors if and only if: either * there is an integer r>1 such that both a*b^n and -c are perfect rth powers or * a*b^n*c is of the form 4*m^4 with integer m If (a*b^n+c)/gcd(a+c,b-1) (where a>=1, c != 0, gcd(a,c) = 1, gcd(b,c) = 1) has algebra factors, then it must be composite, the only exception is when it is either GFN (generalized Fermat number) base b or GRU (generalized repunit number) base b, in these two cases this number may be prime, the only condition is the n is power of 2 if it is GFN, and the n is prime if it is GRU GFNs and GRUs are the only simple families x{y}z (where x and z are strings of base b digits, y is base b digit) in base b which are also cyclotomic numbers (i.e. numbers of the form Phi(n,b)/gcd(Phi(n,b),n), where Phi is cyclotomic polynomial) or Zsigmondy numbers Zs(n,b,1) (see Zsigmondy's theorem) GFNs and GRUs in bases 2<=b<=36: Code: base GFN family GRU family 2 1{0}1 {1} 3 {1}2 {1} 4 1{0}1 1{3}, {2}3 5 {2}3 {1} 6 1{0}1 {1} 7 {3}4 {1} 8 2{0}1, 4{0}1 1{7}, 3{7} 9 {4}5 1{4}, {6}7 10 1{0}1 {1} 11 {5}6 {1} 12 1{0}1 {1} 13 {6}7 {1} 14 1{0}1 {1} 15 {7}8 {1} 16 1{0}1 1{F}, 7{F}, {A}B, 2{A}B 17 {8}9 {1} 18 1{0}1 {1} 19 {9}A {1} 20 1{0}1 {1} 21 {A}B {1} 22 1{0}1 {1} 23 {B}C {1} 24 1{0}1 {1} 25 {C}D 1{6}, {K}L 26 1{0}1 {1} 27 1{D}E, 4{D}E 1{D}, 4{D} 28 1{0}1 {1} 29 {E}F {1} 30 1{0}1 {1} 31 {F}G {1} 32 2{0}1, 4{0}1, 8{0}1, G{0}1 1{V}, 3{V}, 7{V}, F{V} 33 {G}H {1} 34 1{0}1 {1} 35 {H}I {1} 36 1{0}1 1{7}, {U}V Note: we do not include the case where the "ground base" of the GFNs or GRUs is either perfect power or of the form -4*m^4 with integer m, since such numbers have algebra factors and are composite for all n or are prime only for very small n, such families for bases 2<=b<=36 are: Code: base GFN family GRU family 4 {1} 8 1{0}1 {1} 9 {1} 16 {1}, 1{5}, {C}D 25 {1} 27 {D}E {1} 32 1{0}1 {1} 36 {1} Note: the "ground base" of the GFNs or GRUs need not to be b (when b is perfect power), it may be root of b, it may also be negative integer which is root of b Last fiddled with by sweety439 on 2021-03-26 at 13:23
 2021-01-09, 21:18 #117 sweety439   Nov 2016 B0416 Posts These bases 2<=b<=1024 have unsolved families which are GFNs: {31, 32, 37, 38, 50, 55, 62, 63, 67, 68, 77, 83, 86, 89, 91, 92, 93, 97, 98, 99, 104, 107, 109, 117, 122, 123, 125, 127, 128, 133, 135, 137, 143, 144, 147, 149, 151, 155, 161, 168, 177, 179, 182, 183, 186, 189, 193, 197, 200, 202, 207, 211, 212, 213, 214, 215, 216, 217, 218, 223, 225, 227, 233, 235, 241, 244, 246, 247, 249, 252, 255, 257, 258, 263, 265, 269, 273, 277, 281, 283, 285, 286, 287, 291, 293, 294, 297, 298, 302, 303, 304, 307, 308, 311, 319, 322, 324, 327, 338, 343, 344, 347, 351, 354, 355, 356, 357, 359, 361, 362, 367, 368, 369, 377, 380, 381, 383, 385, 387, 389, 390, 393, 394, 397, 398, 401, 402, 404, 407, 410, 411, 413, 416, 417, 421, 422, 423, 424, 437, 439, 443, 446, 447, 450, 454, 457, 458, 465, 467, 468, 469, 473, 475, 480, 481, 482, 483, 484, 489, 493, 495, 497, 500, 509, 511, 512, 514, 515, 518, 524, 528, 530, 533, 534, 538, 541, 547, 549, 552, 555, 558, 563, 564, 572, 574, 578, 580, 590, 591, 593, 597, 601, 602, 603, 604, 608, 611, 615, 619, 620, 621, 622, 625, 626, 627, 629, 632, 633, 635, 637, 638, 645, 647, 648, 650, 651, 653, 655, 659, 662, 663, 666, 667, 668, 670, 671, 673, 675, 678, 679, 683, 684, 687, 691, 692, 693, 694, 697, 698, 706, 707, 709, 712, 717, 720, 722, 724, 731, 733, 734, 735, 737, 741, 743, 744, 746, 749, 752, 753, 754, 755, 757, 759, 762, 765, 766, 767, 770, 771, 773, 775, 777, 783, 785, 787, 792, 793, 794, 797, 801, 802, 806, 807, 809, 812, 813, 814, 817, 818, 823, 825, 836, 840, 842, 844, 848, 849, 851, 853, 854, 865, 867, 868, 870, 872, 873, 877, 878, 887, 888, 889, 893, 896, 897, 899, 902, 903, 904, 907, 908, 911, 915, 922, 923, 924, 926, 927, 932, 933, 937, 938, 939, 941, 942, 943, 944, 945, 947, 948, 953, 954, 957, 958, 961, 964, 967, 968, 974, 975, 977, 978, 980, 983, 987, 988, 993, 994, 998, 999, 1000, 1002, 1003, 1005, 1006, 1009, 1014, 1016, 1017, 1024} Such families are: * 4:{0}:1, 16:{0}:1 for b = 32 * 12:{62}:63 for b = 125 * 16:{0}:1 for b = 128 * 36:{0}:1 for b = 216 * 24:{171}:172 for b = 343 * 2:{0}:1, 4:{0}:1, 16:{0}:1, 32:{0}:1, 256:{0}:1 for b = 512 * 10:{0}:1, 100:{0}:1 for b = 1000 * 4:{0}:1, 16:{0}:1, 256:{0}:1 for b = 1024 * 1:{0}:1 for other even bases b * {((b-1)/2)}:((b+1)/2) for other odd bases b These bases 2<=b<=1024 have unsolved families which are GRUs: {185, 243, 269, 281, 380, 384, 385, 394, 452, 465, 511, 574, 601, 631, 632, 636, 711, 713, 759, 771, 795, 861, 866, 881, 938, 948, 951, 956, 963, 1005, 1015} Such families are: * 40:{121} for b = 243 * {1} for other bases b Last fiddled with by sweety439 on 2021-03-25 at 10:46
 2021-01-09, 21:44 #118 carpetpool     "Sam" Nov 2016 2·163 Posts Here's a suggestion: Instead of constantly posting search limits and reservations that can be done in minutes, use a doc or pdf instead. Attach or provide the link in a single post. If you need to edit the link or update the attachment, edit the post instead of creating a new post. It saves time and space! The updates you see on CRUS or other prime search projects take at least several months, if not years to complete. Now your searches and updates on the other hand, can be done in minutes. Why post something that is so trivial that anyone who wants to do it can do it in such a short amount of time?
2021-01-10, 12:09   #119
sweety439

Nov 2016

282010 Posts

Quote:
 Originally Posted by carpetpool Here's a suggestion: Instead of constantly posting search limits and reservations that can be done in minutes, use a doc or pdf instead. Attach or provide the link in a single post. If you need to edit the link or update the attachment, edit the post instead of creating a new post. It saves time and space! The updates you see on CRUS or other prime search projects take at least several months, if not years to complete. Now your searches and updates on the other hand, can be done in minutes. Why post something that is so trivial that anyone who wants to do it can do it in such a short amount of time?

Well, the proofs for base 2, 3, 4 are really trivial, but they are part of the project, I want to store these proofs, and the pdf file was made recently

 2021-01-10, 19:27 #120 sweety439   Nov 2016 1011000001002 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=12: Code: 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
 2021-01-10, 19:28 #121 sweety439   Nov 2016 22·3·5·47 Posts There are totally 106 minimal primes (start with 2 digits) in base 12, there are 77 such primes in base 10

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