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 2007-01-26, 07:03 #12 robert44444uk     Jun 2003 Oxford, UK 2·7·137 Posts Approach to solution Thinking a bit about this: The simplest covering set comprises 2 primes with order base b of 2. One provides cover for n 1,3,5,7... and the other for 2,4,6,8... It is simple to prove that there are no two such primes. Covering n=1 implies that (2.i^1+1)modp == 0modp and (2.i^3+1)modp==0modp or 0modp == (2i^3-2i)modp 0modp == 2.i.(i-1).(i+1)modp therefore i=1 and (2.1^2+1)modp==3mod(p)==0mod(p) which is only solvable for p=3, which is only 1 prime, and in position 1,3,5... this is base=1mod3 and position 2,4,6... is 2mod3 A second covering set can be described using 3 and two other primes q and r which cover either position 1 and 5 or 2 and 6, with 3 covering 2,4,6.. or 1,3,5.. . Such q,r are of the form 4j+1, where j is an integer. Let us look at 3 covering positions 2,4,6.. and q covering 1,5,9... Then (2.b^1+1)modq==(2.b^5+1)modq is solvable for 5, in position 1 as 2mod5 and position 2 as 3mod5 Trying to find prime r of the form 4j+1 requires a solution to the equation A 4th root of unity modr equals either (r-1)/2 or the solution to (2.x^3+1)modr == 0modr I checked the first few 4j+1 primes and found no solutions except 5: 1. r 2. a 4th root of unity modr 3. another 4th root of unity modr 4. (r-1)/2 5. solution to (2.x^3+1)modr==0modr **= no solution 5 2 3 2 3 13 5 8 6 ** 17 4 13 8 2 29 12 17 14 10 37 6 31 18 ** 41 9 32 20 8 53 23 30 26 50 61 11 50 30 ** 73 27 46 36 ** 89 34 55 44 50 97 22 75 48 ** 101 10 91 50 66 109 33 76 54 62 113 15 98 56 53 137 37 100 68 130 149 44 105 74 83 157 28 129 78 15 173 80 93 86 101 181 19 162 90 ** 193 81 112 96 ** 197 14 183 98 75 229 107 122 114 7 233 89 144 116 195 241 64 177 120 ** 257 16 241 128 225 269 82 187 134 19 Makes me think that there may be no r that can satisfy the criteria, and that any covering set (if one exists) has to be more complex. Perhaps someone can find an r or prove that I am comparing apples and oranges. Last fiddled with by robert44444uk on 2007-01-26 at 07:13
2007-01-26, 07:21   #13
robert44444uk

Jun 2003
Oxford, UK

2·7·137 Posts

Quote:
 Originally Posted by Citrix I don't think that 2 can be a sierpinki/riesel number for any base. Nor can any of the low k values.
4 is the lowest k I have found, which is sierpinski and riesel quite frequently for bases where b^2-1 produces two primitive prime factors.

For example 4*14^n+/-1 never prime. Primitive primes factors of 14^2-1 are 3 and 5.

Maybe k=1 can be sierpinski/ riesel. May have a look at that.

Last fiddled with by robert44444uk on 2007-01-26 at 07:23

 2017-02-09, 15:51 #14 sweety439   Nov 2016 22×691 Posts For fixed k, find the smallest base b such that all numbers of the form k*b^n+1 (k*b^n-1) are composite. If k is of the form 2^n-1 (2^n+1), except k=1 (k=9), it is conjectured for every nontrivial base b, there is a prime of the form k*b^n+1 (k*b^n-1). However, for all other k's, there is a base b such that all numbers of the form k*b^n+1 (k*b^n-1) are composite. S = conjectured smallest base b such that k is a Sierpinski number. k S remaining bases b with no known primes 1 none {38, 50, 62, 68, 86, 92, 98, 104, 122, 144, 168, 182, 186, 200, 202, 212, 214, 218, 244, 246, 252, 258, 286, 294, 298, 302, 304, 308, 322, 324, 338, 344, 354, 356, 362, 368, 380, 390, 394, 398, 402, 404, 410, 416, 422, 424, 446, 450, 454, 458, 468, 480, 482, 484, 500, 514, 518, 524, 528, 530, 534, 538, 552, 558, 564, 572, 574, 578, 580, 590, 602, 604, 608, 620, 622, 626, 632, 638, 648, 650, 662, 666, 668, 670, 678, 684, 692, 694, 698, 706, 712, 720, 722, 724, 734, 744, 746, 752, 754, 762, 766, 770, 792, 794, 802, 806, 812, 814, 818, 836, 840, 842, 844, 848, 854, 868, 870, 872, 878, 888, 896, 902, 904, 908, 922, 924, 926, 932, 938, 942, 944, 948, 954, 958, 964, 968, 974, 978, 980, 988, 994, 998, 1002, 1006, 1014, 1016, 1026, ...} (b=m^r with odd r>1 proven composite by full algebraic factors) 2 201446503145165177 (?) {218, 236, 365, 383, 461, 512, 542, 647, 773, 801, 836, 878, 908, 914, 917, 947, 1004, ...} 3 none {718, 912, ...} 4 14 proven 5 140324348 {308, 326, 512, 824, ...} 6 34 proven 7 none {1004, ...} 8 20 proven (b=8 proven composite by full algebraic factors) 9 177744 {592, 724, 884, ...} 10 32 proven 11 14 proven 12 142 {12} 13 20 proven 14 38 proven 15 none {398, 650, 734, 874, 876, 1014, ...} 16 38 {32} 17 278 {68, 218} 18 322 {18, 74, 227, 239, 293} 19 14 proven 20 56 proven 21 54 proven 22 68 {22} 23 32 proven 24 114 {79} 25 38 proven 26 14 proven 27 90 {62} 28 86 {41} 29 20 proven 30 898 31 none 32 92 {87} (b=32 proven composite by full algebraic factors) R = conjectured smallest base b such that k is a Riesel number. k R remaining bases b with no known primes 1 none proven 2 none {303, 522, 578, 581, 992, 1019, ...} 3 none {588, 972, ...} 4 14 proven (b=9 proven composite by full algebraic factors) 5 none {338, 998, ...} 6 34 proven (b=24 proven composite by partial algebraic factors) 7 9162668342 {308, 392, 398, 518, 548, 638, 662, 848, 878, ...} 8 20 proven 9 none {378, 438, 536, 566, 570, 592, 636, 688, 718, 808, 830, 852, 926, 990, 1010, ...} (b=m^2 proven composite by full algebraic factors, b=4 mod 5 proven composite by partial algebraic factors) 10 32 proven 11 14 proven 12 142 proven 13 20 proven 14 8 proven 15 8241218 {454, 552, 734, 856, ...} 16 50 proven (b=9 proven composite by full algebraic factors, b=33 proven composite by partial algebraic factors) 17 none {98, 556, 650, 662, 734, ...} 18 203 {174} (b=50 proven composite by partial algebraic factors) 19 14 proven 20 56 proven 21 54 proven 22 68 {38, 62} 23 32 proven 24 114 proven 25 38 proven (b=36 proven composite by full algebraic factors, b=12 proven composite by partial algebraic factors) 26 14 proven 27 90 {34} (b=8 and 64 proven composite by full algebraic factors, b=12 proven composite by partial algebraic factors) 28 86 {74} 29 20 proven 30 898 31 362 {80, 84, 122, 278, 350} 32 92 {54, 71, 77}
 2017-02-09, 15:52 #15 sweety439   Nov 2016 22×691 Posts S = conjectured smallest base b such that k is a Sierpinski number. k S cover set 1 none 2 201446503145165177 (?) {3, 5, 17, 257, 641, 65537, 6700417} period=64 3 none 4 14 {3, 5} period=2 5 140324348 {3, 13, 17, 313, 11489} period=16 6 34 {5, 7} period=2 7 none 8 20 {3, 7} period=2 9 177744 {5, 17, 41, 193} period=8 10 32 {3, 11} period=2 11 14 {3, 5} period=2 12 142 {11, 13} period=2 13 20 {3, 7} period=2 14 38 {3, 13} period=2 15 none 16 38 {3, 5, 17} period=4 17 278 {3, 5, 29} period=4 18 322 {17, 19} period=2 19 14 {3, 5} period=2 20 56 {3, 19} period=2 21 54 {5, 11} period=2 22 68 {3, 23} period=2 23 32 {3, 11} period=2 24 114 {5, 23} period=2 25 38 {3, 13} period=2 26 14 {3, 5} period=2 27 90 {7, 13} period=2 28 86 {3, 29} period=2 29 20 {3, 7} period=2 30 898 {29, 31} period=2 31 none 32 92 {3, 31} period=2 33 5592 {5, 17, 109} period=4 34 14 {3, 5} period=2 35 50 {3, 17} period=2 36 68 {5, 7, 13, 31, 37} period=12 37 56 {3, 19} period=2 38 98 {3, 5, 17} period=4 39 94 {5, 19} period=2 40 122 {3, 41} period=2 41 14 {3, 5} period=2 42 1762 {41, 43} period=2 43 32 {3, 11} period=2 44 128 {3, 43} period=2 45 252 {11, 23} period=2 46 140 {3, 47} period=2 47 8 {3, 5, 13} period=4 48 328 {7, 47} period=2 49 14 {3, 5} period=2 50 20 {3, 7} period=2 51 64 {5, 13} period=2 52 158 {3, 53} period=2 53 38 {3, 13} period=2 54 264 {5, 53} period=2 55 20 {3, 7} period=2 56 14 {3, 5} period=2 57 202 {7, 29} period=2 58 176 {3, 59} period=2 59 144 {5, 29} period=2 60 3598 {59, 61} period=2 61 92 {3, 31} period=2 62 182 {3, 61} period=2 63 none 64 29 {3, 5} period=2 R = conjectured smallest base b such that k is a Riesel number. k R cover set 1 none 2 none 3 none 4 14 {3, 5} period=2 5 none 6 34 {5, 7} period=2 7 9162668342 {3, 5, 17, 1201, 169553} period=16 8 20 {3, 7} period=2 9 none 10 32 {3, 11} period=2 11 14 {3, 5} period=2 12 142 {11, 13} period=2 13 20 {3, 7} period=2 14 8 {3, 5, 13} period=4 15 8241218 {7, 17, 113, 1489} period=8 16 50 {3, 17} period=2 17 none 18 203 {5, 13, 17} period=4 19 14 {3, 5} period=2 20 56 {3, 19} period=2 21 54 {5, 11} period=2 22 68 {3, 23} period=2 23 32 {3, 11} period=2 24 114 {5, 23} period=2 25 38 {3, 13} period=2 26 14 {3, 5} period=2 27 90 {7, 13} period=2 28 86 {3, 29} period=2 29 20 {3, 7} period=2 30 898 {29, 31} period=2 31 362 {3, 7, 13, 37, 331} period=12 32 92 {3, 31} period=2 33 none 34 14 {3, 5} period=2 35 50 {3, 17} period=2 36 184 {5, 37} period=2 37 56 {3, 19} period=2 38 110 {3, 37} period=2 39 94 {5, 19} period=2 40 122 {3, 41} period=2 41 14 {3, 5} period=2 42 1762 {41, 43} period=2 43 32 {3, 11} period=2 44 128 {3, 43} period=2 45 252 {11, 23} period=2 46 140 {3, 47} period=2 47 68 {3, 23} period=2 48 328 {7, 47} period=2 49 14 {3, 5} period=2 50 20 {3, 7} period=2 51 64 {5, 13} period=2 52 158 {3, 53} period=2 53 38 {3, 13} period=2 54 264 {5, 53} period=2 55 20 {3, 7} period=2 56 14 {3, 5} period=2 57 202 {7, 29} period=2 58 176 {3, 59} period=2 59 86 {3, 29} period=2 60 3598 {59, 61} period=2 61 68 {3, 7, 13, 31} period=6 62 182 {3, 61} period=2 63 36858 {5, 31, 397} period=4 64 14 {3, 5} period=2
2017-02-09, 15:55   #17
sweety439

Nov 2016

22·691 Posts

Quote:
 Originally Posted by Citrix Has there been any work done to find S/R for fixed k and variable base? eg. What is the lowest base=b for k=2 such that 2*b^n+1 is never prime? What k's can never be sierpinski numbers? Is there any proof to this. How does one generate the sierpinski base for a given k? (The same questions for Riesel side)
This base is probably 201446503145165177.

Last fiddled with by sweety439 on 2017-02-09 at 15:55

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