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Old 2017-01-20, 09:19   #1
sweety439
 
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Nov 2016

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Default The reverse Sierpinski/Riesel problem

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 {171, 173, 269, 293, 347, 432, 490, 659, 661, 695, 712, 738, 795, 830}
31 none {38, 74, 116, 152, 174, 182, 242, 248, 254, 272, 278, 332, 448, 454, 458, 486, 494, 570, 578, 584, 614, 620, 632, 662, 714, 722, 728, 734, 758, 786, 794, 812, 824, 828, 842, 898, 938, 1014, 1028, ...}
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 {193, 247, 254, 305, 495, 501, 514, 535, 537, 569, 654, 659, 661, 683, 753, 764, 774, 809, 869}
31 362 {80, 84, 122, 278, 350}
32 92 {54, 71, 77}

Last fiddled with by sweety439 on 2019-03-01 at 18:10
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