A Sierpinski/Riesel-like problem

[sweety439]

The CK for S66 is known to be 21314443 (I have checked it, no k<21314443 have covering set)

The CK for S120, R66, R120 are still running ....

[sweety439]

Originally Posted by sweety439

Update newest file for Sierpinski problems to include the newest test limit for S53 k=4 and the lower&upper bounds for the CK for S66 and S120 and the covering sets for the CK for S66 and S120

Update newest file for Riesel problems to include the lower&upper bounds for the CK for R66 and R120 and the covering sets for the CK for R66 and R120

Reservations for the Sierpinski/Riesel problems:

S2, all k except 65536: PrimeGrid's Seventeen or Bust (status)
S2, k=65536: FermatSearch (status)
S24, all k not == 22 mod 23: rajula to n=500K
S32, k=4: FermatSearch (status)
S53, k=4: pepi37 to n=1.7M
S128, k=16: FermatSearch (status)
S512, all k except 5: FermatSearch (status)
S1024, k=4 and k=16: FermatSearch (status)
All remain GFN's except Fermat numbers base 2: (S6, k=1296; S10, k=100; S12, k=12; etc.): Generalized Fermat Prime Search (status)
All remain half GFN's (S15, k=225; S31, k=1; S37, k=37; etc.): Richard Fischer (status)

R2, all k except 351134 and 478214: PrimeGrid's Riesel Problem (status)
R2, k=351134 and k=478214: PRPnet 1400 to n=7M
R6, k=1597: masser
R36, k=9582: masser
R36, all k not == 1 mod 5 or == 1 mod 7 except 9582: VBCurtis to 400K

[sweety439]

Also, there are many reservations for bases > 128 (except 256, 512, 1024) or k's > CK:

S2 primes 78557<k<271129: Prime Sierpinski Problem
S2 odd composites 78557<k<271129: Extended Sierpinski Problem
S2 k=90646, k=101746, k=6136576, k=103991296, and k=157444096: Proth Sierpinski Problem
S4 k=18534: Jean Penne
S4 k=64494: PRPnet 1400
S5 k=6436, 7528, 10918, 26798, 29914, 31712, 36412, 41738, 44348, 44738, 45748, 51208, 58642, 60394, 62698, 64258, 67612, 67748, 71492, 74632, 76724, 83936, 84284, 90056, 92906, 93484, 105464, 126134, 139196, 152588: Sierpinski/Riesel base 5 project
S9 k=2036: BOINC to n=5M
S10 k=7666: Cruelty
S19 for many large k's: Xentar
S9409 k=97: Fan Ming
S67607 k=2: "Divides Phi" category
and many other reservations in CRUS
and many GFN reservations (including S216 k=36 and S1000 k=10, etc.) by GFN search
and many half GFN reservations (including S243 k=27 and S343 k=49, etc.) by Fischer

R4 k=14361 and k=19464: PRPnet 1400 to n=3.5M
R5 k=3622, 4906, 23906, 26222, 35248, 52922, 63838, 64598, 68132, 71146, 76354, 81134, 92936, 102818, 102952, 109238, 109862, 127174, 131848, 134266, 136804, 143632, 145462, 145484, 146756, 147844, 151042, 152428, 154844, 159388, 164852, 170386, 170908, 177742, 182398, 187916, 189766, 190334, 195872, 201778, 204394, 206894, 213988, 231674, 239062, 239342, 246238, 248546, 259072, 265702, 267298, 271162, 273662, 285598, 285728, 298442, 304004, 313126, 318278, 325922, 335414, 338866: Sierpinski/Riesel base 5 project
R10 k=4421: Cruelty
R10 k=5428, k=11197, and k=16750: Patrick De Geest
R243 k=81: Paul Bourdelais
R396 k=156816: Paul Bourdelais
R598 k=357604: Paul Bourdelais
k=1 for remain Riesel bases <= 1000: Michael Stocker
k=2 for remain Riesel bases <= 2048: LaurV
k=base-1 for remain Riesel bases <= 2049: Steven Harvey
and many other reservations in CRUS

[sweety439]

Originally Posted by sweety439

The CK for S66 is known to be 21314443 (I have checked it, no k<21314443 have covering set)

The CK for S120, R66, R120 are still running ....

The CK for R66 is 63717671

covering set: {7, 67, 613, 4423}, period=6

[sweety439]

All n must be >= 1.

k-values that make a full covering set with all or partial algebraic factors are excluded from the conjectures.

k-values that are a multiple of base (b) and where (k+-1)/gcd(k+-1,b-1) (+ for Sierpinski, - for Riesel) is not prime are included in the conjectures but excluded from testing. Such k-values will have the same prime as k / b.

[sweety439]

Originally Posted by sweety439

All Sierpinski/Riesel bases listed "NA" have CK > 5M (i.e. 5M is the lower bound for these Sierpinski/Riesel bases)

upper bounds for these Sierpinski/Riesel bases <= 600:

S66: 21314443 (if not exactly this number, then must be == 4 mod 5 or == 12 mod 13)
S120: 374876369 (if not exactly this number, then must be == 6 mod 7 or == 16 mod 17)
S156: 18406311208 (if not exactly this number, then must be == 4 mod 5 or == 30 mod 31)
S210: 147840103 (if not exactly this number, then must be == 10 mod 11 or == 18 mod 19)
S280: 82035074042274 (if not exactly this number, then must be == 2 mod 3 or == 30 mod 31)
S330: 16636723 (if not exactly this number, then must be == 6 mod 7 or == 46 mod 47)
S358: 27478218 (if not exactly this number, then must be == 2 mod 3 or == 6 mod 7 or == 16 mod 17)
S456: 14836963 (if not exactly this number, then must be == 4 mod 5 or == 6 mod 7 or == 12 mod 13)
S462: 6880642 (if not exactly this number, then must be == 460 mod 461)
S546: 45119296 (if not exactly this number, then must be == 4 mod 5 or == 108 mod 109)

R66: 101954772 (if not exactly this number, then must be == 1 mod 5 or == 1 mod 13)
R120: 166616308 (if not exactly this number, then must be == 1 mod 7 or == 1 mod 17)
R156: 2113322677 (if not exactly this number, then must be == 1 mod 5 or == 1 mod 31)
R180: 7674582 (if not exactly this number, then must be == 1 mod 179)
R210: 80176412 (if not exactly this number, then must be == 1 mod 11 or == 1 mod 19)
R280: 513613045571841 (if not exactly this number, then must be == 1 mod 3 or == 1 mod 31)
R330: 16527822 (if not exactly this number, then must be == 1 mod 7 or == 1 mod 47)
R358: 27606383 (if not exactly this number, then must be == 1 mod 3 or == 1 mod 7 or == 1 mod 17)
R420: 6548233 (if not exactly this number, then must be == 1 mod 419)
R456: 76303920 (if not exactly this number, then must be == 1 mod 5 or == 1 mod 7 or == 1 mod 13)
R546: 11732602 (if not exactly this number, then must be == 1 mod 5 or == 1 mod 109)
R570: 12511182 (if not exactly this number, then must be == 1 mod 569)

The CK for R180 is known to be 7674582
The CK for S462 is known to be 6880642
I have checked them.

Note: For the conjecture of R180, these k-values proven composite by partial algebra factors:

* k = m^2 with m = = 19 or 162 mod 181
* k = 5*m^2 with m = = 67 or 114 mod 181

[sweety439]

All Sierpinski/Riesel bases listed "NA" have CK > 5M (i.e. 5M is the lower bound for these Sierpinski/Riesel bases)

upper bounds for these Sierpinski/Riesel bases <= 1024:

S66: 21314443 (if not exactly this number, then must be == 4 mod 5 or == 12 mod 13)
S120: 374876369 (if not exactly this number, then must be == 6 mod 7 or == 16 mod 17)
S156: 18406311208 (if not exactly this number, then must be == 4 mod 5 or == 30 mod 31)
S210: 147840103 (if not exactly this number, then must be == 10 mod 11 or == 18 mod 19)
S280: 82035074042274 (if not exactly this number, then must be == 2 mod 3 or == 30 mod 31)
S330: 16636723 (if not exactly this number, then must be == 6 mod 7 or == 46 mod 47)
S358: 27478218 (if not exactly this number, then must be == 2 mod 3 or == 6 mod 7 or == 16 mod 17)
S456: 14836963 (if not exactly this number, then must be == 4 mod 5 or == 6 mod 7 or == 12 mod 13)
S462: 6880642 (if not exactly this number, then must be == 460 mod 461)
S546: 45119296 (if not exactly this number, then must be == 4 mod 5 or == 108 mod 109)
S630: 24015859 (if not exactly this number, then must be == 16 mod 17 or == 36 mod 37)
S690: 395800653 (if not exactly this number, then must be == 12 mod 13 or == 52 mod 53)
S726: 10923176 (if not exactly this number, then must be == 4 mod 5 or == 28 mod 29)
S756: 67836285 (if not exactly this number, then must be == 4 mod 5 or == 150 mod 151)
S876: 258073912 (if not exactly this number, then must be == 4 mod 5 or == 6 mod 7)
S910: 4989054506113 (if not exactly this number, then must be == 2 mod 3 or == 100 mod 101)
S960: 197938246042 (if not exactly this number, then must be == 6 mod 7 or == 136 mod 137)
S966: 924230558 (if not exactly this number, then must be == 4 mod 5 or == 192 mod 193)
S1008: 12730554 (if not exactly this number, then must be == 18 mod 19 or == 52 mod 53)
S1020: 95696289 (if not exactly this number, then must be == 1018 mod 1019)

R66: 101954772 (if not exactly this number, then must be == 1 mod 5 or == 1 mod 13)
R120: 166616308 (if not exactly this number, then must be == 1 mod 7 or == 1 mod 17)
R156: 2113322677 (if not exactly this number, then must be == 1 mod 5 or == 1 mod 31)
R180: 7674582 (if not exactly this number, then must be == 1 mod 179)
R210: 80176412 (if not exactly this number, then must be == 1 mod 11 or == 1 mod 19)
R280: 513613045571841 (if not exactly this number, then must be == 1 mod 3 or == 1 mod 31)
R330: 16527822 (if not exactly this number, then must be == 1 mod 7 or == 1 mod 47)
R358: 27606383 (if not exactly this number, then must be == 1 mod 3 or == 1 mod 7 or == 1 mod 17)
R420: 6548233 (if not exactly this number, then must be == 1 mod 419)
R456: 76303920 (if not exactly this number, then must be == 1 mod 5 or == 1 mod 7 or == 1 mod 13)
R546: 11732602 (if not exactly this number, then must be == 1 mod 5 or == 1 mod 109)
R570: 12511182 (if not exactly this number, then must be == 1 mod 569)
R630: 24412760 (if not exactly this number, then must be == 1 mod 17 or == 1 mod 37)
R690: 42053568 (if not exactly this number, then must be == 1 mod 13 or == 1 mod 53)
R726: 12751579 (if not exactly this number, then must be == 1 mod 5 or == 1 mod 29)
[R728 has CK=212722, didn't found this CK since it has a large prime (105997) in the covering set]
R756: 54604682 (if not exactly this number, then must be == 1 mod 5 or == 1 mod 151)
R876: 51768432 (if not exactly this number, then must be == 1 mod 5 or == 1 mod 7)
R910: 5005381602981 (if not exactly this number, then must be == 1 mod 3 or == 1 mod 101)
R946: 2156122023 (if not exactly this number, then must be == 1 mod 3 or == 1 mod 5 or == 1 mod 7)
R960: 61681833328 (if not exactly this number, then must be == 1 mod 7 or == 1 mod 137)
R966: 699327630 (if not exactly this number, then must be == 1 mod 5 or == 1 mod 193)
R1020: 94655888 (if not exactly this number, then must be == 1 mod 1019)

[sweety439]

Changed the CK for S728 from 4606127 to 953974, I didn't found this CK since it has a large prime (105997) in the covering set

[sweety439]

Originally Posted by sweety439

The CK for R180 is known to be 7674582
The CK for S462 is known to be 6880642
I have checked them.

Note: For the conjecture of R180, these k-values proven composite by partial algebra factors:

* k = m^2 with m = = 19 or 162 mod 181
* k = 5*m^2 with m = = 67 or 114 mod 181

The CK for R420 is known to be 6548233
I have checked it.

Note: For the conjecture of R420, these k-values proven composite by partial algebra factors:

* k = m^2 with m = = 29 or 392 mod 421
* k = 105*m^2 with m = = 58 or 363 mod 421

[sweety439]

Originally Posted by sweety439

Riesel k=9: {107, 207, 237, 325, 347, 378, 438, 483, 536, 566, 570, 592, 636, 688, 705, 711, 718, 823, 830, 835, 852, 893, 907, 926, 927, 995, 1010} (totally 27 bases)
Riesel k=10: {80, 233, 262, 284, 307, 505, 530, 551, 611, 691, 712, 724, 883, 899, 912, 980} (totally 16 bases)
Riesel k=11: {65, 123, 137, 163, 173, 207, 214, 221, 227, 235, 247, 263, 283, 293, 317, 331, 375, 377, 422, 444, 452, 458, 471, 487, 533, 542, 555, 603, 627, 638, 663, 668, 691, 723, 752, 793, 804, 823, 843, 857, 863, 872, 907, 911, 923, 949, 950, 962, 987} (totally 49 bases)
Riesel k=12: {263, 615, 912, 978} (totally 4 bases)

Sierpinski k=1: {31, 38, 50, 55, 62, 63, 67, 68, 77, 83, 86, 89, 91, 92, 97, 98, 99, 104, 107, 109, 122, 123, 127, 135, 137, 143, 144, 147, 149, 151, 155, 161, 168, 179, 182, 183, 186, 189, 197, 200, 202, 207, 211, 212, 214, 215, 218, 223, 227, 233, 235, 241, 244, 246, 247, 249, 252, 255, 257, 258, 263, 265, 269, 281, 283, 285, 286, 287, 291, 293, 294, 298, 302, 303, 304, 307, 308, 311, 319, 322, 324, 327, 338, 344, 347, 351, 354, 355, 356, 359, 362, 367, 368, 369, 377, 380, 383, 387, 389, 390, 394, 398, 401, 402, 404, 407, 410, 411, 413, 416, 417, 422, 423, 424, 437, 439, 443, 446, 447, 450, 454, 458, 467, 468, 469, 473, 475, 480, 482, 483, 484, 489, 493, 495, 497, 500, 509, 511, 514, 515, 518, 524, 528, 530, 533, 534, 538, 547, 549, 552, 555, 558, 563, 564, 572, 574, 578, 580, 590, 591, 593, 597, 601, 602, 603, 604, 608, 611, 615, 619, 620, 622, 626, 627, 629, 632, 635, 637, 638, 645, 647, 648, 650, 651, 653, 655, 659, 662, 663, 666, 667, 668, 670, 671, 675, 678, 679, 683, 684, 687, 691, 692, 694, 698, 706, 707, 709, 712, 720, 722, 724, 731, 734, 735, 737, 741, 743, 744, 746, 749, 752, 753, 754, 755, 759, 762, 766, 767, 770, 771, 773, 775, 783, 785, 787, 792, 794, 797, 802, 806, 807, 809, 812, 813, 814, 818, 823, 825, 836, 840, 842, 844, 848, 849, 851, 853, 854, 867, 868, 870, 872, 873, 878, 887, 888, 889, 893, 896, 899, 902, 903, 904, 907, 908, 911, 915, 922, 923, 924, 926, 927, 932, 937, 938, 939, 941, 942, 943, 944, 945, 947, 948, 953, 954, 958, 961, 964, 967, 968, 974, 975, 977, 978, 980, 983, 987, 988, 993, 994, 998, 999, 1002, 1003, 1006, 1009, 1014, 1016} (totally 317 bases)

Sierpinski k=2: {365, 383, 461, 512, 542, 647, 773, 801, 829, 836, 859, 878, 908, 914, 917, 947, 1004, 1006} (totally 18 bases)

Sierpinski k=3: {83, 123, 191, 303, 323, 333, 363, 403, 453, 461, 483, 499, 511, 523, 683, 711, 718, 723, 743, 751, 779, 783, 807, 813, 823, 827, 847, 912, 979, 997, 1003, 1005, 1011, 1023} (totally 34 bases)

Sierpinski k=4: {32, 53, 155, 174, 204, 212, 230, 266, 281, 332, 334, 335, 356, 371, 386, 395, 467, 512, 593, 611, 661, 731, 767, 776, 803, 848, 851, 861, 875, 926, 941, 971, 981, 1024} (totally 34 bases)

Sierpinski k=5: {137, 183, 187, 193, 243, 305, 308, 415, 439, 512, 533, 545, 663, 667, 675, 761, 795, 809, 824, 835, 898, 941, 955, 997, 999, 1006} (totally 26 bases)

Sierpinski k=6: {212, 239, 509, 579, 625, 729, 774, 799, 894, 993, 999} (totally 11 bases)

[sweety439]

Originally Posted by sweety439

All Sierpinski/Riesel bases listed "NA" have CK > 5M (i.e. 5M is the lower bound for these Sierpinski/Riesel bases)

upper bounds for these Sierpinski/Riesel bases <= 1024:

S66: 21314443 (if not exactly this number, then must be == 4 mod 5 or == 12 mod 13)
S120: 374876369 (if not exactly this number, then must be == 6 mod 7 or == 16 mod 17)
S156: 18406311208 (if not exactly this number, then must be == 4 mod 5 or == 30 mod 31)
S210: 147840103 (if not exactly this number, then must be == 10 mod 11 or == 18 mod 19)
S280: 82035074042274 (if not exactly this number, then must be == 2 mod 3 or == 30 mod 31)
S330: 16636723 (if not exactly this number, then must be == 6 mod 7 or == 46 mod 47)
S358: 27478218 (if not exactly this number, then must be == 2 mod 3 or == 6 mod 7 or == 16 mod 17)
S456: 14836963 (if not exactly this number, then must be == 4 mod 5 or == 6 mod 7 or == 12 mod 13)
S462: 6880642 (if not exactly this number, then must be == 460 mod 461)
S546: 45119296 (if not exactly this number, then must be == 4 mod 5 or == 108 mod 109)
S630: 24015859 (if not exactly this number, then must be == 16 mod 17 or == 36 mod 37)
S690: 395800653 (if not exactly this number, then must be == 12 mod 13 or == 52 mod 53)
S726: 10923176 (if not exactly this number, then must be == 4 mod 5 or == 28 mod 29)
S756: 67836285 (if not exactly this number, then must be == 4 mod 5 or == 150 mod 151)
S876: 258073912 (if not exactly this number, then must be == 4 mod 5 or == 6 mod 7)
S910: 4989054506113 (if not exactly this number, then must be == 2 mod 3 or == 100 mod 101)
S960: 197938246042 (if not exactly this number, then must be == 6 mod 7 or == 136 mod 137)
S966: 924230558 (if not exactly this number, then must be == 4 mod 5 or == 192 mod 193)
S1008: 12730554 (if not exactly this number, then must be == 18 mod 19 or == 52 mod 53)
S1020: 95696289 (if not exactly this number, then must be == 1018 mod 1019)

R66: 101954772 (if not exactly this number, then must be == 1 mod 5 or == 1 mod 13)
R120: 166616308 (if not exactly this number, then must be == 1 mod 7 or == 1 mod 17)
R156: 2113322677 (if not exactly this number, then must be == 1 mod 5 or == 1 mod 31)
R180: 7674582 (if not exactly this number, then must be == 1 mod 179)
R210: 80176412 (if not exactly this number, then must be == 1 mod 11 or == 1 mod 19)
R280: 513613045571841 (if not exactly this number, then must be == 1 mod 3 or == 1 mod 31)
R330: 16527822 (if not exactly this number, then must be == 1 mod 7 or == 1 mod 47)
R358: 27606383 (if not exactly this number, then must be == 1 mod 3 or == 1 mod 7 or == 1 mod 17)
R420: 6548233 (if not exactly this number, then must be == 1 mod 419)
R456: 76303920 (if not exactly this number, then must be == 1 mod 5 or == 1 mod 7 or == 1 mod 13)
R546: 11732602 (if not exactly this number, then must be == 1 mod 5 or == 1 mod 109)
R570: 12511182 (if not exactly this number, then must be == 1 mod 569)
R630: 24412760 (if not exactly this number, then must be == 1 mod 17 or == 1 mod 37)
R690: 42053568 (if not exactly this number, then must be == 1 mod 13 or == 1 mod 53)
R726: 12751579 (if not exactly this number, then must be == 1 mod 5 or == 1 mod 29)
[R728 has CK=212722, didn't found this CK since it has a large prime (105997) in the covering set]
R756: 54604682 (if not exactly this number, then must be == 1 mod 5 or == 1 mod 151)
R876: 51768432 (if not exactly this number, then must be == 1 mod 5 or == 1 mod 7)
R910: 5005381602981 (if not exactly this number, then must be == 1 mod 3 or == 1 mod 101)
R946: 2156122023 (if not exactly this number, then must be == 1 mod 3 or == 1 mod 5 or == 1 mod 7)
R960: 61681833328 (if not exactly this number, then must be == 1 mod 7 or == 1 mod 137)
R966: 699327630 (if not exactly this number, then must be == 1 mod 5 or == 1 mod 193)
R1020: 94655888 (if not exactly this number, then must be == 1 mod 1019)

Since 280, 358, 910, and 946 are == 1 mod 3, the covering sets of these two bases (in both sides) may have the prime 3 and the period is divisible by 3, thus may have much small CK's than the CK's listed above, and the bases <= 1024 having the largest CK may be 960, than 156 and 966.