[QUOTE=sweety439;550196]Also, for Sierpinski k=2500 (while n == 0 mod 4 have algebra factors)

These bases have (odd n has factor of 3 and n == 2 mod 4 has factor of 17):

38, 47, 89, 98, 140, 149, 191, 200, 242, 251, 293, 302, 344, 353, 395, 404, 446, 455, 497, 506, 548, 557, 599, 608, 650, 659, 701, 710, 752, 761, 803, 812, 854, 863, 905, 914, 956, 965, 1007, 1016, 1058, 1067, 1109, 1118, 1160, 1169, 1211, 1220, 1262, 1271, 1313, 1322, 1364, 1373, 1415, 1424, 1466, 1475, 1517, 1526, 1568, 1577, 1619, 1628, 1670, 1679, 1721, 1730, 1772, 1781, 1823, 1832, 1874, 1883, 1925, 1934, 1976, 1985, 2027, 2036, 2078, 2087, 2129, 2138, 2180, 2189, 2231, 2240, 2282, 2291, 2333, 2342, 2384, 2393, 2435, 2444, 2486, 2495, 2537, 2546, 2588, 2597, 2639, 2648, 2690, 2699, 2741, 2750, 2792, 2801, 2843, 2852, 2894, 2903, 2945, 2954, 2996, 3005, 3047, 3056, 3098, 3107, 3149, 3158, 3200, 3209, 3251, 3260, 3302, 3311, 3353, 3362, 3404, 3413, 3455, 3464, 3506, 3515, 3557, 3566, 3608, 3617, 3659, 3668, 3710, 3719, 3761, 3770, 3812, 3821, 3863, 3872, 3914, 3923, 3965, 3974, 4016, 4025, 4067, 4076, 4118, 4127, 4169, 4178, 4220, 4229, 4271, 4280, 4322, 4331, 4373, 4382, 4424, 4433, 4475, 4484, 4526, 4535, 4577, 4586, 4628, 4637, 4679, 4688, 4730, 4739, 4781, 4790, 4832, 4841, 4883, 4892, 4934, 4943, 4985, 4994, 5036, 5045, 5087, 5096, 5138, 5147, 5189, 5198, 5240, 5249, 5291, 5300, 5342, 5351, 5393, 5402, 5444, 5453, 5495, 5504, 5546, 5555, 5597, 5606, 5648, 5657, 5699, 5708, 5750, 5759, 5801, 5810, 5852, 5861, 5903, 5912, 5954, 5963, 6005, 6014, 6056, 6065, 6107, 6116, 6158, 6167, 6209, 6218, 6260, 6269, 6311, 6320, 6362, 6371, 6413, 6422, 6464, 6473, 6515, 6524, 6566, 6575, 6617, 6626, 6668, 6677, 6719, 6728, 6770, 6779, 6821, 6830, 6872, 6881, 6923, 6932, 6974, 6983, 7025, 7034, 7076, 7085, 7127, 7136, 7178, 7187, 7229, 7238, 7280, 7289, 7331, 7340, 7382, 7391, 7433, 7442, 7484, 7493, 7535, 7544, 7586, 7595, 7637, 7646, 7688, 7697, 7739, 7748, 7790, 7799, 7841, 7850, 7892, 7901, 7943, 7952, 7994, 8003, 8045, 8054, 8096, 8105, 8147, 8156, 8198, 8207, 8249, 8258, 8300, 8309, 8351, 8360, 8402, 8411, 8453, 8462, 8504, 8513, 8555, 8564, 8606, 8615, 8657, 8666, 8708, 8717, 8759, 8768, 8810, 8819, 8861, 8870, 8912, 8921, 8963, 8972, 9014, 9023, 9065, 9074, 9116, 9125, 9167, 9176, 9218, 9227, 9269, 9278, 9320, 9329, 9371, 9380, 9422, 9431, 9473, 9482, 9524, 9533, 9575, 9584, 9626, 9635, 9677, 9686, 9728, 9737, 9779, 9788, 9830, 9839, 9881, 9890, 9932, 9941, 9983, 9992, ...

These bases have (odd n has factor of 7 and n == 2 mod 4 has factor of 17):

13, 55, 132, 174, 251, 293, 370, 412, 489, 531, 608, 650, 727, 769, 846, 888, 965, 1007, 1084, 1126, 1203, 1245, 1322, 1364, 1441, 1483, 1560, 1602, 1679, 1721, 1798, 1840, 1917, 1959, 2036, 2078, 2155, 2197, 2274, 2316, 2393, 2435, 2512, 2554, 2631, 2673, 2750, 2792, 2869, 2911, 2988, 3030, 3107, 3149, 3226, 3268, 3345, 3387, 3464, 3506, 3583, 3625, 3702, 3744, 3821, 3863, 3940, 3982, 4059, 4101, 4178, 4220, 4297, 4339, 4416, 4458, 4535, 4577, 4654, 4696, 4773, 4815, 4892, 4934, 5011, 5053, 5130, 5172, 5249, 5291, 5368, 5410, 5487, 5529, 5606, 5648, 5725, 5767, 5844, 5886, 5963, 6005, 6082, 6124, 6201, 6243, 6320, 6362, 6439, 6481, 6558, 6600, 6677, 6719, 6796, 6838, 6915, 6957, 7034, 7076, 7153, 7195, 7272, 7314, 7391, 7433, 7510, 7552, 7629, 7671, 7748, 7790, 7867, 7909, 7986, 8028, 8105, 8147, 8224, 8266, 8343, 8385, 8462, 8504, 8581, 8623, 8700, 8742, 8819, 8861, 8938, 8980, 9057, 9099, 9176, 9218, 9295, 9337, 9414, 9456, 9533, 9575, 9652, 9694, 9771, 9813, 9890, 9932, ...[/QUOTE]

Such situation also exists for Sierpinski k=324:

These bases have (odd n has factor of 19 and n == 2 mod 4 has factor of 17):

132, 208, 455, 531, 778, 854, 1101, 1177, 1424, 1500, 1747, 1823, 2070, 2146, 2393, 2469, 2716, 2792, 3039, 3115, 3362, 3438, 3685, 3761, 4008, 4084, 4331, 4407, 4654, 4730, 4977, 5053, 5300, 5376, 5623, 5699, 5946, 6022, 6269, 6345, 6592, 6668, 6915, 6991, 7238, 7314, 7561, 7637, 7884, 7960, 8207, 8283, 8530, 8606, 8853, 8929, 9176, 9252, 9499, 9575, 9822, 9898, ...

[FONT="Arial Black"][COLOR="Red"]Stop quoting a previous post in full for no reason.[/COLOR][/FONT]

[QUOTE=sweety439;549958]Checking whether a k-value makes a full covering set with algebraic factors not always very easy. The way I do it is to look for patterns in the factors of the various n-values for specific k-values. If there are algebraic factors, it's most common for them to be in a pattern of f*(f+2), i.e.:
11*13
179*181
etc.

In other cases there may be a consistent steady increase in the differences of their factors, which is especially tricky to find but indicates the existence of algebraic factors.

e.g. for the case R15 k=47
n-value : factors
1 : 2^5 · 11
2 : 17 · 311
3 : 2^4 · 4957
4 : 31 · 38377
6 : 11 · 43 · 565919
8 : 199 · 1627 · 186019
10 : 17 · 61 · 13067776451
12 : 37 · 82406457849451
20 : 15061 · 236863181 · 2190492030407

Analysis:
For n=1 & 3 (and all odd n), all values are divisible by 2 so we only consider even n's.
For n=4, the two prime factors does not close.
For n=6 & 10, multiplying the 2 lower prime factors together does not come close to the higher prime factor so little chance of algebraic factors.
For n=12, the large lowest prime factor that bears no relation to the other prime factor means that there is unlikely to be a pattern to the occurrences of large prime factors so there must be a prime at some point.

R33 k=257:
n-value : factors
1 : 5 · 53
2 : 2 · 4373
3 : 397 · 727
4 : 2^2 · 2381107
5 : 5^3 · 7 · 359207
7 : 11027 · 31040117
15 : 13337 · 706661 · 51076716238627
19 : 38231 · 14932493857679888742000509
For n=15 & 19 same explanation as R15 k=47

R36 k=1555:
n-value : factors
1 : 11 · 727
2 : 31 · 37 · 251
3 : 67 · 154691
4 : 37 · 127 · 271 · 293
7 : 4943 · 3521755879
9 : 59 · 382386761790283
For n=7 & 9 same explanation as R15 k=47


The prime factors for n=12, n=15, and n=7 respectively make it clear to me that these k-values should all yield primes at some point so you are correct to include them as remaining.

The higher-math folks may be able to chime in and answer why there are an abnormally large # of k's that are perfect squares that end up remaining even though they don't have known algebraic factors for most bases. IMHO, it's because there ARE algebraic factors for a subset of the universe of n-values on them but not for all of the n-values. Hence they are frequently lower weight than the other k's but NOT zero weight and so should eventually yield a prime.[/QUOTE]

These k's cannot have any algebra factors since there is no n such that 47*15^n (or 257*33^n, 1555*36^n) is perfect power, however, we can check whether a k-value makes a full covering set with algebraic factors, like these examples:

R58 k=400:

since 400 is square, all even n have algebra factors, and we only want to know whether it has a covering set of primes for all odd n, if so, then this k makes a full covering set with algebraic factors and be excluded from the conjecture; if not, then this k does not make a full covering set with algebraic factors and be included from the conjecture.

n-value : factors
1 : 11 · 37
3 : 7^2 · 27943
5 : 3 · 23 · 66753803
7 : 61 · 254010257987
9 : 7 · 184913 · 40269069377
11 : 3^2 · 11 · 1627649 · 1088170916957
13 : 947 · 894431 · 696389525163251
15 : 7 · 71 · 7193 · 555058285756249213567
25 : 1171 · 10639 · 450563 · 211297508330411 · 720737916824065571
43 : 9901 · (a 73-digit prime)
79 : 257 · 4990187 · 365846287 · (a 123-digit prime)

and it does not appear to be any covering set of primes, so there must be a prime at some point.

R93 k=125:

since 125 is cube, all n divisible by 3 have algebra factors, and we only want to know whether it has a covering set of primes for all n not divisible by 3, if so, then this k makes a full covering set with algebraic factors and be excluded from the conjecture; if not, then this k does not make a full covering set with algebraic factors and be included from the conjecture.

n-value : factors
1 : 2 · 1453
2 : 11 · 24571
4 : 271 · 8626061
5 : 2 · 4201 · 4861 · 5323
7 : 2^4 · 11 · 10683609208231
8 : 109 · 52981 · 30280773239
10 : 1259 · 38851 · 30920860779409
14 : 131381 · 748966512379 · 1149784204819
16 : 431 · 3881 · 834208399 · 701264295413691479
20 : 79 · 394653205936499 · 2347827938794175966096911
34 : 469429 · (a 63-digit prime)
40 : 271 · (a 78-digit prime)
64 : 751 · 766651 · (a 119-digit composite with no known prime factor)
70 : 191 · (a 138-digit prime)
74 : 179 · (a 145-digit prime)
86 : (a 171-digit composite with no known prime factor)

and it does not appear to be any covering set of primes, so there must be a prime at some point.

R96 k=1681:

since 1681 is square, all even n have algebra factors, and we only want to know whether it has a covering set of primes for all odd n, if so, then this k makes a full covering set with algebraic factors and be excluded from the conjecture; if not, then this k does not make a full covering set with algebraic factors and be included from the conjecture.

n-value : factors
1 : 5^2 · 1291
3 : 43 · 53 · 130517
5 : 23 · 311 · 383235427
7 : 29 · 23159 · 37616513449
9 : 11117 · 20943593541080351
13 : 67273 · 309220057 · 950638256285203
19 : 164429 · 94139680772968423679537510579981183
25 : 34584287 · 4026877213339 · 87002417657719496636646465818327
33 : 28051 · 77261 · (a 59-digit prime)
37 : (a 28-digit prime) · (a 49-digit prime)
55 : 53 · 191 · (a 108-digit prime)
59 : 313 · 11953 · (a 113-digit prime)

and it does not appear to be any covering set of primes, so there must be a prime at some point.

Update newest status of [URL="https://docs.google.com/document/d/e/2PACX-1vQ96fx2qG6Gqfp56Z66Yqy3bPQhMZ0SI_7iGmmMO4oXiwusyjRcihnROLwg2uPHHv4mo7yKpahXG1PL/pub"]Sierpinski problems[/URL]

(S53 and S122)

[QUOTE=sweety439;546593]In Riesel conjectures, if k=m^2 and m and b satisfy at least one of these conditions, then this k should be excluded from the Riesel base b problem, since it has algebraic factors for even n and it has a single prime factor for odd n, thus proven composite by partial algebraic factors

list all such mod <= 2048

[CODE]
m b
= 2 or 3 mod 5 = 4 mod 5
= 5 or 8 mod 13 = 12 mod 13
= 3 or 5 mod 8 = 9 mod 16
= 4 or 13 mod 17 = 16 mod 17
= 12 or 17 mod 29 = 28 mod 29
= 7 or 9 mod 16 = 17 mod 32
= 6 or 31 mod 37 = 36 mod 37
= 9 or 32 mod 41 = 40 mod 41
= 23 or 30 mod 53 = 52 mod 53
= 11 or 50 mod 61 = 60 mod 61
= 15 or 17 mod 32 = 33 mod 64
= 27 or 46 mod 73 = 72 mod 73
= 34 or 55 mod 89 = 88 mod 89
= 22 or 75 mod 97 = 96 mod 97
= 10 or 91 mod 101 = 100 mod 101
= 33 or 76 mod 109 = 108 mod 109
= 15 or 98 mod 113 = 112 mod 113
= 31 or 33 mod 64 = 65 mod 128
= 37 or 100 mod 137 = 136 mod 137
= 44 or 105 mod 149 = 148 mod 149
= 28 or 129 mod 157 = 156 mod 157
= 80 or 93 mod 173 = 172 mod 173
= 19 or 162 mod 181 = 180 mod 181
= 81 or 112 mod 193 = 192 mod 193
= 14 or 183 mod 197 = 196 mod 197
= 107 or 122 mod 229 = 228 mod 229
= 89 or 144 mod 233 = 232 mod 233
= 64 or 177 mod 241 = 240 mod 241
= 63 or 65 mod 128 = 129 mod 256
= 16 or 241 mod 257 = 256 mod 257
= 82 or 187 mod 269 = 268 mod 269
= 60 or 217 mod 277 = 276 mod 277
= 53 or 228 mod 281 = 280 mod 281
= 138 or 155 mod 293 = 292 mod 293
= 25 or 288 mod 313 = 312 mod 313
= 114 or 203 mod 317 = 316 mod 317
= 148 or 189 mod 337 = 336 mod 337
= 136 or 213 mod 349 = 348 mod 349
= 42 or 311 mod 353 = 352 mod 353
= 104 or 269 mod 373 = 372 mod 373
= 115 or 274 mod 389 = 388 mod 389
= 63 or 334 mod 397 = 396 mod 397
= 20 or 381 mod 401 = 400 mod 401
= 143 or 266 mod 409 = 408 mod 409
= 29 or 392 mod 421 = 420 mod 421
= 179 or 254 mod 433 = 432 mod 433
= 67 or 382 mod 449 = 448 mod 449
= 109 or 348 mod 457 = 456 mod 457
= 48 or 413 mod 461 = 460 mod 461
= 208 or 301 mod 509 = 508 mod 509
= 127 or 129 mod 256 = 257 mod 512
= 235 or 286 mod 521 = 520 mod 521
= 52 or 489 mod 541 = 540 mod 541
= 118 or 439 mod 557 = 556 mod 557
= 86 or 483 mod 569 = 568 mod 569
= 24 or 553 mod 577 = 576 mod 577
= 77 or 516 mod 593 = 592 mod 593
= 125 or 476 mod 601 = 600 mod 601
= 35 or 578 mod 613 = 612 mod 613
= 194 or 423 mod 617 = 616 mod 617
= 154 or 487 mod 641 = 640 mod 641
= 149 or 504 mod 653 = 652 mod 653
= 106 or 555 mod 661 = 660 mod 661
= 58 or 615 mod 673 = 672 mod 673
= 26 or 651 mod 677 = 676 mod 677
= 135 or 566 mod 701 = 700 mod 701
= 96 or 613 mod 709 = 708 mod 709
= 353 or 380 mod 733 = 732 mod 733
= 87 or 670 mod 757 = 756 mod 757
= 39 or 722 mod 761 = 760 mod 761
= 62 or 707 mod 769 = 768 mod 769
= 317 or 456 mod 773 = 772 mod 773
= 215 or 582 mod 797 = 796 mod 797
= 318 or 491 mod 809 = 808 mod 809
= 295 or 526 mod 821 = 820 mod 821
= 246 or 583 mod 829 = 828 mod 829
= 333 or 520 mod 853 = 852 mod 853
= 207 or 650 mod 857 = 856 mod 857
= 151 or 726 mod 877 = 876 mod 877
= 387 or 494 mod 881 = 880 mod 881
= 324 or 605 mod 929 = 928 mod 929
= 196 or 741 mod 937 = 936 mod 937
= 97 or 844 mod 941 = 940 mod 941
= 442 or 511 mod 953 = 952 mod 953
= 252 or 725 mod 977 = 976 mod 977
= 161 or 836 mod 997 = 996 mod 997
= 469 or 540 mod 1009 = 1008 mod 1009
= 45 or 968 mod 1013 = 1012 mod 1013
= 374 or 647 mod 1021 = 1020 mod 1021
= 255 or 257 mod 512 = 513 mod 1024
= 355 or 678 mod 1033 = 1032 mod 1033
= 426 or 623 mod 1049 = 1048 mod 1049
= 103 or 958 mod 1061 = 1060 mod 1061
= 249 or 820 mod 1069 = 1068 mod 1069
= 530 or 563 mod 1093 = 1092 mod 1093
= 341 or 756 mod 1097 = 1096 mod 1097
= 354 or 755 mod 1109 = 1108 mod 1109
= 214 or 903 mod 1117 = 1116 mod 1117
= 168 or 961 mod 1129 = 1128 mod 1129
= 140 or 1013 mod 1153 = 1152 mod 1153
= 243 or 938 mod 1181 = 1180 mod 1181
= 186 or 1007 mod 1193 = 1192 mod 1193
= 49 or 1152 mod 1201 = 1200 mod 1201
= 495 or 718 mod 1213 = 1212 mod 1213
= 78 or 1139 mod 1217 = 1216 mod 1217
= 597 or 632 mod 1229 = 1228 mod 1229
= 546 or 691 mod 1237 = 1236 mod 1237
= 585 or 664 mod 1249 = 1248 mod 1249
= 113 or 1164 mod 1277 = 1276 mod 1277
= 479 or 810 mod 1289 = 1288 mod 1289
= 36 or 1261 mod 1297 = 1296 mod 1297
= 51 or 1250 mod 1301 = 1300 mod 1301
= 257 or 1064 mod 1321 = 1320 mod 1321
= 614 or 747 mod 1361 = 1360 mod 1361
= 668 or 705 mod 1373 = 1372 mod 1373
= 366 or 1015 mod 1381 = 1380 mod 1381
= 452 or 957 mod 1409 = 1408 mod 1409
= 620 or 809 mod 1429 = 1428 mod 1429
= 542 or 891 mod 1433 = 1432 mod 1433
= 497 or 956 mod 1453 = 1452 mod 1453
= 465 or 1016 mod 1481 = 1480 mod 1481
= 225 or 1264 mod 1489 = 1488 mod 1489
= 432 or 1061 mod 1493 = 1492 mod 1493
= 88 or 1461 mod 1549 = 1548 mod 1549
= 339 or 1214 mod 1553 = 1552 mod 1553
= 610 or 987 mod 1597 = 1596 mod 1597
= 40 or 1561 mod 1601 = 1600 mod 1601
= 523 or 1086 mod 1609 = 1608 mod 1609
= 127 or 1486 mod 1613 = 1612 mod 1613
= 166 or 1455 mod 1621 = 1620 mod 1621
= 316 or 1321 mod 1637 = 1636 mod 1637
= 783 or 874 mod 1657 = 1656 mod 1657
= 220 or 1449 mod 1669 = 1668 mod 1669
= 92 or 1601 mod 1693 = 1692 mod 1693
= 414 or 1283 mod 1697 = 1696 mod 1697
= 390 or 1319 mod 1709 = 1708 mod 1709
= 473 or 1248 mod 1721 = 1720 mod 1721
= 410 or 1323 mod 1733 = 1732 mod 1733
= 59 or 1682 mod 1741 = 1740 mod 1741
= 713 or 1040 mod 1753 = 1752 mod 1753
= 775 or 1002 mod 1777 = 1776 mod 1777
= 724 or 1065 mod 1789 = 1788 mod 1789
= 824 or 977 mod 1801 = 1800 mod 1801
= 61 or 1800 mod 1861 = 1860 mod 1861
= 737 or 1136 mod 1873 = 1872 mod 1873
= 137 or 1740 mod 1877 = 1876 mod 1877
= 331 or 1558 mod 1889 = 1888 mod 1889
= 218 or 1683 mod 1901 = 1900 mod 1901
= 712 or 1201 mod 1913 = 1912 mod 1913
= 598 or 1335 mod 1933 = 1932 mod 1933
= 589 or 1360 mod 1949 = 1948 mod 1949
= 259 or 1714 mod 1973 = 1972 mod 1973
= 834 or 1159 mod 1993 = 1992 mod 1993
= 412 or 1585 mod 1997 = 1996 mod 1997
= 229 or 1788 mod 2017 = 2016 mod 2017
= 992 or 1037 mod 2029 = 2028 mod 2029
= 511 or 513 mod 1024 = 1025 mod 2048
[/CODE][/QUOTE]

If

k is square and k == -1 mod p and b == -1 mod p for some odd prime p (this situation only exists for p == 1 mod 4)

or

k is square and k == 2^(r-1)+1 mod 2^r and b == 2^(r-1)+1 mod 2^r for some r >= 2 (this situation only exists for r >= 4)

Then this k proven composite by partial algebraic factors (has algebraic factors (difference of two squares) for even n and divisible by a prime (p or 2, respectively) for odd n)

[QUOTE=sweety439;550297]If

k is square and k == -1 mod p and b == -1 mod p for some odd prime p (this situation only exists for p == 1 mod 4)

or

k is square and k == 2^(r-1)+1 mod 2^r and b == 2^(r-1)+1 mod 2^r for some r >= 2 (this situation only exists for r >= 4)

Then this k proven composite by partial algebraic factors (has algebraic factors (difference of two squares) for even n and divisible by a prime (p or 2, respectively) for odd n)[/QUOTE]

Also,

if

k*b is square and k*b == -1 mod p and b == -1 mod p for some odd prime p (this situation only exists for p == 1 mod 4)

or

k*b is square and k*b == 2^(r-1)+1 mod 2^r and b == 2^(r-1)+1 mod 2^r for some r >= 2 (this situation only exists for r >= 4)

Then this k proven composite by partial algebraic factors (divisible by a prime (p or 2, respectively) for even n and has algebraic factors (difference of two squares) for odd n)

All such examples for bases <= 64 with k below the CK: (square bases are not counted, since they already have full algebra factors for these square k)

R12, k = 3*m^2 and m = = 3 or 10 mod 13 (k = 27 and 300 below the CK)

R28, k = 7*m^2 and m = = 5 or 24 mod 29 (k = 175 below the CK)

R33, k = 33*m^2 and m = = 4 or 13 mod 17 (k = 528 below the CK)

R40, k = 10*m^2 and m = = 18 or 23 mod 41 (k = 3240 and 5290 below the CK)

R52, k = 13*m^2 and m = = 7 or 46 mod 53 (k = 637 below the CK)

R54, k = 6*m^2 and m = = 1 or 4 mod 5 (k = 6 below the CK)

R60, k = 15*m^2 and m = = 22 or 39 mod 61 (k = 7260 below the CK)

The smallest Riesel base such that the second situation (k*b is square and k*b == 2^(r-1)+1 mod 2^r and b == 2^(r-1)+1 mod 2^r for some r >= 2 (this situation only exists for r >= 4)) exists for k below the CK is 153 (square bases are not counted, since they already have full algebra factors for these square k), such k is 17

[QUOTE=sweety439;550208]These k's cannot have any algebra factors since there is no n such that 47*15^n (or 257*33^n, 1555*36^n) is perfect power, however, we can check whether a k-value makes a full covering set with algebraic factors, like these examples:

R58 k=400:

since 400 is square, all even n have algebra factors, and we only want to know whether it has a covering set of primes for all odd n, if so, then this k makes a full covering set with algebraic factors and be excluded from the conjecture; if not, then this k does not make a full covering set with algebraic factors and be included from the conjecture.

n-value : factors
1 : 11 · 37
3 : 7^2 · 27943
5 : 3 · 23 · 66753803
7 : 61 · 254010257987
9 : 7 · 184913 · 40269069377
11 : 3^2 · 11 · 1627649 · 1088170916957
13 : 947 · 894431 · 696389525163251
15 : 7 · 71 · 7193 · 555058285756249213567
25 : 1171 · 10639 · 450563 · 211297508330411 · 720737916824065571
43 : 9901 · (a 73-digit prime)
79 : 257 · 4990187 · 365846287 · (a 123-digit prime)

and it does not appear to be any covering set of primes, so there must be a prime at some point.

R93 k=125:

since 125 is cube, all n divisible by 3 have algebra factors, and we only want to know whether it has a covering set of primes for all n not divisible by 3, if so, then this k makes a full covering set with algebraic factors and be excluded from the conjecture; if not, then this k does not make a full covering set with algebraic factors and be included from the conjecture.

n-value : factors
1 : 2 · 1453
2 : 11 · 24571
4 : 271 · 8626061
5 : 2 · 4201 · 4861 · 5323
7 : 2^4 · 11 · 10683609208231
8 : 109 · 52981 · 30280773239
10 : 1259 · 38851 · 30920860779409
14 : 131381 · 748966512379 · 1149784204819
16 : 431 · 3881 · 834208399 · 701264295413691479
20 : 79 · 394653205936499 · 2347827938794175966096911
34 : 469429 · (a 63-digit prime)
40 : 271 · (a 78-digit prime)
64 : 751 · 766651 · (a 119-digit composite with no known prime factor)
70 : 191 · (a 138-digit prime)
74 : 179 · (a 145-digit prime)
86 : (a 171-digit composite with no known prime factor)

and it does not appear to be any covering set of primes, so there must be a prime at some point.

R96 k=1681:

since 1681 is square, all even n have algebra factors, and we only want to know whether it has a covering set of primes for all odd n, if so, then this k makes a full covering set with algebraic factors and be excluded from the conjecture; if not, then this k does not make a full covering set with algebraic factors and be included from the conjecture.

n-value : factors
1 : 5^2 · 1291
3 : 43 · 53 · 130517
5 : 23 · 311 · 383235427
7 : 29 · 23159 · 37616513449
9 : 11117 · 20943593541080351
13 : 67273 · 309220057 · 950638256285203
19 : 164429 · 94139680772968423679537510579981183
25 : 34584287 · 4026877213339 · 87002417657719496636646465818327
33 : 28051 · 77261 · (a 59-digit prime)
37 : (a 28-digit prime) · (a 49-digit prime)
55 : 53 · 191 · (a 108-digit prime)
59 : 313 · 11953 · (a 113-digit prime)

and it does not appear to be any covering set of primes, so there must be a prime at some point.[/QUOTE]

For Riesel side ((k*b^n-1)/gcd(k-1,b-1)), if k is r-th power with prime r, then all n divisible by r have algebra factors, and we only want to know whether it has a covering set of primes for all n not divisible by r, if so, then this k makes a full covering set with algebraic factors and be excluded from the conjecture; if not, then this k does not make a full covering set with algebraic factors and be included from the conjecture, and there must be a prime at some point.

For Sierpinski side ((k*b^n+1)/gcd(k+1,b-1)), if k is r-th power with odd prime r, then all n divisible by r have algebra factors, and we only want to know whether it has a covering set of primes for all n not divisible by r, if so, then this k makes a full covering set with algebraic factors and be excluded from the conjecture; if not, then this k does not make a full covering set with algebraic factors and be included from the conjecture, and there must be a prime at some point.

For Sierpinski side ((k*b^n+1)/gcd(k+1,b-1)), if k is of the form 4*m^4, then all n divisible by 4 have algebra factors, and we only want to know whether it has a covering set of primes for all n not divisible by 4, if so, then this k makes a full covering set with algebraic factors and be excluded from the conjecture; if not, then this k does not make a full covering set with algebraic factors and be included from the conjecture, and there must be a prime at some point.

[QUOTE=sweety439;550355]For Riesel side ((k*b^n-1)/gcd(k-1,b-1)), if k is r-th power with prime r, then all n divisible by r have algebra factors, and we only want to know whether it has a covering set of primes for all n not divisible by r

For Sierpinski side ((k*b^n+1)/gcd(k+1,b-1)), if k is r-th power with odd prime r, then all n divisible by r have algebra factors, and we only want to know whether it has a covering set of primes for all n not divisible by r

For Sierpinski side ((k*b^n+1)/gcd(k+1,b-1)), if k is of the form 4*m^4, then all n divisible by 4 have algebra factors, and we only want to know whether it has a covering set of primes for all n not divisible by 4[/QUOTE]

Also,

For Riesel side ((k*b^n-1)/gcd(k-1,b-1)), if k*b^s is r-th power with prime r, then all n == s mod r have algebra factors, and we only want to know whether it has a covering set of primes for all n not == s mod r, if so, then this k makes a full covering set with algebraic factors and be excluded from the conjecture; if not, then this k does not make a full covering set with algebraic factors and be included from the conjecture, and there must be a prime at some point.

For Sierpinski side ((k*b^n+1)/gcd(k+1,b-1)), if k*b^s is r-th power with odd prime r, then all n == s mod r have algebra factors, and we only want to know whether it has a covering set of primes for all n not == s mod r, if so, then this k makes a full covering set with algebraic factors and be excluded from the conjecture; if not, then this k does not make a full covering set with algebraic factors and be included from the conjecture, and there must be a prime at some point.

For Sierpinski side ((k*b^n+1)/gcd(k+1,b-1)), if k*b^s is of the form 4*m^4, then all n == s mod 4 have algebra factors, and we only want to know whether it has a covering set of primes for all n not == s mod 4, if so, then this k makes a full covering set with algebraic factors and be excluded from the conjecture; if not, then this k does not make a full covering set with algebraic factors and be included from the conjecture, and there must be a prime at some point.

[QUOTE=sweety439;550355]For Riesel side ((k*b^n-1)/gcd(k-1,b-1)), if k is r-th power with prime r, then all n divisible by r have algebra factors, and we only want to know whether it has a covering set of primes for all n not divisible by r, if so, then this k makes a full covering set with algebraic factors and be excluded from the conjecture; if not, then this k does not make a full covering set with algebraic factors and be included from the conjecture, and there must be a prime at some point.

For Sierpinski side ((k*b^n+1)/gcd(k+1,b-1)), if k is r-th power with odd prime r, then all n divisible by r have algebra factors, and we only want to know whether it has a covering set of primes for all n not divisible by r, if so, then this k makes a full covering set with algebraic factors and be excluded from the conjecture; if not, then this k does not make a full covering set with algebraic factors and be included from the conjecture, and there must be a prime at some point.

For Sierpinski side ((k*b^n+1)/gcd(k+1,b-1)), if k is of the form 4*m^4, then all n divisible by 4 have algebra factors, and we only want to know whether it has a covering set of primes for all n not divisible by 4, if so, then this k makes a full covering set with algebraic factors and be excluded from the conjecture; if not, then this k does not make a full covering set with algebraic factors and be included from the conjecture, and there must be a prime at some point.[/QUOTE]

The only exception of "there must be a prime at some point" is the case in post [URL="https://mersenneforum.org/showpost.php?p=459405&postcount=265"]#265[/URL]

Why do you insist on quoting whole posts all the time?

[QUOTE=sweety439;550208]These k's cannot have any algebra factors since there is no n such that 47*15^n (or 257*33^n, 1555*36^n) is perfect power, however, we can check whether a k-value makes a full covering set with algebraic factors, like these examples:

R58 k=400:

since 400 is square, all even n have algebra factors, and we only want to know whether it has a covering set of primes for all odd n, if so, then this k makes a full covering set with algebraic factors and be excluded from the conjecture; if not, then this k does not make a full covering set with algebraic factors and be included from the conjecture.

n-value : factors
1 : 11 · 37
3 : 7^2 · 27943
5 : 3 · 23 · 66753803
7 : 61 · 254010257987
9 : 7 · 184913 · 40269069377
11 : 3^2 · 11 · 1627649 · 1088170916957
13 : 947 · 894431 · 696389525163251
15 : 7 · 71 · 7193 · 555058285756249213567
25 : 1171 · 10639 · 450563 · 211297508330411 · 720737916824065571
43 : 9901 · (a 73-digit prime)
79 : 257 · 4990187 · 365846287 · (a 123-digit prime)

and it does not appear to be any covering set of primes, so there must be a prime at some point.

R93 k=125:

since 125 is cube, all n divisible by 3 have algebra factors, and we only want to know whether it has a covering set of primes for all n not divisible by 3, if so, then this k makes a full covering set with algebraic factors and be excluded from the conjecture; if not, then this k does not make a full covering set with algebraic factors and be included from the conjecture.

n-value : factors
1 : 2 · 1453
2 : 11 · 24571
4 : 271 · 8626061
5 : 2 · 4201 · 4861 · 5323
7 : 2^4 · 11 · 10683609208231
8 : 109 · 52981 · 30280773239
10 : 1259 · 38851 · 30920860779409
14 : 131381 · 748966512379 · 1149784204819
16 : 431 · 3881 · 834208399 · 701264295413691479
20 : 79 · 394653205936499 · 2347827938794175966096911
34 : 469429 · (a 63-digit prime)
40 : 271 · (a 78-digit prime)
64 : 751 · 766651 · (a 119-digit composite with no known prime factor)
70 : 191 · (a 138-digit prime)
74 : 179 · (a 145-digit prime)
86 : (a 171-digit composite with no known prime factor)

and it does not appear to be any covering set of primes, so there must be a prime at some point.

R96 k=1681:

since 1681 is square, all even n have algebra factors, and we only want to know whether it has a covering set of primes for all odd n, if so, then this k makes a full covering set with algebraic factors and be excluded from the conjecture; if not, then this k does not make a full covering set with algebraic factors and be included from the conjecture.

n-value : factors
1 : 5^2 · 1291
3 : 43 · 53 · 130517
5 : 23 · 311 · 383235427
7 : 29 · 23159 · 37616513449
9 : 11117 · 20943593541080351
13 : 67273 · 309220057 · 950638256285203
19 : 164429 · 94139680772968423679537510579981183
25 : 34584287 · 4026877213339 · 87002417657719496636646465818327
33 : 28051 · 77261 · (a 59-digit prime)
37 : (a 28-digit prime) · (a 49-digit prime)
55 : 53 · 191 · (a 108-digit prime)
59 : 313 · 11953 · (a 113-digit prime)

and it does not appear to be any covering set of primes, so there must be a prime at some point.[/QUOTE]

Also these cases:

S15 k=343:

since 343 is cube, all n divisible by 3 have algebra factors, and we only want to know whether it has a covering set of primes for all n not divisible by 3, if so, then this k makes a full covering set with algebraic factors and be excluded from the conjecture; if not, then this k does not make a full covering set with algebraic factors and be included from the conjecture.

n-value : factors
1 : 31 · 83
2 : 2^2 · 11 · 877
4 : 2^2 · 809 · 2683
5 : 811 · 160583
7 : 11^2 · 242168453
11 : 31 · 101 · 25357 · 18684739
13 : 397 · 1281101 · 656261029
17 : 11 · 27479311 · 55900668804553
29 : 53 · 197741 · 209188613429183386499227445981
35 : 1337724923 · 18667724069720862256321575167267431
43 : 20943991 · 3055827403675875709696160949928034201885723243
61 : 23539 · (a 61-digit prime)

and it does not appear to be any covering set of primes, so there must be a prime at some point.

S61 k=324:

since 324 is of the form 4*m^4, all n divisible by 4 have algebra factors, and we only want to know whether it has a covering set of primes for all n not divisible by 4, if so, then this k makes a full covering set with algebraic factors and be excluded from the conjecture; if not, then this k does not make a full covering set with algebraic factors and be included from the conjecture.

n-value : factors
1 : 59 · 67
2 : 41 · 5881
3 : 13 · 1131413
5 : 5 · 7 · 1563709723
6 : 13 · 256809250661
7 : 23 · 1255679 · 7051433
13 : 191 · 7860337 · 27268229 · 256289843
14 : 1540873 · 1698953 · 244480646906833
31 : 1888149043321 · 441337391577139 · 1721840403480692512106884569347
34 : 10601 · 174221 · (a 54-digit prime)

and it does not appear to be any covering set of primes, so there must be a prime at some point.