A Sierpinski/Riesel-like problem

[sweety439]

All n must be >= 1.

All MOB k-values need an n>=1 prime (unless they have a covering set of primes, or make a full covering set with all or partial algebraic factors)

MOB k-values such that (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.

There are some MOB k-values such that (k+-1)/gcd(k+-1,b-1) (+ for Sierpinski, - for Riesel) is prime which do not have an easy prime for n>=1:

GFN's and half GFN's: (all with no known primes)

S2 k=65536
S3 k=3433683820292512484657849089281
S4 k=65536
S5 k=625
S6 k=1296
S7 k=2401
S8 k=256
S8 k=65536
S9 k=3433683820292512484657849089281
S10 k=100
S11 k=14641
S12 k=12
S13 k=815730721
S14 k=196
S15 k=225
S16 k=65536

Other k's:

S2 k=55816: first prime at n=14536
S2 k=90646: no prime with n<=6.6M
S2 k=101746: no prime with n<=6.6M
S3 k=621: first prime at n=20820
S4 k=176: first prime at n=228
S5 k=40: first prime at n=1036
S6 k=90546
S7 k=21: first prime at n=124
S9 k=1746: first prime at n=1320
S9 k=2007: first prime at n=3942
S10 k=640: first prime at n=120
S24 k=17496
S155 k=310
S333 k=1998
R2 k=74: first prime at n=2552
R2 k=674: first prime at n=11676
R2 k=1094: first prime at n=652
R4 k=19464: no prime with n<=3.3M
R6 k=103536: first prime at n=6474
R6 k=106056: first prime at n=3038
R10 k=450: first prime at n=11958
R10 k=16750: no prime with n<=200K
R11 k=308: first prime at n=444
R14 k=2954: no prime with n<=50K
R15 k=2940: first prime at n=13254
R15 k=8610: first prime at n=5178
R18 k=324: first prime at n=25665
R21 k=84: first prime at n=88
R23 k=230: first prime at n=6228
R27 k=594: first prime at n=36624
R40 k=520: no prime with n<=1K
R42 k=1764: first prime at n=1317
R48 k=384: no prime with n<=200K
R66 k=1056: no prime with n<=1K
R78 k=7800: no prime with n<=1K
R88 k=3168: first prime at n=205764
R96 k=9216: first prime at n=3341
R120 k=4320: no prime with n<=1K
R210 k=44100: first prime at n=19817
R306 k=93636: first prime at n=26405
R396 k=156816: no prime with n<=50K
R591 k=1182: first prime at n=1190
R954 k=1908: first prime at n=1476
R976 k=1952: first prime at n=1924
R1102 k=2204: first prime at n=52176
R1297 k=2594: first prime at n=19839
R1360 k=2720: first prime at n=74688

[sweety439]

Originally Posted by sweety439

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

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

There are other k's excluded from the Riesel/Sierpinski problems (Riesel is still much more such k's)

* R30 k=1369:

for even n let n=2*q; factors to: (37*30^q - 1) * (37*30^q + 1)

odd n: covering set 7, 13, 19

* R88 k=400:

for even n let n=2*q; factors to: (20*88^q - 1) * (20*88^q + 1)

odd n: covering set 3, 7, 13

* R95 k=324:

for even n let n=2*q; factors to: (18*95^q - 1) * (18*95^q + 1)

odd n: covering set 7, 13, 229

* R498 k=93025:

for even n let n=2*q; factors to: (305*498^q - 1) * (305*498^q + 1)

odd n: covering set 13, 67, 241

* R540 k=61009:

for even n let n=2*q; factors to: (247*540^q - 1) * (247*540^q + 1)

odd n: covering set 17, 1009

* S13 k=2500:

odd n: factor of 7

n = = 2 mod 4: factor of 17

n = = 0 mod 4: let n=4*q and let m=5*13^q; factors to: (2*m^2 + 2m + 1) * (2*m^2 - 2m + 1)

* S55 k=2500:

odd n: factor of 7

n = = 2 mod 4: factor of 17

n = = 0 mod 4: let n=4*q and let m=5*55^q; factors to: (2*m^2 + 2m + 1) * (2*m^2 - 2m + 1)

* S200 k=16:

odd n: factor of 3

n = = 0 mod 4: factor of 17

n = = 2 mod 4: let n=4*q+2 and let m = 20*200^q; factors to: (2*m^2 + 2m + 1) * (2*m^2 - 2m + 1)

* S1101 k=324:

odd n: factor of 19

n = = 2 mod 4: factor of 17

n = = 0 mod 4: let n=4*q and let m=3*1101^q; factors to: (2*m^2 + 2m + 1) * (2*m^2 - 2m + 1)

* S2070 k=324:

odd n: factor of 19

n = = 2 mod 4: factor of 17

n = = 0 mod 4: let n=4*q and let m=3*2070^q; factors to: (2*m^2 + 2m + 1) * (2*m^2 - 2m + 1)

* S225 k=114244:

for even n let k=4*q^4 and let m=q*15^(n/2); factors to: (2*m^2 + 2m + 1) * (2*m^2 - 2m + 1)

odd n: factor of 113

* R10 k=343:

n = = 1 mod 3: factor of 3

n = = 2 mod 3: factor of 37

n = = 0 mod 3: let n=3*q and let m=7*10^q; factors to: (m - 1) * (m^2 + m + 1)

* R957 k=64:

n = = 1 mod 3: factor of 73

n = = 2 mod 3: factor of 19

n = = 0 mod 3: let n=3*q and let m=4*957^q; factors to: (m - 1) * (m^2 + m + 1)

* S63 k=3511808:

n = = 1 mod 3: factor of 37

n = = 2 mod 3: factor of 109

n = = 0 mod 3: let n=3*q and let m=152*63^q; factors to: (m + 1) * (m^2 - m + 1)

* S63 k=27000000:

n = = 1 mod 3: factor of 37

n = = 2 mod 3: factor of 109

n = = 0 mod 3: let n=3*q and let m=300*63^q; factors to: (m + 1) * (m^2 - m + 1)

* R936 k=64:

n = = 0 mod 2: let n=2*q; factors to: (8*936^q - 1) * (8*936^q + 1)

n = = 0 mod 3: let n=3*q; factors to: (4*936^q - 1) * [16*936^(2q) + 4*936^q + 1]

n = = 1 mod 6: factor of 37

n = = 5 mod 6: factor of 109

[sweety439]

Originally Posted by sweety439

There are other k's excluded from the Riesel/Sierpinski problems (Riesel is still much more such k's)

* R30 k=1369:

for even n let n=2*q; factors to: (37*30^q - 1) * (37*30^q + 1)

odd n: covering set 7, 13, 19

* R88 k=400:

for even n let n=2*q; factors to: (20*88^q - 1) * (20*88^q + 1)

odd n: covering set 3, 7, 13

* R95 k=324:

for even n let n=2*q; factors to: (18*95^q - 1) * (18*95^q + 1)

odd n: covering set 7, 13, 229

* S55 k=2500:

odd n: factor of 7

n = = 2 mod 4: factor of 17

n = = 0 mod 4: let n=4q and let m=5*55^q; factors to: (2*m^2 + 2m + 1) * (2*m^2 - 2m + 1)

* S200 k=16:

odd n: factor of 3

n = = 0 mod 4: factor of 17

n = = 2 mod 4: let n = 4*q - 2 and let m = 20^q*10^(q-1); factors to: (2*m^2 + 2m + 1) * (2*m^2 - 2m + 1)

* S225 k=114244:

for even n let k=4*q^4 and let m=q*15^(n/2); factors to: (2*m^2 + 2m + 1) * (2*m^2 - 2m + 1)

odd n: factor of 113

* R10 k=343:

n = = 1 mod 3: factor of 3

n = = 2 mod 3: factor of 37

n = = 0 mod 3: let n=3q and let m=7*10^q; factors to: (m - 1) * (m^2 + m + 1)

* R957 k=64:

n = = 1 mod 3: factor of 73

n = = 2 mod 3: factor of 19

n = = 0 mod 3: let n=3q and let m=4*957^q; factors to: (m - 1) * (m^2 + m + 1)

* S63 k=3511808:

n = = 1 mod 3: factor of 37

n = = 2 mod 3: factor of 109

n = = 0 mod 3: let n=3q and let m=152*63^q; factors to: (m + 1) * (m^2 - m + 1)

* S63 k=27000000:

n = = 1 mod 3: factor of 37

n = = 2 mod 3: factor of 109

n = = 0 mod 3: let n=3q and let m=300*63^q; factors to: (m + 1) * (m^2 - m + 1)

* R936 k=64:

n = = 0 mod 2: let n = 2q; factors to: (8*936^q - 1) * (8*936^q + 1)

n = = 0 mod 3: let n=3q; factors to: (4*936^q - 1) * [16*936^(2q) + 4*936^q + 1]

n = = 1 mod 6: factor of 37

n = = 5 mod 6: factor of 109

Of course,

* In the Riesel case if k and b are both r-th powers for an r>1

* In the Sierpinski case if k and b are both r-th powers for an odd r>1

* In the Sierpinski case if k is of the form 4*m^4, and b is 4th power

Then this k proven composite by full algebraic factors

[sweety439]

The k's that make a full covering set with all or partial algebraic factors are excluded from the conjectures, unless they also have a covering set of primes (e.g. R4 k=361, R8 k=343, R9 k=49, R9 k=121, R16 k=100, R49 k=81, R121 k=100, R125 k=8, S125 k=8, R169 k=16, R169 k=49, R216 k=125, S216 k=125, R256 k=100, R1024 k=81, R1024 k=121, R1024 k=169), thus cannot be the CK

[sweety439]

Originally Posted by sweety439

Of course,

* In the Riesel case if k and b are both r-th powers for an r>1

* In the Sierpinski case if k and b are both r-th powers for an odd r>1

* In the Sierpinski case if k is of the form 4*m^4, and b is 4th power

Then this k proven composite by full algebraic factors

Also, in the Sierpinski case if b = q^m, k = q^r, where q is not of the form t^s with odd s>1, and m and r have no common odd prime factor, and the exponent of highest power of 2 dividing r >= the exponent of highest power of 2 dividing m, and the equation 2^x = r (mod m) has no solution, then this k has no possible prime. (such k's are also excluded from the conjectures)

Examples:

b = q^7, k = q^r, where r = 3, 5, 6 (mod 7).
b = q^14, k = q^r, where r = 6, 10, 12 (mod 14).
b = q^15, k = q^r, where r = 7, 11, 13, 14 (mod 15).
b = q^17, k = q^r, where r = 3, 5, 6, 7, 10, 11, 12, 14 (mod 17).
b = q^21, k = q^r, where r = 5, 10, 13, 17, 19, 20 (mod 21)
b = q^23, k = q^r, where r = 5, 7, 10, 11, 14, 15, 17, 19, 20, 21, 22 (mod 23)
b = q^28, k = q^r, where r = 12, 20, 24 (mod 28)
b = q^30, k = q^r, where r = 14, 22, 26, 28 (mod 30)
b = q^31, k = q^r, where r = 3, 5, 6, 7, 9, 10, 11, 12, 13, 14, 15, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30 (mod 31)
b = q^33, k = q^r, where r = 5, 7, 10, 13, 14, 19, 20, 23, 26, 28 (mod 33)
etc.

(these are all examples for m<=33)

[Uncwilly]

There is no need to quote an entire previous post. Trim your quotes.

[sweety439]

Originally Posted by sweety439

Reserve some 1k bases (S17, S27, S51, S56, R38, R54)

See https://en.wikipedia.org/w/index.php...ldid=966555963

[sweety439]

Originally Posted by sweety439

(79*73^9339-1)/6 is (probable) prime!!!

R73 is proven!!!

Update newest status

https://docs.google.com/document/d/e...8HDWrvEC82/pub

[sweety439]

Originally Posted by sweety439

Of course,

* In the Riesel case if k and b are both r-th powers for an r>1

* In the Sierpinski case if k and b are both r-th powers for an odd r>1

* In the Sierpinski case if k is of the form 4*m^4, and b is 4th power

Then this k proven composite by full algebraic factors

There are many Riesel case for k=64 (since 64 = 4^3 = 8^2, thus n == 0 mod 2 or n == 0 mod 3 have algebra factors:

Bases 619, 1322, 2025, 2728, 3431, 4134, 4837, 5540, 6243, 6946, 7649, 8352, 9055, 9758, ...: n == 1 mod 3: factor of 19, n == 5 mod 6: factor of 37

Bases 429, 1816, 3203, 4590, 5977, 7364, 8751, ...: n == 1 mod 3: factor of 19, n == 2 mod 3: factor of 73

Bases 391, 2462, 4533, 6604, 8675, ...: n == 1 mod 3: factor of 19, n == 5 mod 6: factor of 109

Bases 159, 862, 1565, 2268, 2971, 3674, 4377, 5080, 5783, 6486, 7189, 7892, 8595, 9298, ...: n == 1 mod 6: factor of 37, n == 2 mod 3: factor of 19

Bases 1232, 3933, 6634, 9335, ...: n == 1 mod 6: factor of 37, n == 2 mod 3: factor of 73

Bases 936, 4969, 9002, ...: n == 1 mod 6: factor of 37, n == 5 mod 6: factor of 109

Bases 957, 2344, 3731, 5118, 6505, 7892, 9279, ...: n == 1 mod 3: factor of 73, n == 2 mod 3: factor of 19

Bases 1322, 4023, 6724, 9425, ...: n == 1 mod 3: factor of 73, n == 5 mod 6: factor of 37

Bases 4315, ...: n == 1 mod 3: factor of 73, n == 5 mod 6: factor of 109

Bases 482, 2553, 4624, 6695, 8766, ...: n == 1 mod 6: factor of 109, n == 2 mod 3: factor of 19

Bases 3098, 7131, ...: n == 1 mod 6: factor of 109, n == 5 mod 6: factor of 37

Bases 4079, ...: n == 1 mod 6: factor of 109, n == 2 mod 3: factor of 73

[sweety439]

Originally Posted by sweety439

There are many Riesel case for k=64 (since 64 = 4^3 = 8^2, thus n == 0 mod 2 or n == 0 mod 3 have algebra factors:

Bases 619, 1322, 2025, 2728, 3431, 4134, 4837, 5540, 6243, 6946, 7649, 8352, 9055, 9758, ...: n == 1 mod 3: factor of 19, n == 5 mod 6: factor of 37

Bases 429, 1816, 3203, 4590, 5977, 7364, 8751, ...: n == 1 mod 3: factor of 19, n == 2 mod 3: factor of 73

Bases 391, 2462, 4533, 6604, 8675, ...: n == 1 mod 3: factor of 19, n == 5 mod 6: factor of 109

Bases 159, 862, 1565, 2268, 2971, 3674, 4377, 5080, 5783, 6486, 7189, 7892, 8595, 9298, ...: n == 1 mod 6: factor of 37, n == 2 mod 3: factor of 19

Bases 1232, 3933, 6634, 9335, ...: n == 1 mod 6: factor of 37, n == 2 mod 3: factor of 73

Bases 936, 4969, 9002, ...: n == 1 mod 6: factor of 37, n == 5 mod 6: factor of 109

Bases 957, 2344, 3731, 5118, 6505, 7892, 9279, ...: n == 1 mod 3: factor of 73, n == 2 mod 3: factor of 19

Bases 1322, 4023, 6724, 9425, ...: n == 1 mod 3: factor of 73, n == 5 mod 6: factor of 37

Bases 4315, ...: n == 1 mod 3: factor of 73, n == 5 mod 6: factor of 109

Bases 482, 2553, 4624, 6695, 8766, ...: n == 1 mod 6: factor of 109, n == 2 mod 3: factor of 19

Bases 3098, 7131, ...: n == 1 mod 6: factor of 109, n == 5 mod 6: factor of 37

Bases 4079, ...: n == 1 mod 6: factor of 109, n == 2 mod 3: factor of 73

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, ...

[sweety439]

Originally Posted by sweety439

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, ...

Sierpinski k=2500 also for these square bases, in square bases, all even n have algebra factors, and for these bases (square of numbers == 4 or 13 mod 17) all odd n have factor of 17:

16, 169, 441, 900, 1444, 2209, 3025, 4096, 5184, 6561, 7921, 9604, 11236, 13225, 15129, 17424, 19600, 22201, 24649, 27556, 30276, 33489, 36481, 40000, 43264, 47089, 50625, 54756, 58564, 63001, 67081, 71824, 76176, 81225, 85849, 91204, 96100, 101761, 106929, 112896, 118336, 124609, 130321, 136900, 142884, 149769, 156025, 163216, 169744, 177241, 184041, 191844, 198916, 207025, 214369, 222784, 230400, 239121, 247009, 256036, 264196, 273529, 281961, 291600, 300304, 310249, 319225, 329476, 338724, 349281, 358801, 369664, 379456, 390625, 400689, 412164, 422500, 434281, 444889, 456976, 467856, 480249, 491401, 504100, 515524, 528529, 540225, 553536, 565504, 579121, 591361, 605284, 617796, 632025, 644809, 659344, 672400, 687241, 700569, 715716, 729316, 744769, 758641, 774400, 788544, 804609, 819025, 835396, 850084, 866761, 881721, 898704, 913936, 931225, 946729, 964324, 980100, 998001, ...