Done to base 2500

Some CK in the text files in post [URL="https://mersenneforum.org/showpost.php?p=546592&postcount=772"]#772[/URL]have been fixed:

S970: 139823 --> 134552 (k = 134552 has covering set {3, 7, 44851})
S1150: >1M --> 755222 (k = 755222 has covering set {3, 7, 63031})
S1876: >1M --> 473969 (k = 473969 has covering set {3, 37, 31723})

R1150: >1M --> 190243 (k = 190243 has covering set {3, 7, 63031})
R1414: 64 --> 284 (k = 64 is unlikely to have a covering set, this error is because for this k, all n <= 1000 has a prime factor <= 10000)
R1876: 793972 --> 475846 (k = 475846 has covering set {3, 37, 31723})

The text file in post [URL="https://mersenneforum.org/showpost.php?p=546592&postcount=772"]#772 [/URL]have these errors is because I only tested the primes <= 10000 and the exponents <= 1000, but all these covering sets of these k has a prime > 10000, now I tested the primes <= 100000 and the exponents <= 2100, the fixed files are in the post [URL="https://mersenneforum.org/showpost.php?p=551601&postcount=914"]#914[/URL]

See [URL="https://github.com/xayahrainie4793/Extended-Sierpinski-Riesel-conjectures/commit/cb8f6e8d76fa148cc8219ceef6b979ab7f1d43be#diff-0edded909f81afec33f9596b6ce34d02"]https://github.com/xayahrainie4793/Extended-Sierpinski-Riesel-conjectures/commit/cb8f6e8d76fa148cc8219ceef6b979ab7f1d43be#diff-0edded909f81afec33f9596b6ce34d02[/URL] (Sierpinski) and [URL="https://github.com/xayahrainie4793/Extended-Sierpinski-Riesel-conjectures/commit/0fb2ab83fd1df4e2ca6edb436dad3ee79c43359a#diff-7536d0728a9f5b128918a8665bfad2d0"]https://github.com/xayahrainie4793/Extended-Sierpinski-Riesel-conjectures/commit/0fb2ab83fd1df4e2ca6edb436dad3ee79c43359a#diff-7536d0728a9f5b128918a8665bfad2d0[/URL] (Riesel) for the updating.

Reserve: (for k's > CK)

S4: k = 1238 & 1286
R4: k = 1159 & 1189

See [URL="https://github.com/xayahrainie4793/first-4-Sierpinski-Riesel-conjectures"]https://github.com/xayahrainie4793/first-4-Sierpinski-Riesel-conjectures[/URL] for the status of the first 4 CK for the Sierpinski/Riesel conjectures for the bases b<=64 with smaller CK

[QUOTE=sweety439;551665]Reserve: (for k's > CK)

S4: k = 1238 & 1286
R4: k = 1159 & 1189[/QUOTE]

and found 2 (probable) primes for R4:

(1159*4^5628-1)/3
(1189*4^3404-1)/3

the first 4 conjectures of R4 are all proven!!!

the 2 k's for S4 are still remain.

[QUOTE=sweety439;462037]We can find the 2nd, 3rd, 4th, ... n such that (k*b^n+-1)/gcd(k+-1,b-1) is prime. (i.e. find the 2nd, 3rd, 4th, ... prime of the form (k*b^n+-1)/gcd(k+-1,b-1) for fixed k and fixed Sierpinski/Riesel base b)

For the n's such that (k*b^n+-1)/gcd(k+-1,b-1) is prime: (excluding MOB, since if k is multiple of the base (b), then k and k / b are from the same family, if k = k' * b^r, then the exponent n for this k is (the first number > r for k') - r, and the correspond prime for this k is the correspond prime for n = (the first number > r for k') for k')

S2:

k = 1: 1, 2, 4, 8, 16, ... (sequence is not in OEIS)
k = 3: [URL="https://oeis.org/A002253"]A002253[/URL]
k = 5: [URL="https://oeis.org/A002254"]A002254[/URL]
k = 7: [URL="https://oeis.org/A032353"]A032353[/URL]
k = 9: [URL="https://oeis.org/A002256"]A002256[/URL]
k = 11: [URL="https://oeis.org/A002261"]A002261[/URL]
k = 13: [URL="https://oeis.org/A032356"]A032356[/URL]
k = 15: [URL="https://oeis.org/A002258"]A002258[/URL]
k = 17: [URL="https://oeis.org/A002259"]A002259[/URL]
k = 19: [URL="https://oeis.org/A032359"]A032359[/URL]
k = 21: [URL="https://oeis.org/A032360"]A032360[/URL]
k = 23: [URL="https://oeis.org/A032361"]A032361[/URL]
k = 25: [URL="https://oeis.org/A032362"]A032362[/URL]
k = 27: [URL="https://oeis.org/A032363"]A032363[/URL]
k = 29: [URL="https://oeis.org/A032364"]A032364[/URL]
k = 31: [URL="https://oeis.org/A032365"]A032365[/URL]
k = 33: [URL="https://oeis.org/A032366"]A032366[/URL]
k = 35: [URL="https://oeis.org/A032367"]A032367[/URL]
k = 37: [URL="https://oeis.org/A032368"]A032368[/URL]
k = 39: [URL="https://oeis.org/A002269"]A002269[/URL]
k = 41: [URL="https://oeis.org/A032370"]A032370[/URL]
k = 43: [URL="https://oeis.org/A032371"]A032371[/URL]
k = 45: [URL="https://oeis.org/A032372"]A032372[/URL]
k = 47: [URL="https://oeis.org/A032373"]A032373[/URL]
k = 49: [URL="https://oeis.org/A032374"]A032374[/URL]
k = 51: [URL="https://oeis.org/A032375"]A032375[/URL]
k = 53: [URL="https://oeis.org/A032376"]A032376[/URL]
k = 55: [URL="https://oeis.org/A032377"]A032377[/URL]
k = 57: [URL="https://oeis.org/A002274"]A002274[/URL]
k = 59: [URL="https://oeis.org/A032379"]A032379[/URL]
k = 61: [URL="https://oeis.org/A032380"]A032380[/URL]
k = 63: [URL="https://oeis.org/A032381"]A032381[/URL]

S3:

k = 1: [URL="https://oeis.org/A171381"]A171381[/URL]
k = 2: [URL="https://oeis.org/A003306"]A003306[/URL]
k = 4: [URL="https://oeis.org/A005537"]A005537[/URL]
k = 5: 2, 6, 12, 18, 26, 48, 198, 456, ... (sequence is not in OEIS)
k = 7: 1, 9, 33, 65, 337, ... (sequence is not in OEIS)
k = 8: [URL="https://oeis.org/A005538"]A005538[/URL]
k = 10: [URL="https://oeis.org/A005539"]A005539[/URL]
k = 11: 1, 3, 21, 39, 651, ... (sequence is not in OEIS)
k = 13: 2, 14, 32, 40, 112, ... (sequence is not in OEIS)
k = 14: [URL="https://oeis.org/A216890"]A216890[/URL]
k = 16: 3, 4, 5, 12, 24, 36, 77, 195, 296, 297, 533, 545, 644, 884, 932, ... (sequence is not in OEIS)

S4:

k = 1: 1, 2, 4, 8, ... (sequence is not in OEIS)
k = 2: [URL="https://oeis.org/A127936"]A127936[/URL]
k = 3: 1, 3, 4, 6, 9, 15, 18, 33, 138, 204, 219, 267, ... (sequence is not in OEIS)
k = 5: 1, 3, 6, 12, 15, 18, 36, 72, 81, 84, 117, 522, 1023, 1083, 1206, ... (sequence is not in OEIS)
k = 6: 2, 20, 94, 100, 104, 176, 1594, ... (sequence is not in OEIS)
k = 7: [URL="https://oeis.org/A002255"]A002255[/URL]

S5:

k = 1: 1, 2, 4, ... (sequence is not in OEIS)
k = 2: [URL="https://oeis.org/A058934"]A058934[/URL]
k = 3: 2, 6, 8, 62, 120, 186, 414, 764, ... (sequence is not in OEIS)
k = 4: [URL="https://oeis.org/A204322"]A204322[/URL]
k = 6: [URL="https://oeis.org/A143279"]A143279[/URL]
k = 7: (covering set {2, 3})
k = 8: 1, 1037, ... (sequence is not in OEIS)

S6:

k = 1: 1, 2, 4, ... (sequence is not in OEIS)
k = 2: [URL="https://oeis.org/A120023"]A120023[/URL]
k = 3: [URL="https://oeis.org/A186112"]A186112[/URL]
k = 4: [URL="https://oeis.org/A248613"]A248613[/URL]
k = 5: [URL="https://oeis.org/A247260"]A247260[/URL]
k = 7: 1, 6, 17, 38, 50, 80, 207, 236, 264, 309, 555 ... (sequence is not in OEIS)
k = 8: 4, 10, 16, 32, 40, 70, 254, ... (sequence is not in OEIS)

S10:

k = 1: 1, 2, ... (sequence is not in OEIS)
k = 2: [URL="https://oeis.org/A096507"]A096507[/URL]
k = 3: [URL="https://oeis.org/A056807"]A056807[/URL]
k = 4: [URL="https://oeis.org/A056806"]A056806[/URL]
k = 5: [URL="https://oeis.org/A102940"]A102940[/URL]
k = 6: [URL="https://oeis.org/A056805"]A056805[/URL]
k = 7: [URL="https://oeis.org/A056804"]A056804[/URL]
k = 8: [URL="https://oeis.org/A096508"]A096508[/URL]
k = 9: [URL="https://oeis.org/A056797"]A056797[/URL]
k = 11: [URL="https://oeis.org/A102975"]A102975[/URL]
k = 12: 2, 38, 80, 9230, 25598, 39500, ... (sequence is not in OEIS)
k = 13: [URL="https://oeis.org/A289051"]A289051[/URL]
k = 14: [URL="https://oeis.org/A099017"]A099017[/URL]
k = 15: 1, 4, 7, 8, 18, 19, 73, 143, 192, 408, 533, 792, 3179, 7709, 9554, 52919, 56021, 61604, ... (sequence is not in OEIS)
k = 16: [URL="https://oeis.org/A273002"]A273002[/URL]

R2:

k = 1: [URL="https://oeis.org/A000043"]A000043[/URL]
k = 3: [URL="https://oeis.org/A002235"]A002235[/URL]
k = 5: [URL="https://oeis.org/A001770"]A001770[/URL]
k = 7: [URL="https://oeis.org/A001771"]A001771[/URL]
k = 9: [URL="https://oeis.org/A002236"]A002236[/URL]
k = 11: [URL="https://oeis.org/A001772"]A001772[/URL]
k = 13: [URL="https://oeis.org/A001773"]A001773[/URL]
k = 15: [URL="https://oeis.org/A002237"]A002237[/URL]
k = 17: [URL="https://oeis.org/A001774"]A001774[/URL]
k = 19: [URL="https://oeis.org/A001775"]A001775[/URL]
k = 21: [URL="https://oeis.org/A002238"]A002238[/URL]
k = 23: [URL="https://oeis.org/A050537"]A050537[/URL]
k = 25: [URL="https://oeis.org/A050538"]A050538[/URL]
k = 27: [URL="https://oeis.org/A050539"]A050539[/URL]
k = 29: [URL="https://oeis.org/A050540"]A050540[/URL]
k = 31: [URL="https://oeis.org/A050541"]A050541[/URL]
k = 33: [URL="https://oeis.org/A002240"]A002240[/URL]
k = 35: [URL="https://oeis.org/A050543"]A050543[/URL]
k = 37: [URL="https://oeis.org/A050544"]A050544[/URL]
k = 39: [URL="https://oeis.org/A050545"]A050545[/URL]
k = 41: [URL="https://oeis.org/A050546"]A050546[/URL]
k = 43: [URL="https://oeis.org/A050547"]A050547[/URL]
k = 45: [URL="https://oeis.org/A002242"]A002242[/URL]
k = 47: [URL="https://oeis.org/A050549"]A050549[/URL]
k = 49: [URL="https://oeis.org/A050550"]A050550[/URL]
k = 51: [URL="https://oeis.org/A050551"]A050551[/URL]
k = 53: [URL="https://oeis.org/A050552"]A050552[/URL]
k = 55: [URL="https://oeis.org/A050553"]A050553[/URL]
k = 57: [URL="https://oeis.org/A050554"]A050554[/URL]
k = 59: [URL="https://oeis.org/A050555"]A050555[/URL]
k = 61: [URL="https://oeis.org/A050556"]A050556[/URL]
k = 63: [URL="https://oeis.org/A050557"]A050557[/URL]

R3:

k = 1: [URL="https://oeis.org/A028491"]A028491[/URL]
k = 2: [URL="https://oeis.org/A003307"]A003307[/URL]
k = 4: [URL="https://oeis.org/A005540"]A005540[/URL]
k = 5: 1, 3, 5, 9, 15, 23, 45, 71, 99, 125, 183, 1143, ... (sequence is not in OEIS)
k = 7: 2, 4, 6, 8, 16, 18, 28, 36, 52, 106, 114, 204, 270, 292, 472, 728, 974, ... (sequence is not in OEIS)
k = 8: [URL="https://oeis.org/A005541"]A005541[/URL]
k = 10: [URL="https://oeis.org/A005542"]A005542[/URL]
k = 11: 22, 30, 46, 162, ... (sequence is not in OEIS)
k = 13: 1, 5, 25, 41, 293, 337, 569, 1085, ... (sequence is not in OEIS)
k = 14: 1, 11, 16, 80, 83, 88, 136, 187, 328, 397, 776, 992, 1195, ... (sequence is not in OEIS)
k = 16: 1, 3, 9, 13, 31, 43, 81, 121, 235, 1135, 1245, ... (sequence is not in OEIS)

R4:

k = 1: (proven composite by full algebra factors)
k = 2: [URL="https://oeis.org/A146768"]A146768[/URL]
k = 3: [URL="https://oeis.org/A272057"]A272057[/URL]
k = 5: 1, 2, 4, 5, 6, 7, 9, 16, 24, 27, 36, 74, 92, 124, 135, 137, 210, 670, 719, 761, 819, 877, 942, 1007, 1085, ... (sequence is not in OEIS)
k = 6: 1, 3, 5, 21, 27, 51, 71, 195, 413, ... (sequence is not in OEIS)
k = 7: 2, 3, 5, 12, 14, 41, 57, 66, 284, 296, 338, 786, 894, ... (sequence is not in OEIS)

R5:

k = 1: [URL="https://oeis.org/A004061"]A004061[/URL]
k = 2: [URL="https://oeis.org/A120375"]A120375[/URL]
k = 3: 1, 2, 4, 9, 16, 17, 54, 64, 112, 119, 132, 245, 557, 774, 814, 1020, 1110, ... (sequence is not in OEIS)
k = 4: [URL="https://oeis.org/A046865"]A046865[/URL]
k = 6: [URL="https://oeis.org/A257790"]A257790[/URL]
k = 7: 1, 5, 11, 13, 15, 41, 61, 77, 103, 123, 199, 243, 279, 1033, 1145, ... (sequence is not in OEIS)
k = 8: 2, 4, 8, 10, 28, 262, 356, 704, ... (sequence is not in OEIS)

R6:

k = 1: [URL="https://oeis.org/A004062"]A004062[/URL]
k = 2: [URL="https://oeis.org/A057472"]A057472[/URL]
k = 3: [URL="https://oeis.org/A186106"]A186106[/URL]
k = 4: 1, 3, 25, 31, 43, 97, 171, 213, 273, 449, 575, 701, 893, ... (sequence is not in OEIS)
k = 5: [URL="https://oeis.org/A079906"]A079906[/URL]
k = 7: 1, 2, 3, 13, 21, 28, 30, 32, 36, 48, 52, 76, 734, ... (sequence is not in OEIS)
k = 8: 1, 5, 35, 65, 79, 215, 397, 845, ... (sequence is not in OEIS)

R10:

k = 1: [URL="https://oeis.org/A004023"]A004023[/URL]
k = 2: [URL="https://oeis.org/A002957"]A002957[/URL]
k = 3: [URL="https://oeis.org/A056703"]A056703[/URL]
k = 4: [URL="https://oeis.org/A056698"]A056698[/URL]
k = 5: [URL="https://oeis.org/A056712"]A056712[/URL]
k = 6: [URL="https://oeis.org/A056716"]A056716[/URL]
k = 7: [URL="https://oeis.org/A056701"]A056701[/URL]
k = 8: [URL="https://oeis.org/A056721"]A056721[/URL]
k = 9: [URL="https://oeis.org/A056725"]A056725[/URL]
k = 11: [URL="https://oeis.org/A111391"]A111391[/URL]
k = 12: 5, 3191, 3785, 5513, 14717, ... (sequence is not in OEIS)
k = 13: [URL="https://oeis.org/A056707"]A056707[/URL]
k = 14: 1, 2, 3, 4, 5, 16, 21, 23, 62, 175, 195, 206, 261, 347, 448, 494, 689, 987, 1361, 8299, 13225, 21513, 23275, ... (sequence is not in OEIS)
k = 15: 1, 2, 15, 22, 27, 33, 38, 473, 519, 591, 699, 2273, 2476, 2985, 6281, 6947, 11990, 16828, 17096, 26236, 33459, 34963, ... (sequence is not in OEIS)
k = 16: [URL="https://oeis.org/A056714"]A056714[/URL][/QUOTE]

Also these OEIS sequences:

[URL="https://oeis.org/A078680"]A078680[/URL]: S2, k = index
[URL="https://oeis.org/A040076"]A040076[/URL]: S2, k = index (allow n=0)
[URL="https://oeis.org/A050412"]A050412[/URL]: R2, k = index + 1
[URL="https://oeis.org/A040081"]A040081[/URL]: R2, k = index (allow n=0)
[URL="https://oeis.org/A291437"]A291437[/URL]: S3, k = index * 2 (allow n=0)
[URL="https://oeis.org/A177330"]A177330[/URL]: R4, k = prime(index) if prime(index) == 1 mod 3, n = value/2; k = 2*prime(index) if prime(index) == 2 mod 3, n = (value-1)/2
[URL="https://oeis.org/A250204"]A250204[/URL]: S6, k = index, k != 4 mod 5
[URL="https://oeis.org/A250205"]A250205[/URL]: R6, k = index, k != 1 mod 5
[URL="https://oeis.org/A217377"]A217377[/URL]: R6, k = index * 5 + 1 (allow n=0)
[URL="https://oeis.org/A069568"]A069568[/URL]: R10, k = index * 9 + 1
[URL="https://oeis.org/A083747"]A083747[/URL]: R10, k = index * 9 + 1 (allow n=0)
[URL="https://oeis.org/A090584"]A090584[/URL]: R10, k = index * 3 + 1, k != 1 mod 9 (allow n=0)
[URL="https://oeis.org/A090465"]A090465[/URL]: R10, k = index + 1, k != 1 mod 3 (allow n=0)
[URL="https://oeis.org/A257459"]A257459[/URL]: R10, k = prime(index) * 9 + 1
[URL="https://oeis.org/A232210"]A232210[/URL]: R10, k = prime(index) * 3 + 1, k != 1 mod 9
[URL="https://oeis.org/A257461"]A257461[/URL]: R10, k = prime(index) + 1, k != 1 mod 3

Update all primes for R102 and some primes for S80

Extended Sierpinski problem base b:

Finding and proving the smallest k>=1 such that (k*b^n+1)/gcd(k+1,b-1) is not prime for all integers n>=1. (k-values that make a full covering set with all or partial algebraic factors are excluded from the conjectures)

Extended Riesel problem base b:

Finding and proving the smallest k>=1 such that (k*b^n-1)/gcd(k-1,b-1) is not prime for all integers n>=1. (k-values that make a full covering set with all or partial algebraic factors are excluded from the conjectures)

The reason for why S8 k=27, S16 k=4, R4 k=1, R8 k=1, R16 k=1, R27 k=1, R36 k=1 are excluded from the conjectures is the same as the reason for why in CRUS conjectures, R12 k=1, R14 k=1, R18 k=1, R20 k=1, R24 k=1 are excluded from the conjectures

The reason is although they have a prime, but they can [I]have only this prime[/I], thus excluded from the conjectures, a k-value with no covering set is included from the conjectures if and only if this k-value [I]may[/I] have infinitely many primes

[QUOTE=sweety439;550194]There are many Riesel case for k=64 (since 64 = 4^3 = 8^2, thus n == 0 mod 2 [B]or[/B] 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[/QUOTE]

Also, (for Riesel k=64)

Bases 235, 748, 1261, 1774, 2287, 2800, 3313, 3826, 4339, 4852, 5365, 5878, 6391, 6904, 7417, 7930, 8443, 8956, 9469, 9982, ...: n == 1 mod 3: factor of 3, n == 2 mod 3: factor of 19

Bases 397, 1396, 2395, 3394, 4393, 5392, 6391, 7390, 8389, 9388, ...: n == 1 mod 3: factor of 3, n == 5 mod 6: factor of 37

Bases 721, 2692, 4663, 6634, 8605, ...: n == 1 mod 3: factor of 3, n == 2 mod 3: factor of 73

Bases 1045, 3988, 6931, 9874, ...: n == 1 mod 3: factor of 3, n == 5 mod 6: factor of 109

Bases 334, 847, 1360, 1873, 2386, 2899, 3412, 3925, 4438, 4951, 5464, 5977, 6490, 7003, 7516, 8029, 8542, 9055, 9568, ...: n == 1 mod 3: factor of 19, n == 2 mod 3: factor of 3

Bases 307, 1306, 2305, 3304, 4303, 5302, 6301, 7300, 8299, 9298, ...: n == 1 mod 6: factor of 37, n == 2 mod 3: factor of 3

Bases 1468, 3439, 5410, 7381, 9352, ...: n == 1 mod 3: factor of 73, n == 2 mod 3: factor of 3

Bases 2008, 4951, 7894, ...: n == 1 mod 6: factor of 109, n == 2 mod 3: factor of 3