[QUOTE=sweety439;550553]Found an error of S81: (34*81^734+1)/gcd(34+1,81-1) is prime

Double checking S81....[/QUOTE]

There are two other errors (for k=317 and 389): (317*81^518+1)/gcd(317+1,81-1) and (389*81^871+1)/gcd(389+1,81-1) are primes

Re-update the zip file

Update newest status of [URL="https://docs.google.com/document/d/e/2PACX-1vTsmy_HaE-GxLL6ICfbvUNr9iXdkgQfuVYpPkFPFUjPerzPglR11zObhWRdG7YlLd5judUF8OgSUVsS/pub"]Sierpinski problems[/URL]

S81 has only 7 k remain

These ranges are completed: "[]" for the remaining (b,k) pair such that no smaller k for this b, no smaller b for this k, no smaller b and smaller k, are remaining.

(b>=2, k>=1)

Sierpinski:

[2,21181]

b=2, k<=21180

[3,1187]

b<=4, k<=1186

[5,181]

b<=9, k<=180

[10,100]

b<=11, k<=99

[12,12]

b<=30, k<=11

[31,1]

Riesel:

[2,2293]

b=2, k<=2292

[3,1613]

b<=4, k<=1612

[5,1279]

b<=6, k<=1278

[7,679]

b<=7, k<=678

[8,239]

b<=10, k<=238

[11,201]

b<=14, k<=200

[15,47]

b<=30, k<=46

[31,5]

b<=158, k<=4

[159,3]

b<=184, k<=2

[185,1]

[QUOTE=sweety439;550622]These ranges are completed: "[]" for the remaining (b,k) pair such that no smaller k for this b, no smaller b for this k, no smaller b and smaller k, are remaining.
[/QUOTE]

(Probable) primes with n>=1000 and smaller (b,k):

Sierpinski:

b=2: (see [URL="http://www.prothsearch.com/sierp.html"]http://www.prothsearch.com/sierp.html[/URL])

b=3:

k=41, n=4892 (k=123, n=4891, k=369, n=4890, k=1107, n=4889)
k=523, n=1775
k=621, n=20820
k=821, n=5512
k=823, n=6087
k=935, n=3967

b=4:

k=186, n=10458 (k=744, n=10457)
k=766, n=3196
k=839, n=1217

b=5:

k=40, n=1036
k=61, n=6208

b=6:

k=160, n=3143

b=7:

k=141, n=1044

b=8:

k=173, n=7771

b=9:

k=41, n=2446

b=17:

k=10, n=1356

b=23:

k=8, n=119215
k=10, n=3762

Riesel:

b=2: (see [URL="http://www.prothsearch.com/rieselprob.html"]http://www.prothsearch.com/rieselprob.html[/URL])

b=3:

k=97, n=3131 (k=291, n=3130, k=873, n=3129)
k=119, n=8972 (k=357, n=8971, k=1071, n=8970)
k=302, n=2091 (k=906, n=2090)
k=313, n=24761 (k=939, n=24760)
k=599, n=1240
k=811, n=1126
k=997, n=20847
k=1013, n=1233
k=1093, n=1297
k=1199, n=3876
k=1303, n=1384

b=4:

k=74, n=1276 (k=296, n=1275, k=1184, n=1274)
k=106, n=4553 (k=424, n=4552)
k=373, n=2508 (k=1492, n=2507)
k=659, n=400258
k=674, n=5838
k=751, n=6615
k=1103, n=2203
k=1159, n=5628
k=1171, n=2855
k=1189, n=3404
k=1211, n=12621
k=1524, n=1994

b=5:

k=86, n=2058 (k=430, n=2057)
k=428, n=9704
k=638, n=6974
k=662, n=14628
k=935, n=1560
k=1006, n=4197

b=6:

k=251, n=3008
k=1030, n=1199

b=7:

k=159, n=4896 (k=1113, n=4895)
k=197, n=181761
k=313, n=5907
k=367, n=15118
k=419, n=1052
k=429, n=3815
k=653, n=1051

b=8:

k=74, n=2632
k=151, n=2141
k=191, n=1198
k=203, n=1866
k=236, n=5258

b=9:

k=119, n=4486

b=11:

k=62, n=26202

b=14:

k=5, n=19698 (k=70, n=19697)

b=17:

k=13, n=1123
k=29, n=4904
k=44, n=6488

b=23:

k=30, n=1000

b=26:

k=32, n=9812

b=27:

k=23, n=3742

b=30:

k=25, n=34205

b=42:

k=3, n=2523

b=47:

k=4, n=1555

b=51:

k=1, n=4229

b=72:

k=4, n=1119849

b=91:

k=1, n=4421

b=107:

k=2, n=21910
k=3, n=4900

b=115:

k=4, n=4223

b=135:

k=1, n=1171

b=142:

k=1, n=1231

b=152:

k=1, n=270217

b=170:

k=2, n=166428

b=174:

k=1, n=3251

b=184:

k=1, n=16703

Store the files for R42 k=3 and k=14 (3*14=42, thus they are duals)

these solved R1764 k=14, k=126=3*42, k=588=14*42, but k=3 is still unsloved

[QUOTE=sweety439;550622]Sierpinski:

[2,21181]

b=2, k<=21180

[3,1187]

b<=4, k<=1186

[5,181]

b<=9, k<=180

[10,100]

b<=11, k<=99

[12,12]

b<=30, k<=11

[31,1][/QUOTE]

If GFN's and half GFN's are not counted, then this list become:

[2,21181]

b=2, k<=21180

[3,1187]

b<=4, k<=1186

[5,181]

b<=15, k<=180

[16,89]

b<=16, k<=88

[17,53]

b<=26, k<=52

[27,49]

b<=30, k<=48

[31,43]

b<=40, k<=42

[41,28]

b<=46, k<=27

[47,27]

b<=52, k<=26

[53,4]

b<=82, k<=3

[83,3]

b<=364, k=2

[365,2]

(k=1 is no longer available, since k=1 for all even b are GFN's and for all odd b are half GFN's)

Fixed the files

- remove "color used" section
- change the better name of "the top 10 k's" column: "only sorted by n" --> "sorted by n only"

[URL="https://docs.google.com/document/d/e/2PACX-1vQvru05GPynoscjw69pkmzJkLpR9PNO6aQLoi5cYGkHYvpA_XVRbN4AN6Pz20yEylKMf2aM8UUJejDi/pub"]Sierpinski problems[/URL]

[URL="https://docs.google.com/document/d/e/2PACX-1vSS0MZ5-obO4Ghr04s1XELgNY9PPfjJzbm9iuk5hwFVbQKOEaZ5IxKD9y2GQLvLce281JZE43XMbunQ/pub"]Riesel problems[/URL]

[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]

In fact, base 391 has covering set {3, 7, 19, 109}, and base 429 has covering set {5, 7, 19, 73}

(bases 159, 482, 619, 862, 936, 957 are unlikely to have covering set)

[QUOTE=sweety439;550778]If GFN's and half GFN's are not counted, then this list become:

[2,21181]

b=2, k<=21180

[3,1187]

b<=4, k<=1186

[5,181]

b<=15, k<=180

[16,89]

b<=16, k<=88

[17,53]

b<=26, k<=52

[27,49]

b<=30, k<=48

[31,43]

b<=40, k<=42

[41,28]

b<=46, k<=27

[47,27]

b<=52, k<=26

[53,4]

b<=82, k<=3

[83,3]

b<=364, k=2

[365,2]

(k=1 is no longer available, since k=1 for all even b are GFN's and for all odd b are half GFN's)[/QUOTE]

and the (probable) primes with n>=1000 and smaller (b,k) are:

b=2: (see [URL="http://www.prothsearch.com/sierp.html"]http://www.prothsearch.com/sierp.html[/URL])

b=3:

k=41, n=4892 (k=123, n=4891, k=369, n=4890, k=1107, n=4889)
k=523, n=1775
k=621, n=20820
k=821, n=5512
k=823, n=6087
k=935, n=3967

b=4:

k=186, n=10458 (k=744, n=10457)
k=766, n=3196
k=839, n=1217

b=5:

k=40, n=1036
k=61, n=6208

b=6:

k=160, n=3143

b=7:

k=141, n=1044

b=8:

k=173, n=7771

b=9:

k=41, n=2446

b=13:

k=29, n=10574
k=48, n=6267
k=120, n=1552

b=14:

k=73, n=1182
k=145, n=1176

b=16:

k=23, n=1074

b=17:

k=10, n=1356

b=20:

k=43, n=2956

b=23:

k=8, n=119215
k=10, n=3762

b=26:

k=32, n=318071

b=27:

k=33, n=7876

b=30:

k=12, n=1023

b=31:

k=5, n=1026

b=33:

k=36, n=23615

b=37:

k=19, n=5310

b=38:

k=2, n=2729
k=31, n=1528

b=45:

k=24, n=18522

b=46:

k=17, n=4920

b=101:

k=2, n=192275

b=104:

k=2, n=1233

b=167:

k=2, n=6547

b=206:

k=2, n=46205

b=218:

k=2, n=333925

b=236:

k=2, n=161229

b=257:

k=2, n=12183

b=287:

k=2, n=5467

b=305:

k=2, n=16807

b=353:

k=2, n=2313

Searched to 5M, "NA" if > 5M, almost done to base 2500 ....

[QUOTE=sweety439;551437]Searched to 5M, "NA" if > 5M, almost done to base 2500 ....[/QUOTE]

Test limit:

primes in the covering set <= 100K
exponents <= 2100 (not 2000, to include 2048 = 2^11 to return the true CK for S125, otherwise it will return CK=1 for S125)