[QUOTE=sweety439;558829]The CK for S630 is now known to be 24015859
The CK for R630 is now known to be 24412760[/QUOTE]

The CK for S756 is known to be 67836285
The CK for R756 is known to be 54604682

The CK for S726 is known to be 10923176
The CK for R726 is known to be 12751579

The CK for R690 is known to be 42053568
The CK for S690 is still running ....

S690 has CK = 395800653

[QUOTE=sweety439;559219]test limits for GFN's and half GFN's for 128 < base <= 1024 (except 256, 512, 1024):

base 3:

According to [URL="http://www.prothsearch.com/GFN03.html"]http://www.prothsearch.com/GFN03.html[/URL], the first numbers n>=7 such that (3^(2^n)+1)/2 might be primes are 24, 25, 27, 37, 40, 41, 42, 43, 46, 48, 52, 53, ..., and we have:

* For S243 k=27: the test limit is (2^27-3)/5-1 = 26843544 ((2^n-3)/5 is not integer for n = 24, 25)
* [URL="https://github.com/xayahrainie4793/Extended-Sierpinski-Riesel-conjectures/blob/master/Sierpinski%20CK%20for%20bases%20up%20to%202048.txt"]S729 has only CK = 31[/URL], and no half GFN remain for k<31

base 5:

According to [URL="http://www.prothsearch.com/GFN05.html"]http://www.prothsearch.com/GFN05.html[/URL], the first numbers n>=3 such that (5^(2^n)+1)/2 might be primes are 29, 30, 31, 37, 42, 43, 46, 52, 55, 56, 57, 61, ..., and we have:

* [URL="https://github.com/xayahrainie4793/Extended-Sierpinski-Riesel-conjectures/blob/master/Sierpinski%20CK%20for%20bases%20up%20to%202048.txt"]S625 has only CK = 185[/URL], and no half GFN remain for k<185

base 6:

According to [URL="http://www.prothsearch.com/GFN06.html"]http://www.prothsearch.com/GFN06.html[/URL], the first numbers n>=3 such that 6^(2^n)+1 might be primes are 28, 29, 30, 31, 38, 41, 45, 46, 48, ..., and we have:

* For S216 k=36, the test limit is (2^29-2)/3-1 = 178956969 ((2^n-2)/3 is not integer for n = 28)

base 7:

According to [URL="http://www.prothsearch.com/GFN07.html"]http://www.prothsearch.com/GFN07.html[/URL], the first numbers n>=3 such that (7^(2^n)+1)/2 might be primes are 23, 24, 26, 27, 28, 29, 35, 37, 38, 39, 43, ..., and we have:

* For S343 k=49, the test limit is (2^23-2)/3-1 = 2796201

base 10:

According to [URL="http://www.prothsearch.com/GFN10.html"]http://www.prothsearch.com/GFN10.html[/URL], the first numbers n>=2 such that 10^(2^n)+1 might be primes are 31, 32, 33, 34, 36, 42, 44, 45, 47, 49, ..., and we have:

* For S1000 k=10, the test limit is (2^32-1)/3-1 = 1431655764 ((2^n-1)/3 is not integer for n = 31)

base 11:

No power-of-11 bases between 128 and 1024

base 12:

According to [URL="http://www.prothsearch.com/GFN12.html"]http://www.prothsearch.com/GFN12.html[/URL], the first numbers n>=1 such that 12^(2^n)+1 might be primes are 24, 25, 27, 28, 31, 32, 35, 36, 37, 43, 47, 48, ..., however, according to [URL="http://www.primegrid.com/stats_genefer.php"]http://www.primegrid.com/stats_genefer.php[/URL], n^(2^22)+1 is composite for all n<169020, and 169020>12^4, thus (12^4)^(2^22)+1 = 12^(2^24)+1 is composite, and the first number n>=1 such that 12^(2^n)+1 might be prime is 25

* For S144 k=1, the test limit is (2^24)-1 = 16777215

even bases > 12:

According to [URL="http://www.primegrid.com/stats_genefer.php"]http://www.primegrid.com/stats_genefer.php[/URL], n^(2^22)+1 is composite for all even n<169020, and 169020^(1/4) = 20.276104663..., thus n^(2^24)+1 is composite for all even n<=20, and 169020^(1/2) = 411.120420315..., thus n^(2^23)+1 is composite for all even n<=411

* For even bases <= 20, the test limit of GFN's are 2^25-epsilon
* For 21 <= even bases <= 411, the test limit of GFN's are 2^24-epsilon
* For 412 <= even bases <= 169019, the test limit of GFN's are 2^23-epsilon

odd bases > 12:

According to [URL="http://www.fermatquotient.com/PrimSerien/GenFermOdd.txt"]http://www.fermatquotient.com/PrimSerien/GenFermOdd.txt[/URL], (n^(2^17)+1)/2 is composite for all odd n<11559, and 11559^(1/2) = 107.512789936..., thus (n^(2^18)+1)/2 is composite for all odd n<=107

* For odd bases <= 107, the test limit of half GFN's are 2^19-epsilon (I also checked the (n^(2^18)+1)/2 for 108 <= odd bases <= 128, they are also all composite, thus (n^(2^18)+1)/2 is composite for all odd n<=128)
* For 129 <= odd bases <= 11558, the test limit of half GFN's are 2^18-epsilon[/QUOTE]

test limits for generalized repunits (note that (k*b^n+1)/gcd(k+1,b-1) cannot be prime if k*b^n is either perfect odd power (of the form m^r with odd r>1) or of the form 4*m^4, and (k*b^n-1)/gcd(k-1,b-1) cannot be prime if k*b^n is perfect power (of the form m^r with r>1), unless (k*b^n+1)/gcd(k+1,b-1) and (k*b^n-1)/gcd(k-1,b-1) are generalized repunits (i.e. k is rational power of b (for (k*b^n-1)/gcd(k-1,b-1), or for (k*b^n+1)/gcd(k+1,b-1), but an additional condition for (k*b^n+1)/gcd(k+1,b-1) is (if k = b^(r/s) and gcd(r,s)=1) s is even number):

Base 2:

None of SR2, SR4, SR8, SR16, SR32, SR64, SR128, SR512, SR1024 have generalized repunits remain. (note that the exponent of R128 k=16 is (3217-4)/7 = 459, and the corresponding prime is 2^3217-1, since 3217 is the smallest number in [URL="https://oeis.org/A000043"]A000043[/URL] which is == 4 mod 7, and note that the exponent of R512 k=4 is (19937-2)/9 = 2215, and the corresponding prime is 2^19937-1, since 19937 is the smallest number in [URL="https://oeis.org/A000043"]A000043[/URL] which is == 2 mod 9)

Base 3:

* For R243 k=81, the test limit is (2215303-4)/5 = 443059.8 (unfortunately, there are no known numbers in [URL="https://oeis.org/A028491"]A028491[/URL] which are == 4 mod 5), but the true test limit may be larger, since Paul Bourdelais is searching generalized repunit (probable) primes for small bases (see [URL="http://www.primenumbers.net/prptop/prptop.php"]top PRPs[/URL])

Base 5:

None of SR5, SR25, SR125, SR625 have generalized repunits remain. (note that the exponent of S625 k=125 is (67-3)/4 = 16, and the corresponding prime is (5^67+1)/6, since 67 is the smallest number in [URL="https://oeis.org/A057171"]A057171[/URL] which is == 3 mod 4)

Base 6:

None of SR6, SR36, SR216 have generalized repunits remain. (note that the exponent of R216 k=36 is (29-2)/3 = 9, and the corresponding prime is (6^29-1)/5, since 29 is the smallest number in [URL="https://oeis.org/A004062"]A004062[/URL] which is == 2 mod 3)

Base 7:

None of SR7, SR49, SR343 have generalized repunits remain.

Base 10:

None of SR10, SR100, SR1000 have generalized repunits remain. (note that the exponent of R10 k=100 is 19-2 = 17, and the corresponding prime is (10^19-1)/9, since 19 is the smallest number in [URL="https://oeis.org/A004023"]A004023[/URL] which is > 2)

Base 11:

None of SR11, SR121 have generalized repunits remain.

Base 12:

None of SR12, SR144 have generalized repunits remain.

Bases > 12:

No bases with generalized repunits (k is rational power of b) other than k=1 remain, except R243 k=81, all Riesel bases <= 1024 with k=1 (which is equivalent to generalized repunits base b) remain are 185, 269, 281, 380, 384, 385, 394, 452, 465, 511, 574, 601, 631, 632, 636, 711, 713, 759, 771, 795, 861, 866, 881, 938, 948, 951, 956, 963, 1005, 1015, and the test limit is 100K by Michael Stocker

1. Any k that obtains a full covering set in any manner from ALGEBRAIC factors will be excluded. In many instances, this includes k's where there is a partial covering set of numeric factors (or a single numeric factor) and a partial covering set of algebraic factors that combine to make a full covering set.

2. k-values in the case in post [URL="https://mersenneforum.org/showpost.php?p=459405&postcount=265"]https://mersenneforum.org/showpost.php?p=459405&postcount=265[/URL] will be excluded, since they have [I]no possible[/I] primes but with no NUMERIC covering set, like the k-values in the condition in #1.

3. All k below the lowest k found to have a NUMERIC covering set must have a prime including multiples of the base (MOB) but excluding the conditions in #1 and #2 above.

4. All n must be >= 1.

[QUOTE=sweety439;559495]test limits for generalized repunits (note that (k*b^n+1)/gcd(k+1,b-1) cannot be prime if k*b^n is either perfect odd power (of the form m^r with odd r>1) or of the form 4*m^4, and (k*b^n-1)/gcd(k-1,b-1) cannot be prime if k*b^n is perfect power (of the form m^r with r>1), unless (k*b^n+1)/gcd(k+1,b-1) and (k*b^n-1)/gcd(k-1,b-1) are generalized repunits (i.e. k is rational power of b (for (k*b^n-1)/gcd(k-1,b-1), or for (k*b^n+1)/gcd(k+1,b-1), but an additional condition for (k*b^n+1)/gcd(k+1,b-1) is (if k = b^(r/s) and gcd(r,s)=1) s is even number):

Base 2:

None of SR2, SR4, SR8, SR16, SR32, SR64, SR128, SR512, SR1024 have generalized repunits remain. (note that the exponent of R128 k=16 is (3217-4)/7 = 459, and the corresponding prime is 2^3217-1, since 3217 is the smallest number in [URL="https://oeis.org/A000043"]A000043[/URL] which is == 4 mod 7, and note that the exponent of R512 k=4 is (19937-2)/9 = 2215, and the corresponding prime is 2^19937-1, since 19937 is the smallest number in [URL="https://oeis.org/A000043"]A000043[/URL] which is == 2 mod 9)

Base 3:

* For R243 k=81, the test limit is (2215303-4)/5 = 443059.8 (unfortunately, there are no known numbers in [URL="https://oeis.org/A028491"]A028491[/URL] which are == 4 mod 5), but the true test limit may be larger, since Paul Bourdelais is searching generalized repunit (probable) primes for small bases (see [URL="http://www.primenumbers.net/prptop/prptop.php"]top PRPs[/URL])

Base 5:

None of SR5, SR25, SR125, SR625 have generalized repunits remain. (note that the exponent of S625 k=125 is (67-3)/4 = 16, and the corresponding prime is (5^67+1)/6, since 67 is the smallest number in [URL="https://oeis.org/A057171"]A057171[/URL] which is == 3 mod 4)

Base 6:

None of SR6, SR36, SR216 have generalized repunits remain. (note that the exponent of R216 k=36 is (29-2)/3 = 9, and the corresponding prime is (6^29-1)/5, since 29 is the smallest number in [URL="https://oeis.org/A004062"]A004062[/URL] which is == 2 mod 3)

Base 7:

None of SR7, SR49, SR343 have generalized repunits remain.

Base 10:

None of SR10, SR100, SR1000 have generalized repunits remain. (note that the exponent of R10 k=100 is 19-2 = 17, and the corresponding prime is (10^19-1)/9, since 19 is the smallest number in [URL="https://oeis.org/A004023"]A004023[/URL] which is > 2)

Base 11:

None of SR11, SR121 have generalized repunits remain.

Base 12:

None of SR12, SR144 have generalized repunits remain.

Bases > 12:

No bases with generalized repunits (k is rational power of b) other than k=1 remain, except R243 k=81, all Riesel bases <= 1024 with k=1 (which is equivalent to generalized repunits base b) remain are 185, 269, 281, 380, 384, 385, 394, 452, 465, 511, 574, 601, 631, 632, 636, 711, 713, 759, 771, 795, 861, 866, 881, 938, 948, 951, 956, 963, 1005, 1015, and the test limit is 100K by Michael Stocker[/QUOTE]

the exponent of R18 k=324 is 25667-2 = 25665, and the exponent of R324 k=18 is (25667-1)/2 = 12833, and the corresponding prime of them are both (18^25667-1)/17, since 25667 is the smallest number in [URL="https://oeis.org/A133857"]A133857[/URL] which is either > 2 or == 1 mod 2, however, since the CK for R18 is only 246 and the CK for R324 is only 14, both of them are k's > CK

Also, the exponent of R27 k=9 is (71-2)/3 = 23, and the corresponding prime is (3^71-1)/2, since 71 is the smallest number in [URL="https://oeis.org/A028491"]A028491[/URL] which is == 2 mod 3

Also, the exponent of S961 k=31 is (109-1)/2 = 54, and the corresponding prime is (31^109+1)/32, since 109 is the smallest number in [URL="https://oeis.org/A126856"]A126856[/URL]

[QUOTE=sweety439;559504]1. Any k that obtains a full covering set in any manner from ALGEBRAIC factors will be excluded. In many instances, this includes k's where there is a partial covering set of numeric factors (or a single numeric factor) and a partial covering set of algebraic factors that combine to make a full covering set.

2. All k below the lowest k found to have a NUMERIC covering set must have a prime including multiples of the base (MOB) but excluding the conditions in #1 above.

3. All n must be >= 1.[/QUOTE]

The lowest k found to have a numeric covering set are:

[URL="https://github.com/xayahrainie4793/Extended-Sierpinski-Riesel-conjectures/blob/master/Sierpinski%20CK%20for%20bases%20up%20to%202048.txt"]Sierpinski[/URL] [URL="https://github.com/xayahrainie4793/Extended-Sierpinski-Riesel-conjectures/blob/master/Riesel%20CK%20for%20bases%20up%20to%202048.txt"]Riesel[/URL]

(format: b,the lowest k found such that (k*b^n+-1)/gcd(k+-1,b-1) (+ for Sierpinski, - for Riesel) have a numeric covering set)

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 with 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: first prime at n=37938
S35 k=41650: first prime at n=6038
S59 k=3540: first prime at n=218
S75 k=5700: first prime at n=1636
S77 k=6006: no prime with n<=1728
S81 k=6480: first prime at n=5393
S155 k=310: no prime with n<=2K
S333 k=1998: first prime at n=27900
R2 k=74: first prime at n=2552
R2 k=674: first prime at n=11676
R2 k=1094: first prime at n=652
R2 k=71444: first prime at n=98952
R2 k=340934: first prime at n=55272
R2 k=351134: no prime with n<=6.63M
R2 k=381854: first prime at n=72288
R2 k=478214: no prime with n<=6.63M
R4 k=1524: first prime at n=1994
R4 k=19464: no prime with n<=3.3M
R5 k=26000: first prime at n=6112
R5 k=84200: first prime at n=9356
R5 k=138800: first prime at n=4512
R6 k=103536: first prime at n=6474
R6 k=106056: first prime at n=3038
R7 k=679: no prime with n<=3K
R10 k=40: first prime at n=14
R10 k=100: first prime at n=17
R10 k=230: first prime at n=60
R10 k=450: first prime at n=11958
R10 k=2650: first prime at n=252
R10 k=16750: no prime with n<=200K
R10 k=78880: no prime with n<=5K
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
R15 k=300870: first prime at n=156608
R18 k=324: first prime at n=25665
R21 k=84: first prime at n=88
R23 k=230: first prime at n=6228
R24 k=1824: first prime at n=138568
R24 k=18504: first prime at n=48462
R27 k=594: first prime at n=36624
R31 k=124: first prime at n=1116
R31 k=69998: first prime at n=13618
R40 k=520: no prime with n<=1K
R40 k=209960: no prime with n<=100K
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
R126 k=15750: first prime at n=1495
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
R598 k=357604: no prime with n<=30K
R706 k=497730: first prime at n=4734
R738 k=401402628: first prime at n=3490
R777 k=602952: first prime at n=4997
R789 k=490546548: first prime at n=1314
R815 k=663410: first prime at n=1472
R903 k=735498918: first prime at n=2736
R916 k=838140: first prime at n=3887
R954 k=1908: first prime at n=1476
R976 k=1952: first prime at n=1924
R982 k=963342: first prime at n=2484
R1102 k=2204: first prime at n=52176
R1297 k=2594: first prime at n=19839
R1360 k=2720: first prime at n=74688

Multiples of the base (MOB) are NOT excluded from the conjectures. They are excluded from the TESTING of the conjectures if (k+-1)/gcd(k+-1,b-1) (+ for Sierpinski, - for Riesel) is composite. Previously they were shown as being excluded from the conjectures.

#2 is an important distinction if algebraic factors or MOB take priority. That is, if k=4*b is eliminated because it is a MOB and k=4 has algebraic factors to make a full covering set, which of the two takes priority for k=4*b since it would also have algebraic factors to make a full covering set? The answer is algebraic factors take priority because k=4 cannot ever have a prime and so k=4*b must still be accounted for. SO: k=4*b has to be shown with algebraic factors because it too cannot ever have a prime.

This is certainly mathematical pickiness but to account for all k's, you can't just say a k is eliminated from the conjecture because it is a MOB and (k+-1)/gcd(k+-1,b-1) (+ for Sierpinski, - for Riesel) is composite. The k still remains; it's just not shown as remaining or tested because k/b should eventually (or already has) yield the same prime. But if k/b can never have a prime than you must account for k.

Don't worry if this doesn't make much sense. It only comes into play if a base has some k's with algebraic factors to make a full covering set and the conjecture is large enough where k*b comes in below the conjecture. This is a small percentage of bases; very small on the Sierpinski side.

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, since such k-values will have the same prime as k / b.

However, k-values that are a multiple of base (b) and where (k+-1)/gcd(k+-1,b-1) (+ for Sierpinski, - for Riesel) is prime are included from testing since the exponent n must be >=1 (n can be 1, but cannot be 0 or -1 or -2 or ...), and the same prime n=1 for k / b would be n=0 for this k but n must be >=1 hence it is not allowed so this k must continue to be searched. (of course, k-values that are not a multiple of base (b) are included from testing)

Thus, for S3, k = 42, 45, 57, 60, 66 and 72 are included from testing since although 42, 45, 57, 60, 66 and 72 are multiples of 3, but 42+1, (45+1)/2, (57+1)/2, 60+1, 66+1 and 72+1 are primes. However, k = 48, 51, 54, 63, 69 and 75 are excluded from testing since 48, 51, 54, 63, 69 and 75 are multiples of 3, and 48+1, (51+1)/2, 54+1, (63+1)/2, (69+1)/2 and (75+1)/2 are not primes. Besides, for R3, k = 42, 48, 54, 60, 63, 72 and 75 are included from testing since although 42, 48, 54, 60, 63, 72 and 75 are multiples of 3, but 42-1, 48-1, 54-1, 60-1, (63-1)/2, 72-1 and (75-1)/2 are primes. However, k = 45, 51, 57, 66 and 69 are excluded from testing since 45, 51, 57, 66 and 69 are multiples of 3, and (45-1)/2, (51-1)/2, (57-1)/2, 66-1 and (69-1)/2 are not primes.

Note: Since 1 is not prime, thus for R3, k = 3 is excluded from testing. ((3-1)/2 = 1) However, since 2 is prime, thus for S3, k = 3 is included from testing. ((3+1)/2 = 2)