[QUOTE=sweety439;549645]Reserve some 1k bases (S17, S27, S51, S56, R38, R54)[/QUOTE]

and found the prime 38*51^4881+1

the first 4 conjectures of S51 are all proven!!!

Reserve R56

[QUOTE=sweety439;549656]and found the prime 38*51^4881+1

the first 4 conjectures of S51 are all proven!!!

Reserve R56[/QUOTE]

I skipped these bases since they are already reserved by other projects:

S16: see post [URL="https://mersenneforum.org/showpost.php?p=468772&postcount=463"]#463[/URL], already at n=15K

S38: reserved by [URL="http://www.primegrid.com/stats_genefer.php"]Prime Grid's GFN primes search[/URL], already at n=2^24-1

S50: reserved by [URL="http://www.primegrid.com/stats_genefer.php"]Prime Grid's GFN primes search[/URL], already at n=2^24-1

R10: reserved by [URL="http://www.worldofnumbers.com/em197.htm"]http://www.worldofnumbers.com/em197.htm[/URL] (case d=3, k=817) and [URL="https://www.rose-hulman.edu/~rickert/Compositeseq/"]https://www.rose-hulman.edu/~rickert/Compositeseq/[/URL] (case b=10, d=3, k=817), already at n=554789

R12: see post [URL="https://mersenneforum.org/showpost.php?p=490302&postcount=664"]#664[/URL], already at n=21760

R32: reserved by [URL="http://www.noprimeleftbehind.net/crus/Riesel-conjectures-powers2.htm"]CRUS[/URL] (case R1024, k=29), already at n=500K

R49: reserved by [URL="https://github.com/RaymondDevillers/primes"]https://github.com/RaymondDevillers/primes[/URL] (see the "left49" file) (case b=49 family R{G}), already at n=10K

Conjecture 1 (the strong Sierpinski conjecture): For b>=2, k>=1, if there [B][I]is an n[/I][/B] such that:

(1) k*b^n is neither a perfect odd power (i.e. k*b^n is not of the form m^r with odd r>1) nor of the form 4*m^4.
(2) gcd((k*b^n+1)/gcd(k+1,b-1),(b^(945*2^s)-1)/(b-1)) = 1 for all s, i.e. for all s, every prime factor of (k*b^n+1)/gcd(k+1,b-1) does not divide (b^(945*2^s)-1)/(b-1). (i.e. ord_p(b) is not of the form 2^r (r>=0 if p = 2 or p = 3, r>=1 if p>=5) or m*2^r (m divides 945, r>=0) for every prime factor p of (k*b^n+1)/gcd(k+1,b-1)).
(3) this (k,b) pair is not the case: 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. (the first 6 Sierpinski bases with k's which are this case are 128, 2187, 16384, 32768, 78125 and 131072)

Then there are infinitely many primes of the form (k*b^n+1)/gcd(k+1,b-1).

Conjecture 2 (the strong Riesel conjecture): For b>=2, k>=1, if there [B][I]is an n[/I][/B] such that:

(1) k*b^n is not a perfect power (i.e. k*b^n is not of the form m^r with r>1).
(2) gcd((k*b^n-1)/gcd(k-1,b-1),(b^(945*2^s)-1)/(b-1)) = 1 for all s, i.e. for all s, every prime factor of (k*b^n-1)/gcd(k-1,b-1) does not divide (b^(945*2^s)-1)/(b-1). (i.e. ord_p(b) is not of the form 2^r (r>=0 if p = 2 or p = 3, r>=1 if p>=5) or m*2^r (m divides 945, r>=0) for every prime factor p of (k*b^n-1)/gcd(k-1,b-1)).

Then there are infinitely many primes of the form (k*b^n-1)/gcd(k-1,b-1).

[QUOTE=sweety439;549645]Reserve some 1k bases (S17, S27, S51, S56, R38, R54)[/QUOTE]

(33*27^7876+1)/2 is (probable) prime

the first 4 conjectures of S27 are all proven!!!

reserve R57

(281*57^5610-1)/56 is (probable) prime

the first 4 conjectures of R57 are all proven!!!

reserve R49 (the corresponding page only searched it to 10K, I double check it and reserve it for n>10K)

k is Sierpinski number base b if....

[CODE]
k b
1 (none)
2 (no such b < 201446503145165177)
3 (no such b < 158503)
4 == 14 mod 15
5 == 11 mod 12
6 == 34 mod 35
7 == 5 mod 24 or == 11 mod 12
8 == 20 mod 21 or == 47, 83 mod 195 or == 467 mod 73815 or == 722 mod 1551615
9 == 19 mod 20
10 == 32 mod 33
11 == 5 mod 24 or == 14 mod 15 or == 19 mod 20
12 == 142 mod 143 or == 296, 901 mod 19019 or 562, 828, 900, 1166 mod 1729 or == 563 mod 250705 or == 597, 1143 mod 1885
13 == 20 mod 21 or == 27 mod 28 or == 132, 293 mod 595
14 == 38 mod 39 or == 64 mod 65
15 == 13 mod 14 but not == 1 mod 16
16 == 38, 47, 98, 242 mod 255 or == 50 mod 51 or == 84 mod 85
17 == 11 mod 12 or == 278, 302 mod 435 or == 283, 355, 367, 607, 907 mod 1638 or == 373, 445, 646, 718 mod 819
18 == 322 mod 323 or == 398, 512 mod 1235
19 == 11 mod 12 or == 14 mod 15 or == 29 mod 40
20 == 56 mod 57 or == 132 mod 133
21 == 43 mod 44 or == 54 mod 55
22 == 68 mod 69 or == 160 mod 161
23 (== 21 mod 22 but not == 1 mod 8) or == 32 mod 33 or == 41 mod 48 or == 83 mod 530 or == 182 mod 795
24 == 114 mod 115
25 == 38 mod 39 or == 51 mod 52
[/CODE]

[QUOTE=sweety439;549645]Reserve some 1k bases (S17, S27, S51, S56, R38, R54)[/QUOTE]

No (probable) found for these bases except S27, S51, R57, these bases are likely tested to [B][I]at least[/I][/B] n=10K, bases released.

[QUOTE=sweety439;536172]All tested to n=1024.

k's that proven composite by algebra factors:

R243:

k = m^5
k = m^2 with m = 11 or 50 mod 61

R729:

k = m^2
k = m^3

S243:

k = m^5

S729:

k = m^3[/QUOTE]

In fact, R243 k=81 is already tested to n=443060 with no (probable) prime found, since (81*243^n-1)/gcd(81-1,243-1) = (3^(5*n+4)-1)/2, but no known terms in [URL="https://oeis.org/A028491"]A028491[/URL] is = 4 mod 5

Special cases of (k*b^n+-1)/gcd(k+-1,b-1) (+ for Sierpinski, - for Riesel):

* gcd(k+-1,b-1) (+ for Sierpinski, - for Riesel) = 1: the same as the original Sierpinski/Riesel problem in [URL="https://mersenneforum.org/showthread.php?t=9738"]CRUS[/URL]

* Riesel case k=1: the smallest generalized [URL="https://primes.utm.edu/glossary/page.php?sort=Repunit"]repunit[/URL] prime base b (see [URL="https://oeis.org/A084740"]A084740[/URL] and [URL="http://www.fermatquotient.com/PrimSerien/GenRepu.txt"]http://www.fermatquotient.com/PrimSerien/GenRepu.txt[/URL])

* Riesel case k=b-1: the smallest [URL="http://harvey563.tripod.com/wills.txt"]Williams prime[/URL] base (b-1)

* Sierpinski case k=1 and b even: the smallest [URL="https://primes.utm.edu/glossary/page.php?sort=GeneralizedFermatNumber"]generalized Fermat prime[/URL] base b (see [URL="http://www.noprimeleftbehind.net/crus/GFN-primes.htm"]http://www.noprimeleftbehind.net/crus/GFN-primes.htm[/URL] and [URL="http://jeppesn.dk/generalized-fermat.html"]http://jeppesn.dk/generalized-fermat.html[/URL])

* Sierpinski case k=1 and b odd: the smallest generalized half Fermat prime base b (see [URL="http://www.fermatquotient.com/PrimSerien/GenFermOdd.txt"]http://www.fermatquotient.com/PrimSerien/GenFermOdd.txt[/URL])

we allow n=1 or n=2 or n=3 or n=4 or ..., but not allow n=0 or n=-1 or n=-2 or n=-3 or ... for (k*b^n+-1)/gcd(k+-1,b-1)

k is Riesel number base b if....

[CODE]
k b
1 (none)
2 (none)
3 (none)
4 == 14 mod 15
5 == 11 mod 12
6 == 34 mod 35
7 == 11 mod 12
8 == 20 mod 21 or == 83, 307 mod 455
9 == 19 mod 20 or == 29 mod 40
10 == 32 mod 33
11 == 14 mod 15 or == 19 mod 20
12 == 142 mod 143 or == 307 mod 1595 or == 901 mod 19019
13 == 5 mod 24 or == 20 mod 21 or == 27 mod 28 or == 38, 47 mod 255
14 == 8, 47, 83, 122 mod 195 or == 38 mod 39 or == 64 mod 65
15 == 27 mod 28
16 == 50 mod 51 or == 84 mod 85
[/CODE]