I tested all k<=32 for all Sierpinski/Riesel bases b<=32, for the bases with CK >= 32 and the text file already lists all k's, see the text files in the post [URL="http://mersenneforum.org/showpost.php?p=461547&postcount=340"]#340[/URL], for other bases (bases with CK < 32 or the text file only lists the k's not in CRUS), see the text files in this post. (for the exclusion of the k's for all bases b<=32, see the post [URL="http://mersenneforum.org/showpost.php?p=451367&postcount=104"]#104[/URL] (k's proven composite by full or partial algebra factors) and [URL="http://mersenneforum.org/showpost.php?p=460366&postcount=285"]#285[/URL] (k's with covering set))

The remain k's <= 32 for Sierpinski/Riesel bases b<=32 are:

S12, k = 12.
S13, k = 29.
S18, k = 18.
S22, k = 22.
S31, k = 1 and 31.
S32, k = 4 and 16.
R27, k = 23.
R29, k = 21.
R31, k = 5 and 19.
R32, k = 29.

(S12 k = 12, S18 k = 18, S22 k = 22, S32 k = 4, and S32 k = 16 are GFN's, S31 k = 1 and S31 k = 31 are half GFN's, except these bases and k's, only R32 k = 29 has gcd(k+-1,b-1) = 1, although the CK for R32 is only 10, but due to the status of R1024 k = 29 in CRUS, it has been tested to n=1M with no primes, thus, R32 k = 29 has been tested to n=2M with no primes, since the two tests are the same, 29*32^n-1 can be prime only for even n, since if n is odd, then 29*32^n-1 is divisible by 3)

Some large primes given by CRUS:

10*17^1356+1
8*23^119215+1
32*26^318071+1
12*30^1023+1
5*14^19698-1
30*23^1000-1
32*26^9812-1
25*30^34205-1

Also, the (probable) prime (10*23^3762+1)/11 was found by me, other (probable) prime with n>=1000 found by me are: (for k<=32, b<=32)

(23*16^1074+1)/3
(5*31^1026+1)/6
(13*17^1123-1)/4
(29*17^4904-1)/4

They are already in this project.

[QUOTE=sweety439;449149]For the original Sierpinski/Riesel problem, it is finding and proving the smallest k such that k*b^n+-1 is composite for all integer n>0 and gcd(k+-1, b-1) = 1. Now, I extend to the k's such that gcd(k+-1, b-1) is not 1. Of course, all numbers of the form k*b^n+-1 is divisible by gcd(k+-1, b-1). Thus, I must take out this factor and find and prove the smallest k such that (k*b^n+-1)/gcd(k+-1, b-1) is composite for all integer n>1. (of course, in the base 2 case, this is completely the same as the original Sierpinski/Riesel problem)

In the original Sierpinski/Riesel problems, k-values with all n-values have a single trivial factor are excluded from the conjectures. However, in these problems, we take out this trivial factor, thus all k-values are included from the conjectures. (thus, in this problems, the divisor of k*b^n+-1 is the largest trivial factor of k*b^n+-1, which equals gcd(k+-1, b-1))

The research is form [URL]http://mersenneforum.org/showthread.php?t=21832[/URL].

The Riesel case are also researched in [URL]https://www.rose-hulman.edu/~rickert/Compositeseq/[/URL].

The strong (extended) Sierpinski problem base 4 is proven, with the conjectured k=419. Also, the strong (extended) Riesel problem base 10 is proven, with the conjectured k=334.

For the strong (extended) Riesel problem base 3, for k<=500, I cannot find a prime for k = {119, 313, 357}. For k=291, the prime is the same as the k=97 prime: (97*3^3131-1)/2, since 291 = 97*3, and since 357 = 119*3, the prime for k=357 is the same as the prime for k=119, but both are unknown.

Edit: According to the link, (313*3^24761-1)/2 is a probable prime.

Extended Sierpinski problem base b: Finding and proving the smallest integer k>=1 such that (k*b^n+1)/gcd(k+1, b-1) is not prime for all integer n>=1.

Extended Riesel problem base b: Finding and proving the smallest integer k>=1 such that (k*b^n-1)/gcd(k-1, b-1) is not prime for all integer n>=1.

The last text file is the list of the conjectured smallest strong Sierpinski/Riesel number number to base b for b = 2 to b = 12.[/QUOTE]

Do you understand this problem?

The correspond primes for S2 for k = 1, 2, 3, ... are:

{3, 5, 7, 17, 11, 13, 29, 17, 19, 41, 23, 97, 53, 29, 31, 257, 137, 37, 1217, 41, 43, 89, 47, 97, 101, 53, 109, 113, 59, 61, 7937, 257, ...}

The correspond primes for S3 for k = 1, 2, 3, ... are:

{2, 7, 5, 13, 23, 19, 11, 73, 41, 31, 17, 37, 59, 43, 23, 433, 6197, 163, 29, 61, 68891, 67, 311, 73, 113, 79, 41, 757, 131, 271, 47, 97, ...}

The correspond primes for S4 for k = 1, 2, 3, ... are:

{5, 3, 13, 17, 7, 97, 29, 11, 37, 41, 59, 193, 53, 19, 61, 257, 23, 73, 1217, 107, 337, 89, 31, 97, 101, 139, 109, 113, 619, 7681, 7937, 43, ...}

The correspond primes for S5 for k = 1, 2, 3, ... are: (0 if no such prime exists or this k is excluded from the conjecture)

{3, 11, 19, 101, 13, 31, 0, 41, 23, 251, 0, 61, 163, 71, 19, 401, 43, 2251, 2969, 101, 53, 13751, 29, 601, 313, 131, 4219, 701, 73, 151, 0, 4001, ...}

The correspond primes for S6 for k = 1, 2, 3, ... are:

{7, 13, 19, 5, 31, 37, 43, 10369, 11, 61, 67, 73, 79, 17, 541, 97, 103, 109, 23, 155521, 127, 28513, 139, 29, 151, 157, 163, 1009, 7517, 181, 1117, 193, ...}

The correspond primes for S7 for k = 1, 2, 3, ... are:

{1201, 5, 11, 29, 41, 43, 1201, 19, 529421, 71, 13, 28813, 15607, 229, 53, 113, 139, 127, 67, 47, 6506482422146989923806032135894782377546749491930607652617339526616854149081491252134508288317638363981211, 7547, 154688827, 8233, 613, 61, 26693911031, 197, 7870665723233837, 211, 109, 523, ...}

The correspond primes for S8 for k = 1, 2, 3, ... are: (0 if no such prime exists or this k is excluded from the conjecture)

{0, 17, 193, 257, 41, 7, 449, 0, 73, 641, 89, 97, 7607, 113, 7681, 65537, 137, 1153, 1217, 23, 10753, 1409, 11777, 193, 1601, 13313, 0, 114689, 233, 241, 35740566642812256257, 257, ...}

The correspond primes for S9 for k = 1, 2, 3, ... are: (0 if no such prime exists or this k is excluded from the conjecture)

{5, 19, 7, 37, 23, 487, 71, 73, 41, 811, 223, 109, 59, 127, 17, 1297, 6197, 163, 43, 181, 68891, 199, 233, 17497, 113, 13817467, 61, 2269, 131, 271, 0, 2593, ...}

The correspond primes for S10 for k = 1, 2, 3, ... are:

{11, 7, 31, 41, 17, 61, 71, 89, 9001, 101, 37, 1201, 131, 47, 151, 1601, 19, 181, 191, 67, 211, 22000001, 76667, 241, 251, 29, 271, 281, 97, 3001, 311, 107, ...}

The correspond primes for R2 for k = 1, 2, 3, ... are:

{3, 3, 5, 7, 19, 11, 13, 31, 17, 19, 43, 23, 103, 223, 29, 31, 67, 71, 37, 79, 41, 43, 367, 47, 199, 103, 53, 223, 463, 59, 61, 127, ...}

The correspond primes for R3 for k = 1, 2, 3, ... are:

{13, 5, 13, 11, 7, 17, 31, 23, 13, 29, 172595827849, 107, 19, 41, 67, 47, 229, 53, 769, 59, 31, 197, 103, 71, 37, 233, 1093, 83, 43, 89, 139, 863, ...}

The correspond primes for R4 for k = 1, 2, 3, ... are: (0 if no such prime exists or this k is excluded from the conjecture)

{0, 7, 11, 0, 19, 23, 37, 31, 0, 13, 43, 47, 17, 223, 59, 0, 67, 71, 101, 79, 83, 29, 367, 383, 0, 103, 107, 37, 463, 479, 41, 127, ...}

The correspond primes for R5 for k = 1, 2, 3, ... are: (0 if no such prime exists or this k is excluded from the conjecture)

{31, 1249, 7, 19, 31, 29, 17, 199, 11, 1249, 137, 59, 0, 349, 37, 79, 0, 89, 47, 499, 131, 109, 7187, 599, 31, 16249, 67, 139, 181, 149, 48437, 12499999, ...}

The correspond primes for R6 for k = 1, 2, 3, ... are:

{7, 11, 17, 23, 29, 7, 41, 47, 53, 59, 13, 71, 467, 83, 89, 19, 101, 107, 113, 719, 151, 131, 137, 863, 149, 31, 971, 167, 173, 179, 37, 191, ...}

The correspond primes for R7 for k = 1, 2, 3, ... are:

{2801, 13, 73, 457, 17, 41, 2801, 19207, 31, 23, 269, 83, 743, 97, 367, 37, 59, 881, 7603, 139, 73, 359, 563, 167, 29, 181, 661, 457, 101, 10289, 8413470255870653, 223, ...}

The correspond primes for R8 for k = 1, 2, 3, ... are: (0 if no such prime exists or this k is excluded from the conjecture)

{0, 127, 23, 31, 20479, 47, 3583, 0, 71, 79, 198158383604301823, 6143, 103, 0, 17, 127, 1087, 1151, 151, 1279, 167, 1609, 1471, 191, 199, 1663, 0, 223, 284694975049, 239, 266287972351, 131071, ...}

The correspond primes for R9 for k = 1, 2, 3, ... are: (0 if no such prime exists or this k is excluded from the conjecture)

{0, 17, 13, 0, 11, 53, 31, 71, 0, 89, 172595827849, 107, 29, 602654093, 67, 0, 19, 13121, 769, 179, 47, 197, 103, 1033121303, 0, 233, 1093, 251, 587, 269, 139, 2591, ...}

The correspond primes for R10 for k = 1, 2, 3, ... are:

{11, 19, 29, 13, 499, 59, 23, 79, 89, 11, 109, 1199999, 43, 139, 149, 53, 1699, 179, 211, 199, 2099, 73, 229, 239, 83, 25999, 269, 31, 289999, 2999, 103, 319999999999999999999999999999, ...}

[QUOTE=sweety439;461576]I tested all k<=32 for all Sierpinski/Riesel bases b<=32, for the bases with CK >= 32 and the text file already lists all k's, see the text files in the post [URL="http://mersenneforum.org/showpost.php?p=461547&postcount=340"]#340[/URL], for other bases (bases with CK < 32 or the text file only lists the k's not in CRUS), see the text files in this post. (for the exclusion of the k's for all bases b<=32, see the post [URL="http://mersenneforum.org/showpost.php?p=451367&postcount=104"]#104[/URL] (k's proven composite by full or partial algebra factors) and [URL="http://mersenneforum.org/showpost.php?p=460366&postcount=285"]#285[/URL] (k's with covering set))

The remain k's <= 32 for Sierpinski/Riesel bases b<=32 are:

S12, k = 12.
S13, k = 29.
S18, k = 18.
S22, k = 22.
S31, k = 1 and 31.
S32, k = 4 and 16.
R27, k = 23.
R29, k = 21.
R31, k = 5 and 19.
R32, k = 29.

(S12 k = 12, S18 k = 18, S22 k = 22, S32 k = 4, and S32 k = 16 are GFN's, S31 k = 1 and S31 k = 31 are half GFN's, except these bases and k's, only R32 k = 29 has gcd(k+-1,b-1) = 1, although the CK for R32 is only 10, but due to the status of R1024 k = 29 in CRUS, it has been tested to n=1M with no primes, thus, R32 k = 29 has been tested to n=2M with no primes, since the two tests are the same, 29*32^n-1 can be prime only for even n, since if n is odd, then 29*32^n-1 is divisible by 3)

Some large primes given by CRUS:

10*17^1356+1
8*23^119215+1
32*26^318071+1
12*30^1023+1
5*14^19698-1
30*23^1000-1
32*26^9812-1
25*30^34205-1

Also, the (probable) prime (10*23^3762+1)/11 was found by me, other (probable) prime with n>=1000 found by me are: (for k<=32, b<=32)

(23*16^1074+1)/3
(5*31^1026+1)/6
(13*17^1123-1)/4
(29*17^4904-1)/4

They are already in this project.[/QUOTE]

Found the (probable) prime (23*27^3742-1)/2.

But still found no (probable) prime for S13 k=29 and R29 k=21.

[QUOTE=sweety439;461750]Found the (probable) prime (23*27^3742-1)/2.

But still found no (probable) prime for S13 k=29 and R29 k=21.[/QUOTE]

No, R29 k=21 has covering set {2, 5}.

Thus, the only two remain k<=32 for Sierpinski/Riesel bases b<=32 (excluding GFN's and half GFN's) are S13 k=29 and R32 k=29. (R32 k=29 is searched to n=2M by CRUS)

[QUOTE=sweety439;461694]Do you understand this problem?

The correspond primes for S2 for k = 1, 2, 3, ... are:

{3, 5, 7, 17, 11, 13, 29, 17, 19, 41, 23, 97, 53, 29, 31, 257, 137, 37, 1217, 41, 43, 89, 47, 97, 101, 53, 109, 113, 59, 61, 7937, 257, ...}

The correspond primes for S3 for k = 1, 2, 3, ... are:

{2, 7, 5, 13, 23, 19, 11, 73, 41, 31, 17, 37, 59, 43, 23, 433, 6197, 163, 29, 61, 68891, 67, 311, 73, 113, 79, 41, 757, 131, 271, 47, 97, ...}

The correspond primes for S4 for k = 1, 2, 3, ... are:

{5, 3, 13, 17, 7, 97, 29, 11, 37, 41, 59, 193, 53, 19, 61, 257, 23, 73, 1217, 107, 337, 89, 31, 97, 101, 139, 109, 113, 619, 7681, 7937, 43, ...}

The correspond primes for S5 for k = 1, 2, 3, ... are: (0 if no such prime exists or this k is excluded from the conjecture)

{3, 11, 19, 101, 13, 31, 0, 41, 23, 251, 0, 61, 163, 71, 19, 401, 43, 2251, 2969, 101, 53, 13751, 29, 601, 313, 131, 4219, 701, 73, 151, 0, 4001, ...}

The correspond primes for S6 for k = 1, 2, 3, ... are:

{7, 13, 19, 5, 31, 37, 43, 10369, 11, 61, 67, 73, 79, 17, 541, 97, 103, 109, 23, 155521, 127, 28513, 139, 29, 151, 157, 163, 1009, 7517, 181, 1117, 193, ...}

The correspond primes for S7 for k = 1, 2, 3, ... are:

{1201, 5, 11, 29, 41, 43, 1201, 19, 529421, 71, 13, 28813, 15607, 229, 53, 113, 139, 127, 67, 47, 6506482422146989923806032135894782377546749491930607652617339526616854149081491252134508288317638363981211, 7547, 154688827, 8233, 613, 61, 26693911031, 197, 7870665723233837, 211, 109, 523, ...}

The correspond primes for S8 for k = 1, 2, 3, ... are: (0 if no such prime exists or this k is excluded from the conjecture)

{0, 17, 193, 257, 41, 7, 449, 0, 73, 641, 89, 97, 7607, 113, 7681, 65537, 137, 1153, 1217, 23, 10753, 1409, 11777, 193, 1601, 13313, 0, 114689, 233, 241, 35740566642812256257, 257, ...}

The correspond primes for S9 for k = 1, 2, 3, ... are: (0 if no such prime exists or this k is excluded from the conjecture)

{5, 19, 7, 37, 23, 487, 71, 73, 41, 811, 223, 109, 59, 127, 17, 1297, 6197, 163, 43, 181, 68891, 199, 233, 17497, 113, 13817467, 61, 2269, 131, 271, 0, 2593, ...}

The correspond primes for S10 for k = 1, 2, 3, ... are:

{11, 7, 31, 41, 17, 61, 71, 89, 9001, 101, 37, 1201, 131, 47, 151, 1601, 19, 181, 191, 67, 211, 22000001, 76667, 241, 251, 29, 271, 281, 97, 3001, 311, 107, ...}

The correspond primes for R2 for k = 1, 2, 3, ... are:

{3, 3, 5, 7, 19, 11, 13, 31, 17, 19, 43, 23, 103, 223, 29, 31, 67, 71, 37, 79, 41, 43, 367, 47, 199, 103, 53, 223, 463, 59, 61, 127, ...}

The correspond primes for R3 for k = 1, 2, 3, ... are:

{13, 5, 13, 11, 7, 17, 31, 23, 13, 29, 172595827849, 107, 19, 41, 67, 47, 229, 53, 769, 59, 31, 197, 103, 71, 37, 233, 1093, 83, 43, 89, 139, 863, ...}

The correspond primes for R4 for k = 1, 2, 3, ... are: (0 if no such prime exists or this k is excluded from the conjecture)

{0, 7, 11, 0, 19, 23, 37, 31, 0, 13, 43, 47, 17, 223, 59, 0, 67, 71, 101, 79, 83, 29, 367, 383, 0, 103, 107, 37, 463, 479, 41, 127, ...}

The correspond primes for R5 for k = 1, 2, 3, ... are: (0 if no such prime exists or this k is excluded from the conjecture)

{31, 1249, 7, 19, 31, 29, 17, 199, 11, 1249, 137, 59, 0, 349, 37, 79, 0, 89, 47, 499, 131, 109, 7187, 599, 31, 16249, 67, 139, 181, 149, 48437, 12499999, ...}

The correspond primes for R6 for k = 1, 2, 3, ... are:

{7, 11, 17, 23, 29, 7, 41, 47, 53, 59, 13, 71, 467, 83, 89, 19, 101, 107, 113, 719, 151, 131, 137, 863, 149, 31, 971, 167, 173, 179, 37, 191, ...}

The correspond primes for R7 for k = 1, 2, 3, ... are:

{2801, 13, 73, 457, 17, 41, 2801, 19207, 31, 23, 269, 83, 743, 97, 367, 37, 59, 881, 7603, 139, 73, 359, 563, 167, 29, 181, 661, 457, 101, 10289, 8413470255870653, 223, ...}

The correspond primes for R8 for k = 1, 2, 3, ... are: (0 if no such prime exists or this k is excluded from the conjecture)

{0, 127, 23, 31, 20479, 47, 3583, 0, 71, 79, 198158383604301823, 6143, 103, 0, 17, 127, 1087, 1151, 151, 1279, 167, 1609, 1471, 191, 199, 1663, 0, 223, 284694975049, 239, 266287972351, 131071, ...}

The correspond primes for R9 for k = 1, 2, 3, ... are: (0 if no such prime exists or this k is excluded from the conjecture)

{0, 17, 13, 0, 11, 53, 31, 71, 0, 89, 172595827849, 107, 29, 602654093, 67, 0, 19, 13121, 769, 179, 47, 197, 103, 1033121303, 0, 233, 1093, 251, 587, 269, 139, 2591, ...}

The correspond primes for R10 for k = 1, 2, 3, ... are:

{11, 19, 29, 13, 499, 59, 23, 79, 89, 11, 109, 1199999, 43, 139, 149, 53, 1699, 179, 211, 199, 2099, 73, 229, 239, 83, 25999, 269, 31, 289999, 2999, 103, 319999999999999999999999999999, ...}[/QUOTE]

There are k's with a prime, but since these k's can still have this prime because of the algebra factors, thus these k's are excluded from the conjectures.

Such k<=32 for Sierpinski/Riesel bases b<=10 are:

S8 k = 27: The prime is 31, but k = 27 is excluded from S8.
R4 k = 1: The prime is 5, but k = 1 is excluded from R4.
R8 k = 1: The prime is 73, but k = 1 is excluded from R8.
R8 k = 8: The prime is 73, but k = 1 is excluded from R8.

[QUOTE=sweety439;461753]No, R29 k=21 has covering set {2, 5}.

Thus, the only two remain k<=32 for Sierpinski/Riesel bases b<=32 (excluding GFN's and half GFN's) are S13 k=29 and R32 k=29. (R32 k=29 is searched to n=2M by CRUS)[/QUOTE]

(29*13^10574+1)/6 is (probable) prime!!!

Now, the only remain k<=32 for Sierpinski/Riesel bases b<=32 is R32 k=29, and it has been tested to n=2M without finding any prime.

Update the current text files.

The top primes are: (only sorted by n)

[CODE]
base k n
R32 29 ? (>2M)
S26 32 318071
S23 8 119215
R30 25 34205
R14 5 19698
S13 29 10574
R26 32 9812
R17 29 4904
S23 10 3762
R27 23 3742
S17 10 1356
R17 13 1123
S16 23 1074
S31 5 1026
S30 12 1023
R23 30 1000
[/CODE]

[QUOTE=sweety439;461694]Do you understand this problem?

The correspond primes for S2 for k = 1, 2, 3, ... are:

{3, 5, 7, 17, 11, 13, 29, 17, 19, 41, 23, 97, 53, 29, 31, 257, 137, 37, 1217, 41, 43, 89, 47, 97, 101, 53, 109, 113, 59, 61, 7937, 257, ...}

The correspond primes for S3 for k = 1, 2, 3, ... are:

{2, 7, 5, 13, 23, 19, 11, 73, 41, 31, 17, 37, 59, 43, 23, 433, 6197, 163, 29, 61, 68891, 67, 311, 73, 113, 79, 41, 757, 131, 271, 47, 97, ...}

The correspond primes for S4 for k = 1, 2, 3, ... are:

{5, 3, 13, 17, 7, 97, 29, 11, 37, 41, 59, 193, 53, 19, 61, 257, 23, 73, 1217, 107, 337, 89, 31, 97, 101, 139, 109, 113, 619, 7681, 7937, 43, ...}

The correspond primes for S5 for k = 1, 2, 3, ... are: (0 if no such prime exists or this k is excluded from the conjecture)

{3, 11, 19, 101, 13, 31, 0, 41, 23, 251, 0, 61, 163, 71, 19, 401, 43, 2251, 2969, 101, 53, 13751, 29, 601, 313, 131, 4219, 701, 73, 151, 0, 4001, ...}

The correspond primes for S6 for k = 1, 2, 3, ... are:

{7, 13, 19, 5, 31, 37, 43, 10369, 11, 61, 67, 73, 79, 17, 541, 97, 103, 109, 23, 155521, 127, 28513, 139, 29, 151, 157, 163, 1009, 7517, 181, 1117, 193, ...}

The correspond primes for S7 for k = 1, 2, 3, ... are:

{1201, 5, 11, 29, 41, 43, 1201, 19, 529421, 71, 13, 28813, 15607, 229, 53, 113, 139, 127, 67, 47, 6506482422146989923806032135894782377546749491930607652617339526616854149081491252134508288317638363981211, 7547, 154688827, 8233, 613, 61, 26693911031, 197, 7870665723233837, 211, 109, 523, ...}

The correspond primes for S8 for k = 1, 2, 3, ... are: (0 if no such prime exists or this k is excluded from the conjecture)

{0, 17, 193, 257, 41, 7, 449, 0, 73, 641, 89, 97, 7607, 113, 7681, 65537, 137, 1153, 1217, 23, 10753, 1409, 11777, 193, 1601, 13313, 0, 114689, 233, 241, 35740566642812256257, 257, ...}

The correspond primes for S9 for k = 1, 2, 3, ... are: (0 if no such prime exists or this k is excluded from the conjecture)

{5, 19, 7, 37, 23, 487, 71, 73, 41, 811, 223, 109, 59, 127, 17, 1297, 6197, 163, 43, 181, 68891, 199, 233, 17497, 113, 13817467, 61, 2269, 131, 271, 0, 2593, ...}

The correspond primes for S10 for k = 1, 2, 3, ... are:

{11, 7, 31, 41, 17, 61, 71, 89, 9001, 101, 37, 1201, 131, 47, 151, 1601, 19, 181, 191, 67, 211, 22000001, 76667, 241, 251, 29, 271, 281, 97, 3001, 311, 107, ...}

The correspond primes for R2 for k = 1, 2, 3, ... are:

{3, 3, 5, 7, 19, 11, 13, 31, 17, 19, 43, 23, 103, 223, 29, 31, 67, 71, 37, 79, 41, 43, 367, 47, 199, 103, 53, 223, 463, 59, 61, 127, ...}

The correspond primes for R3 for k = 1, 2, 3, ... are:

{13, 5, 13, 11, 7, 17, 31, 23, 13, 29, 172595827849, 107, 19, 41, 67, 47, 229, 53, 769, 59, 31, 197, 103, 71, 37, 233, 1093, 83, 43, 89, 139, 863, ...}

The correspond primes for R4 for k = 1, 2, 3, ... are: (0 if no such prime exists or this k is excluded from the conjecture)

{0, 7, 11, 0, 19, 23, 37, 31, 0, 13, 43, 47, 17, 223, 59, 0, 67, 71, 101, 79, 83, 29, 367, 383, 0, 103, 107, 37, 463, 479, 41, 127, ...}

The correspond primes for R5 for k = 1, 2, 3, ... are: (0 if no such prime exists or this k is excluded from the conjecture)

{31, 1249, 7, 19, 31, 29, 17, 199, 11, 1249, 137, 59, 0, 349, 37, 79, 0, 89, 47, 499, 131, 109, 7187, 599, 31, 16249, 67, 139, 181, 149, 48437, 12499999, ...}

The correspond primes for R6 for k = 1, 2, 3, ... are:

{7, 11, 17, 23, 29, 7, 41, 47, 53, 59, 13, 71, 467, 83, 89, 19, 101, 107, 113, 719, 151, 131, 137, 863, 149, 31, 971, 167, 173, 179, 37, 191, ...}

The correspond primes for R7 for k = 1, 2, 3, ... are:

{2801, 13, 73, 457, 17, 41, 2801, 19207, 31, 23, 269, 83, 743, 97, 367, 37, 59, 881, 7603, 139, 73, 359, 563, 167, 29, 181, 661, 457, 101, 10289, 8413470255870653, 223, ...}

The correspond primes for R8 for k = 1, 2, 3, ... are: (0 if no such prime exists or this k is excluded from the conjecture)

{0, 127, 23, 31, 20479, 47, 3583, 0, 71, 79, 198158383604301823, 6143, 103, 0, 17, 127, 1087, 1151, 151, 1279, 167, 1609, 1471, 191, 199, 1663, 0, 223, 284694975049, 239, 266287972351, 131071, ...}

The correspond primes for R9 for k = 1, 2, 3, ... are: (0 if no such prime exists or this k is excluded from the conjecture)

{0, 17, 13, 0, 11, 53, 31, 71, 0, 89, 172595827849, 107, 29, 602654093, 67, 0, 19, 13121, 769, 179, 47, 197, 103, 1033121303, 0, 233, 1093, 251, 587, 269, 139, 2591, ...}

The correspond primes for R10 for k = 1, 2, 3, ... are:

{11, 19, 29, 13, 499, 59, 23, 79, 89, 11, 109, 1199999, 43, 139, 149, 53, 1699, 179, 211, 199, 2099, 73, 229, 239, 83, 25999, 269, 31, 289999, 2999, 103, 319999999999999999999999999999, ...}[/QUOTE]

The forms for the k's and the bases are:

S2:

k = 1: 1*2^n+1
k = 2: 2*2^n+1
k = 3: 3*2^n+1
k = 4: 4*2^n+1
k = 5: 5*2^n+1
k = 6: 6*2^n+1
k = 7: 7*2^n+1
k = 8: 8*2^n+1
...
k*2^n+1 for all k

S3:

k = 1: (1*3^n+1)/2
k = 2: 2*3^n+1
k = 3: (3*3^n+1)/2
k = 4: 4*3^n+1
k = 5: (5*3^n+1)/2
k = 6: 6*3^n+1
k = 7: (7*3^n+1)/2
k = 8: 8*3^n+1
...
k*3^n+1 for all k = 0 (mod 2)
(k*3^n+1)/2 for all k = 1 (mod 2)

S4:

k = 1: 1*4^n+1
k = 2: (2*4^n+1)/3
k = 3: 3*4^n+1
k = 4: 4*4^n+1
k = 5: (5*4^n+1)/3
k = 6: 6*4^n+1
k = 7: 7*4^n+1
k = 8: (8*4^n+1)/3
...
k*4^n+1 for all k = 0, 1 (mod 3)
(k*4^n+1)/3 for all k = 2 (mod 3)

S5:

k = 1: (1*5^n+1)/2
k = 2: 2*5^n+1
k = 3: (3*5^n+1)/4
k = 4: 4*5^n+1
k = 5: (5*5^n+1)/2
k = 6: 6*5^n+1
k = 7: (7*5^n+1)/4
k = 8: 8*5^n+1
...
k*5^n+1 for all k = 0, 2 (mod 4)
(k*5^n+1)/2 for all k = 1 (mod 4)
(k*5^n+1)/4 for all k = 3 (mod 4)

S6:

k = 1: 1*6^n+1
k = 2: 2*6^n+1
k = 3: 3*6^n+1
k = 4: (4*6^n+1)/5
k = 5: 5*6^n+1
k = 6: 6*6^n+1
k = 7: 7*6^n+1
k = 8: 8*6^n+1
k = 9: (9*6^n+1)/5
k = 10: 10*6^n+1
k = 11: 11*6^n+1
k = 12: 12*6^n+1
...
k*6^n+1 for all k = 0, 1, 2, 3 (mod 5)
(k*6^n+1)/5 for all k = 4 (mod 5)

S7:

k = 1: (1*7^n+1)/2
k = 2: (2*7^n+1)/3
k = 3: (3*7^n+1)/2
k = 4: 4*7^n+1
k = 5: (5*7^n+1)/6
k = 6: 6*7^n+1
k = 7: (7*7^n+1)/2
k = 8: (8*7^n+1)/3
k = 9: (9*7^n+1)/2
k = 10: 10*7^n+1
k = 11: (11*7^n+1)/6
k = 12: 12*7^n+1
...
k*7^n+1 for all k = 0, 4 (mod 6)
(k*7^n+1)/2 for all k = 1, 3 (mod 6)
(k*7^n+1)/3 for all k = 2 (mod 6)
(k*7^n+1)/6 for all k = 5 (mod 6)

R2:

k = 1: 1*2^n-1
k = 2: 2*2^n-1
k = 3: 3*2^n-1
k = 4: 4*2^n-1
k = 5: 5*2^n-1
k = 6: 6*2^n-1
k = 7: 7*2^n-1
k = 8: 8*2^n-1
...
k*2^n-1 for all k

R3:

k = 1: (1*3^n-1)/2
k = 2: 2*3^n-1
k = 3: (3*3^n-1)/2
k = 4: 4*3^n-1
k = 5: (5*3^n-1)/2
k = 6: 6*3^n-1
k = 7: (7*3^n-1)/2
k = 8: 8*3^n-1
...
k*3^n-1 for all k = 0 (mod 2)
(k*3^n-1)/2 for all k = 1 (mod 2)

R4:

k = 1: (1*4^n-1)/3
k = 2: 2*4^n-1
k = 3: 3*4^n-1
k = 4: (4*4^n-1)/3
k = 5: 5*4^n-1
k = 6: 6*4^n-1
k = 7: (7*4^n-1)/3
k = 8: 8*4^n-1
...
k*4^n-1 for all k = 0, 2 (mod 3)
(k*4^n-1)/3 for all k = 1 (mod 3)

R5:

k = 1: (1*5^n-1)/4
k = 2: 2*5^n-1
k = 3: (3*5^n-1)/2
k = 4: 4*5^n-1
k = 5: (5*5^n-1)/4
k = 6: 6*5^n-1
k = 7: (7*5^n-1)/2
k = 8: 8*5^n-1
...
k*5^n-1 for all k = 0, 2 (mod 4)
(k*5^n-1)/2 for all k = 3 (mod 4)
(k*5^n-1)/4 for all k = 1 (mod 4)

R6:

k = 1: (1*6^n-1)/5
k = 2: 2*6^n-1
k = 3: 3*6^n-1
k = 4: 4*6^n-1
k = 5: 5*6^n-1
k = 6: (6*6^n-1)/5
k = 7: 7*6^n-1
k = 8: 8*6^n-1
k = 9: 9*6^n-1
k = 10: 10*6^n-1
k = 11: (11*6^n-1)/5
k = 12: 12*6^n-1
...
k*6^n-1 for all k = 0, 2, 3, 4 (mod 5)
(k*6^n-1)/5 for all k = 1 (mod 5)

R7:

k = 1: (1*7^n-1)/6
k = 2: 2*7^n-1
k = 3: (3*7^n-1)/2
k = 4: (4*7^n-1)/3
k = 5: (5*7^n-1)/2
k = 6: 6*7^n-1
k = 7: (7*7^n-1)/6
k = 8: 8*7^n-1
k = 9: (9*7^n-1)/2
k = 10: (10*7^n-1)/3
k = 11: (11*7^n-1)/2
k = 12: 12*7^n-1
...
k*7^n-1 for all k = 0, 2 (mod 6)
(k*7^n-1)/2 for all k = 3, 5 (mod 6)
(k*7^n-1)/3 for all k = 4 (mod 6)
(k*7^n-1)/6 for all k = 1 (mod 6)

Why some forms can only contain one prime? Or contain no prime?

The reason is: (see these examples)

Example 1: R4 k=1, this form is (1*4^n-1)/3

(1*4^n-1)/3 = ((1*2^n-1)/3) * (1*2^n+1) for even n
(1*4^n-1)/3 = (1*2^n-1) * ((1*2^n+1)/3) for odd n

If (1*4^n-1)/3 is prime, then one of the two factors must be 1, thus, the only prime candidate is n=2.

Example 2: R8 k=1, this form is (1*8^n-1)/7

(1*8^n-1)/7 = ((1*2^n-1)/7) * (1*4^n+1*2^n+1) for n divisible by 3
(1*8^n-1)/7 = (1*2^n-1) * ((1*4^n+1*2^n+1)/7) for n not divisible by 3

If (1*8^n-1)/7 is prime, then one of the two factors must be 1, thus, the only prime candidate is n=3.

Example 3: R9 k=1, this form is (1*9^n-1)/8

(1*9^n-1)/8 = ((1*3^n-1)/4) * ((1*3^n+1)/2) for even n
(1*9^n-1)/8 = ((1*3^n-1)/2) * ((1*3^n+1)/4) for odd n

If (1*9^n-1)/8 is prime, then one of the two factors must be 1, thus, the only prime candidate is n=2. However, (1*9^2-1)/8 = 10 is not prime, thus, there is no prime of the form (1*9^n-1)/8.

Example 4: S8 k=27, this form is (27*8^n+1)/7

(27*8^n+1)/7 = ((3*2^n+1)/7) * (9*4^n-3*2^n+1) for n = 1 (mod 3)
(27*8^n+1)/7 = (3*2^n+1) * ((9*4^n-3*2^n+1)/7) for n = 0, 2 (mod 3)

If (27*8^n+1)/7 is prime, then one of the two factors must be 1, thus, the only prime candidate is n=1.

Update the full text file for R28.

[QUOTE=sweety439;461788]The forms for the k's and the bases are:

S2:

k = 1: 1*2^n+1
k = 2: 2*2^n+1
k = 3: 3*2^n+1
k = 4: 4*2^n+1
k = 5: 5*2^n+1
k = 6: 6*2^n+1
k = 7: 7*2^n+1
k = 8: 8*2^n+1
...
k*2^n+1 for all k

S3:

k = 1: (1*3^n+1)/2
k = 2: 2*3^n+1
k = 3: (3*3^n+1)/2
k = 4: 4*3^n+1
k = 5: (5*3^n+1)/2
k = 6: 6*3^n+1
k = 7: (7*3^n+1)/2
k = 8: 8*3^n+1
...
k*3^n+1 for all k = 0 (mod 2)
(k*3^n+1)/2 for all k = 1 (mod 2)

S4:

k = 1: 1*4^n+1
k = 2: (2*4^n+1)/3
k = 3: 3*4^n+1
k = 4: 4*4^n+1
k = 5: (5*4^n+1)/3
k = 6: 6*4^n+1
k = 7: 7*4^n+1
k = 8: (8*4^n+1)/3
...
k*4^n+1 for all k = 0, 1 (mod 3)
(k*4^n+1)/3 for all k = 2 (mod 3)

S5:

k = 1: (1*5^n+1)/2
k = 2: 2*5^n+1
k = 3: (3*5^n+1)/4
k = 4: 4*5^n+1
k = 5: (5*5^n+1)/2
k = 6: 6*5^n+1
k = 7: (7*5^n+1)/4
k = 8: 8*5^n+1
...
k*5^n+1 for all k = 0, 2 (mod 4)
(k*5^n+1)/2 for all k = 1 (mod 4)
(k*5^n+1)/4 for all k = 3 (mod 4)

S6:

k = 1: 1*6^n+1
k = 2: 2*6^n+1
k = 3: 3*6^n+1
k = 4: (4*6^n+1)/5
k = 5: 5*6^n+1
k = 6: 6*6^n+1
k = 7: 7*6^n+1
k = 8: 8*6^n+1
k = 9: (9*6^n+1)/5
k = 10: 10*6^n+1
k = 11: 11*6^n+1
k = 12: 12*6^n+1
...
k*6^n+1 for all k = 0, 1, 2, 3 (mod 5)
(k*6^n+1)/5 for all k = 4 (mod 5)

S7:

k = 1: (1*7^n+1)/2
k = 2: (2*7^n+1)/3
k = 3: (3*7^n+1)/2
k = 4: 4*7^n+1
k = 5: (5*7^n+1)/6
k = 6: 6*7^n+1
k = 7: (7*7^n+1)/2
k = 8: (8*7^n+1)/3
k = 9: (9*7^n+1)/2
k = 10: 10*7^n+1
k = 11: (11*7^n+1)/6
k = 12: 12*7^n+1
...
k*7^n+1 for all k = 0, 4 (mod 6)
(k*7^n+1)/2 for all k = 1, 3 (mod 6)
(k*7^n+1)/3 for all k = 2 (mod 6)
(k*7^n+1)/6 for all k = 5 (mod 6)

R2:

k = 1: 1*2^n-1
k = 2: 2*2^n-1
k = 3: 3*2^n-1
k = 4: 4*2^n-1
k = 5: 5*2^n-1
k = 6: 6*2^n-1
k = 7: 7*2^n-1
k = 8: 8*2^n-1
...
k*2^n-1 for all k

R3:

k = 1: (1*3^n-1)/2
k = 2: 2*3^n-1
k = 3: (3*3^n-1)/2
k = 4: 4*3^n-1
k = 5: (5*3^n-1)/2
k = 6: 6*3^n-1
k = 7: (7*3^n-1)/2
k = 8: 8*3^n-1
...
k*3^n-1 for all k = 0 (mod 2)
(k*3^n-1)/2 for all k = 1 (mod 2)

R4:

k = 1: (1*4^n-1)/3
k = 2: 2*4^n-1
k = 3: 3*4^n-1
k = 4: (4*4^n-1)/3
k = 5: 5*4^n-1
k = 6: 6*4^n-1
k = 7: (7*4^n-1)/3
k = 8: 8*4^n-1
...
k*4^n-1 for all k = 0, 2 (mod 3)
(k*4^n-1)/3 for all k = 1 (mod 3)

R5:

k = 1: (1*5^n-1)/4
k = 2: 2*5^n-1
k = 3: (3*5^n-1)/2
k = 4: 4*5^n-1
k = 5: (5*5^n-1)/4
k = 6: 6*5^n-1
k = 7: (7*5^n-1)/2
k = 8: 8*5^n-1
...
k*5^n-1 for all k = 0, 2 (mod 4)
(k*5^n-1)/2 for all k = 3 (mod 4)
(k*5^n-1)/4 for all k = 1 (mod 4)

R6:

k = 1: (1*6^n-1)/5
k = 2: 2*6^n-1
k = 3: 3*6^n-1
k = 4: 4*6^n-1
k = 5: 5*6^n-1
k = 6: (6*6^n-1)/5
k = 7: 7*6^n-1
k = 8: 8*6^n-1
k = 9: 9*6^n-1
k = 10: 10*6^n-1
k = 11: (11*6^n-1)/5
k = 12: 12*6^n-1
...
k*6^n-1 for all k = 0, 2, 3, 4 (mod 5)
(k*6^n-1)/5 for all k = 1 (mod 5)

R7:

k = 1: (1*7^n-1)/6
k = 2: 2*7^n-1
k = 3: (3*7^n-1)/2
k = 4: (4*7^n-1)/3
k = 5: (5*7^n-1)/2
k = 6: 6*7^n-1
k = 7: (7*7^n-1)/6
k = 8: 8*7^n-1
k = 9: (9*7^n-1)/2
k = 10: (10*7^n-1)/3
k = 11: (11*7^n-1)/2
k = 12: 12*7^n-1
...
k*7^n-1 for all k = 0, 2 (mod 6)
(k*7^n-1)/2 for all k = 3, 5 (mod 6)
(k*7^n-1)/3 for all k = 4 (mod 6)
(k*7^n-1)/6 for all k = 1 (mod 6)[/QUOTE]

Sierpinski problem base b: Find and prove the smallest k>=1 such that (k*b^n+1)/d is not prime for all n, where d is the largest number dividing k*b^n+1 for all n.
Riesel problem base b: Find and prove the smallest k>=1 such that (k*b^n-1)/d is not prime for all n, where d is the largest number dividing k*b^n-1 for all n.