Now, all extended Sierpinski/Riesel bases b<=36 were completely started or started to at least k=10000. Besides, all started k's for all extended Sierpinski/Riesel bases b<=36 were tested to at least n=1000.

S33, k=67, 203, 319, 407 and R33, k=257, 339 and S36, k=1814 are likely tested to n=5000, no (probable) prime found. Note that S36, k=1814 testing is the same as S6 for the same k-value.

There may be full algebra factors or partial algebra factors of (k*b^n+-1)/gcd(k+-1,b-1).

In fact, full algebra factors only occurs in these situations:

(1) Riesel case, there is r>1 such that both b and k are perfect r-th powers.
(2) Sierpinski case, there is odd r>1 such that both b and k are perfect r-th powers.
(3) Sierpinski case, b is a perfect 4th power, and k is of the form 4*m^4.

If (k*b^n+-1)/gcd(k+-1,b-1) has partial algebra factors, then only this n's can have algebra factors: (there must be covering set for all other n's)

(1) Riesel case, k*b^n is a perfect power.
(2) Sierpinski case, k*b^n is of the form m^r with odd r>1.
(3) Sierpinski case, k*b^n is of the form 4*m^4.

Some bases 37<=b<=64 were also completed, see the text files.

In these files, "algebra" means this k has full/partial algebra factors, "NA" means this k has no known (probable) prime.

All k's in these bases are tested to at least n=1000.

Some primes are given by CRUS.

The (probable) prime (1*51^4229-1)/50 (R51, k=1) is given by [URL]http://oeis.org/A084740[/URL] (or [URL]http://www.fermatquotient.com/PrimSerien/GenRepu.txt[/URL]).

These files does not include bases SR2, SR3, SR6, SR15, SR22, SR24, SR28, R30, R36, SR40, SR42, SR46, SR48, SR52, SR58, SR60 and SR63. Usually, if b+1 is a prime or prime power, then base b has larger conjectured k.

Algebraic factors for bases 33<=b<=64:

S64: all k=m^3 has full algebra factors.
R33: all k=m^2 and m = 4 or 13 mod 17 and all k=m^2 and m = 15 or 17 mod 32 and all k=33*m^2 and m = 4 or 13 mod 17 and all k=33*m^2 and m = 15 or 17 mod 32 has partial algebra factors.
R34: all k=m^2 and m = 2 or 3 mod 5 and all k=34*m^2 and m = 2 or 3 mod 5 has partial algebra factors.
R36: all k=m^2 has full algebra factors.
R38: all k=m^2 and m = 5 or 8 mod 13 and all k=38*m^2 and m = 5 or 8 mod 13 has partial algebra factors.
R39: all k=m^2 and m = 2 or 3 mod 5 and all k=39*m^2 and m = 2 or 3 mod 5 has partial algebra factors.
R40: all k=m^2 and m = 9 or 32 mod 41 and all k=10*m^2 and m = 18 or 23 mod 41 has partial algebra factors.
R44: all k=m^2 and m = 2 or 3 mod 5 and all k=11*m^2 and m = 1 or 4 mod 5 has partial algebra factors.
R49: all k=m^2 has full algebra factors.
R50: all k=m^2 and m = 4 or 13 mod 17 and all k=2*m^2 and m = 3 or 14 mod 17 has partial algebra factors.
R51: all k=m^2 and m = 5 or 8 mod 13 and all k=51*m^2 and m = 5 or 8 mod 13 has partial algebra factors.
R52: all k=m^2 and m = 23 or 30 mod 53 and all k=13*m^2 and m = 7 or 46 mod 53 has partial algebra factors.
R54: all k=m^2 and m = 2 or 3 mod 5 and all k=6*m^2 and m = 1 or 4 mod 5 has partial algebra factors.
R57: all k=m^2 and m = 12 or 17 mod 29 and all k=m^2 and m = 3 or 5 mod 16 and all k=57*m^2 and m = 12 or 17 mod 29 and all k=57*m^2 and m = 3 or 5 mod 16 has partial algebra factors.
R59: all k=m^2 and m = 2 or 3 mod 5 and all k=59*m^2 and m = 2 or 3 mod 5 has partial algebra factors.
R60: all k=m^2 and m = 11 or 50 mod 61 and all k=15*m^2 and m = 22 or 39 mod 61 has partial algebra factors.
R64: all k=m^2 and all k=m^3 has full algebra factors.

(for the algebra factors for bases <= 32, see post [URL="http://mersenneforum.org/showpost.php?p=451367&postcount=104"]#104)[/URL]

The remain k with no known (probable) prime in these bases are:

Sierpinski bases:

[CODE]
base remain k
S4 none
S5 none
S7 none
S8 none
S9 none
S10 100, 269
S11 none
S12 12, (144)
S13 none
S14 none
S16 none
S17 none
S18 18, (324)
S19 none
S20 none
S21 none
S23 none
S25 71
S26 65, 155
S27 none
S29 none
S30 278, 588
S31 1, (31), 43, 51, 73, 77, 107, 117, 149, 181, 189, 209
S32 4
S33 67, 203, 319, 407
S34 none
S35 none
S36 1296, 1814
S37 19, 37
S38 1
S39 none
S41 none
S43 none
S44 none
S45 none
S47 none
S49 none
S50 1
S51 none
S53 4
S54 none
S55 1
S56 none
S57 none
S59 none
S61 23, 43, 62
S62 1
S64 11
[/CODE]

Riesel bases:

[CODE]
base remain k
R4 none
R5 none
R7 197
R8 none
R9 none
R10 none
R11 none
R12 none
R13 none
R14 none
R16 none
R17 none
R18 none
R19 none
R20 none
R21 none
R23 none
R25 none
R26 none
R27 none
R29 none
R31 5, 19, 51, 73, 97
R32 none
R33 257, 339
R34 none
R35 none
R37 none
R38 none
R39 none
R41 none
R43 13
R44 none
R45 none
R47 none
R49 none
R50 none
R51 none
R53 none
R54 none
R55 none
R56 none
R57 none
R59 none
R61 10, 13, 37, 53, 77, 100
R62 none
R64 none
[/CODE]

Completed extended Sierpinski/Riesel conjectures to base 46 and base 58, tested to n=1000.

These are the text files for the status.

k's remain at n=1000 for these bases:

S46: {17, 140, 278, 283, 347, 563, 619, 729, 845} (k=782 is included in the conjecture but excluded from testing, since it will have the same prime as k=17)
S58: {20, 21, 58, 106, 122, 176, 178, 222, 228, 241, 266, 296, 297, 392, 393, 431, 437, 461}
R46: {86, 93, 100, 281, 338, 386, 436, 561, 576, 800, 870}
R58: {71, 130, 169, 176, 178, 312, 319, 382, 400, 421, 456, 473, 487, 493, 499}

Reserve extended Sierpinski/Riesel conjectures to base 63, also test to n=1000.

Extended SR63 were done, tested to n=1000.

These are the text files for the status.

k's remain at n=1000 for these bases:

S63: {1, 9, 83, 101, 103, 113, 133, 143, 185, 223, 237, 267, 283, 307, 309, 335, 343, 365, 367, 381, 391, 411, 425, 463, 467, 471, 487, 509, 549, 581, 587, 603, 605, 637, 643, 645, 673, 677, 681, 687, 689, 701, 789, 803, 807, 821, 825, 827, 881, 888, 903, 937, 951, 963, 983, 989, 1021, 1027, 1043, 1047, 1049, 1063, 1067, 1103, 1108, 1121, 1174, 1189, 1201, 1207, 1263, 1267, 1283, 1321, 1341, 1367, 1401, 1433, 1461, 1463, 1467, 1481, 1523, 1553, 1563, 1581} (k=63 and k=567 are included in the conjecture but excluded from testing, since it will have the same prime as k=1 and k=9)
R63: {37, 64, 65, 93, 129, 139, 177, 211, 231, 237, 251, 271, 281, 291, 333, 372, 417, 457, 471, 473, 491, 493, 497, 513, 587, 599, 633, 669, 677, 679, 687, 691, 695, 717, 733, 771, 817, 819, 821, 831, 853}

If you don't know the CK for the extended Sierpinski/Riesel bases, there is a text file for them.

SR48 were also done to n=1000, I only tested the k = 46 (or 1) mod 47, since other k's are already in CRUS.

Both sides have one k remain at n=1000 for k = 46 (or 1) mod 47: k=563 for Sierpinski, and k=1599 for Riesel.

The full list of the remain k's for extended S48: {29, 36, 62, 153, 561, 563, 622, 701, 937, 1077, 1086, 1114, 1121, 1168}

The full list of the remain k's for extended R48: {313, 384, 708, 909, 916, 1093, 1457, 1599, 1686, 1877, 1896, 1898, 2071, 2148, 2172, 2402, 2589, 2682, 2927, 2939, 3044, 3067}

Note that in extend Sierpinski/Riesel problem for base 48, the conjectured k's are the same as the conjectured k's for the original Sierpinski/Riesel problem for the same base. Thus, for base 48 (but not for all bases), the extend Sierpinski/Riesel problem covers the original Sierpinski/Riesel problem.

Add the extended SR2 text files for k<=10000 (tested to n=5000). Note that the extended SR2 problems are completely the same as the original SR2 problems.

Now, for all Sierpinski/Riesel bases b<=64, only these bases are completely not started:

SR40, SR42, SR52, SR60.

Besides, only these bases are partial started:

SR15, R36.