The proven extended Sierpinski/Riesel conjectures for bases b<=32 are:

[CODE]
Base CK
S4 419
S5 7
S7 209
S8 47
S9 31
S11 5
S12 521 (only the weak conjecture is proven)
S13 15
S14 4
S16 38
S17 31
S18 398 (only the weak conjecture is proven)
S19 9
S20 8
S21 23
S23 5
S27 13
S29 4
S32 10 (only the weak conjecture is proven)
R4 361
R5 13
R8 14
R9 41
R10 334
R11 5
R12 376
R13 29
R14 4
R16 100
R17 49 (with probable primes that have not been certified)
R18 246
R19 9
R20 8
R21 45
R23 5
R25 105
R26 149
R27 13
R29 4
R32 10
[/CODE]

The largest 5 (probable) primes for proven extended Sierpinski/Riesel bases b<=32 are: (only sorted by exponent) (k's that are multiple of b and where the number ((k+1)/gcd(k+1,b-1) for extended Sierpinski conjectures, (k-1)/gcd(k-1,b-1) for extended Riesel conjectures) is not prime are included in the conjectures but excluded from testing, since such k's will have the same prime as k / b. Thus, these k's should not be in the top5 prime lists)

[CODE]
S4:

186 (10458)
94 (291)
176 (228)
129 (207)
89 (167)

S5:

4 (2)
3 (2)
6 (1)
5 (1)
2 (1)

S7:

141 (1044)
121 (252)
101 (216)
21 (124)
181 (80)

S8: (note that we exclude k=27 for S8, although it has a prime, but it has algebra factors and it can has only this prime. Thus, k=27 should not be in the top5 prime lists)

31 (20)
46 (4)
40 (4)
37 (4)
28 (4)

S9:

26 (6)
21 (4)
24 (3)
17 (3)
28 (2)

S11:

4 (2)
1 (2)
3 (1)
2 (1)

S12:

404 (714558)
378 (2388)
261 (644)
407 (367)
354 (291)

S13:

11 (564)
8 (4)
13 (3)
3 (2)
2 (2)

S14:

1 (2)
3 (1)
2 (1)

S16: (note that we exclude k=4 for S16, although it has a prime, but it has algebra factors and it can has only this prime. Thus, k=4 should not be in the top5 prime lists)

23 (1074)
33 (7)
35 (4)
18 (4)
10 (3)

S17:

10 (1356)
7 (190)
2 (47)
29 (41)
20 (13)

S18:

122 (292318)
381 (24108)
291 (2415)
37 (457)
362 (258)

S19:

5 (78)
6 (14)
4 (3)
1 (2)
8 (1)

S20:

6 (15)
7 (2)
4 (2)
1 (2)
5 (1)

S21:

12 (10)
21 (3)
19 (2)
11 (2)
8 (2)

S23:

4 (342)
1 (4)
3 (3)
2 (1)

S27:

9 (10)
7 (3)
12 (2)
5 (2)
2 (2)

S29:

3 (2)
1 (2)
2 (1)

S32:

9 (13)
7 (4)
5 (3)
2 (3)
8 (1)

R4: (note that we exclude k=1 for R4, although it has a prime, but it has algebra factors and it can has only this prime. Thus, k=1 should not be in the top5 prime lists)

106 (4553)
74 (1276)
219 (206)
191 (113)
312 (51)

R5:

2 (4)
1 (3)
11 (2)
8 (2)
12 (1)

R8: (note that we exclude k=1 and k=8 for R8, although they have a prime, but they have algebra factors and they can have only this prime. Thus, k=1 and k=8 should not be in the top5 prime lists)

11 (18)
5 (4)
12 (3)
7 (3)
2 (2)

R9:

11 (11)
24 (8)
14 (8)
38 (3)
18 (3)

R10:

121 (483)
109 (136)
98 (90)
230 (60)
289 (35)

R11:

1 (17)
3 (2)
2 (2)
4 (1)

R12:

298 (1676)
157 (285)
46 (194)
304 (40)
259 (40)

R13:

25 (15)
28 (14)
20 (10)
1 (5)
22 (3)

R14:

2 (4)
1 (3)
3 (1)

R16: (note that we exclude k=1 for R16, although it has a prime, but it has algebra factors and it can has only this prime. Thus, k=1 should not be in the top5 prime lists)

74 (638)
78 (26)
48 (15)
58 (12)
31 (12)

R17:

44 (6488)
29 (4904)
13 (1123)
36 (243)
10 (117)

R18:

151 (418)
78 (172)
50 (110)
79 (63)
237 (44)

R19:

1 (19)
7 (2)
3 (2)
8 (1)
6 (1)

R20:

2 (10)
1 (3)
6 (2)
5 (2)
7 (1)

R21:

29 (98)
34 (17)
43 (10)
32 (4)
5 (4)

R23:

3 (6)
2 (6)
4 (5)
1 (5)

R25:

86 (1029)
58 (26)
72 (24)
67 (24)
79 (21)

R26:

115 (520277)
32 (9812)
121 (1509)
73 (537)
80 (382)

R27: (note that we exclude k=1 for R27, although it has a prime, but it has algebra factors and it can has only this prime. Thus, k=1 should not be in the top5 prime lists)

9 (23)
11 (10)
12 (2)
7 (2)
6 (2)

R29:

2 (136)
1 (5)
3 (1)

R32:

3 (11)
2 (6)
9 (3)
8 (2)
5 (2)
[/CODE]

The remain k's and the largest 5 (probable) primes found for nearly-proven extended Sierpinski/Riesel bases b<=32 are: (only sorted by exponent) (k's that are multiple of b and where the number ((k+1)/gcd(k+1,b-1) for extended Sierpinski conjectures, (k-1)/gcd(k-1,b-1) for extended Riesel conjectures) is not prime are included in the conjectures but excluded from testing, since such k's will have the same prime as k / b. Thus, these k's should not be in the top5 prime lists)

[CODE]
S2:

remain k: 21181, 22699, 24737, 55459, 67607 (all at n=31M) (k=65536 is a GFN without known prime)

largest 5 (probable) primes found:

10223 (31172165)
19249 (13018586)
27653 (9167433)
28433 (7830457)
33661 (7031232)

S10:

remain k: 269 (at n=20K) (k=100 is a GFN without known prime)

largest 5 (probable) primes found:

804 (5470)
342 (338)
485 (230)
912 (215)
815 (190)

S25:

remain k: 71 (at n=10K)

largest 5 (probable) primes found:

61 (3104)
40 (518)
59 (48)
77 (27)
68 (15)

S26:

remain k: 65 and 155 (both at n=560K)

largest 5 (probable) primes found:

32 (318071)
217 (11454)
95 (1683)
178 (1154)
138 (827)

S30:

remain k: 278 and 588 (both at n=500K)

largest 5 (probable) primes found:

699 (11837)
242 (5064)
659 (4936)
311 (1760)
559 (1654)

S31:

remain k: 43, 51, 73, 77, 107, 117, 149, 181, 189, 209 (all at n=2K) (k=1 is a half GFN without known prime)

largest 5 (probable) primes found:

191 (1553)
5 (1026)
113 (178)
121 (118)
145 (78)

R3:

remain k: 1613, 1831, 1937, 3131, 3589, 5755, 6787, 7477, 7627, 7939, 8713, 8777, 9811, 10651, 11597 (all at n=50K)

largest 5 (probable) primes found:

8059 (47256)
11753 (36665)
6119 (28580)
7511 (26022)
8451 (24758)

R6:

remain k: 1597, 6236, 9491, 37031, 49771, 50686, 53941, 55061, 57926, 76761, 79801, 83411 (k=1597 at n=5M, other k's at n=25K)

largest 5 (probable) primes found:

36772 (1723287)
43994 (569498)
77743 (560745)
51017 (528803)
57023 (483561)

R7:

remain k: 197 (at n=25K)

largest 5 (probable) primes found:

367 (15118)
313 (5907)
159 (4896)
429 (3815)
419 (1052)

R31:

remain k: 5, 19, 51, 73, 97 (all at n=2K)

largest 5 (probable) primes found:

123 (1872)
124 (1116)
113 (643)
49 (637)
115 (464)
[/CODE]

Started extended Sierpinski base 33 testing.

There are 4 k's remaining in this base: 67, 203, 319, and 407.

Status is in the text file.

Base 34 and 35 Sierpinski and Riesel problems are very easily to prove, since the conjectured k for these bases are very low. (SR34 has both CK=6, and SR35 has both CK=5)

These are the text file for all the primes in these problems.

Note that in R34, k=4 can be proven composite by partial algebra factors. Generally, for Riesel bases b = 4 mod 5, 4*b^n-1/gcd(4-1,b-1) has algebraic factors for even n and divisible by 5 for odd n.

Extended Riesel base 33 was also done.

There are only 2 k's remain in this base: 257 and 339.

In this base, if k=m^2 and m = 4 or 13 mod 17 (this includes k = 16, 169, and 441), then (k*33^n-1)/gcd(k-1,33-1) has algebraic factors for even n and divisible by 17 for odd n, if k=m^2 and m = 15 or 17 mod 32 (this includes k = 225 and 289), then (k*33^n-1)/gcd(k-1,33-1) has algebraic factors for even n and divisible by 2 for odd n, if k=33*m^2 and m = 4 or 13 mod 17 (this includes k = 528), then then (k*33^n-1)/gcd(k-1,33-1) has algebraic factors for odd n and divisible by 17 for even n, if k=33*m^2 and m = 15 or 17 mod 32 (this does not include any k < CK), then (k*33^n-1)/gcd(k-1,33-1) has algebraic factors for odd n and divisible by 2 for even n. Thus, these k's can be proven composite by partial algebraic factors.

Extended Sierpinski base 36 was also done.

Although this base has a larger conjectured k (1886), amazingly, there are only 2 k's remain in this base: 1296 and 1814, and the testing of both of them are the same as extended Sierpinski base 6 for the same k-value.

I didn't test extended Riesel base 36, since this base has a very high conjectured k. (note that in extended Riesel base 36, all square k's can be proven composite by full algebraic factors, and although k=1 and k=36 have a prime, but they have algebra factors and they can have only this prime, thus, they are still excluded in this problem)

Completed extended R36 for k<=3000 (tested to n=1000). (note that in this base, all square k's can be proven composite by full algebraic factors, and although k=1 and k=36 have a prime, but they have algebra factors and they can have only this prime, thus, they are still excluded in this problem)

Reserving 3001<=k<=10000 (also tested to n=1000).

Completed extended SR15 for k<=4000 (tested to n=1500).

Completed extended R36 for k<=10000 (tested to n=1000).

Completed extended SR15 for k<=10000 (tested to n=1500).