A Sierpinski/Riesel-like problem

[sweety439]

Conjecture 1 (the strong Sierpinski conjecture): For b>=2, k>=1, if there is an n 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^(9*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^(9*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), 3*2^r (r>=0) or 9*2^r (r>=0) for every prime factor p of (k*b^n+1)/gcd(k+1,b-1)).
(3) this k is not excluded from this Sierpinski base b by the post #265. (the first 6 Sierpinski bases with k's excluded by the post #265 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 is an n 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^(9*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^(9*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), 3*2^r (r>=0) or 9*2^r (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).

[sweety439]

Originally Posted by sweety439

Conjecture 1 (the strong Sierpinski conjecture): For b>=2, k>=1, if there is an n 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^(9*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^(9*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), 3*2^r (r>=0) or 9*2^r (r>=0) for every prime factor p of (k*b^n+1)/gcd(k+1,b-1)).
(3) this k is not excluded from this Sierpinski base b by the post #265. (the first 6 Sierpinski bases with k's excluded by the post #265 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 is an n 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^(9*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^(9*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), 3*2^r (r>=0) or 9*2^r (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).

If these two conjectures are true, then all conjectures in this project and all conjectures in CRUS are also true, since these two conjectures cover those conjectures. Eventually, these two conjectures also cover the 2nd, 3rd, 4th, ... conjectures for those conjectures.

Since if there is no such n, then this k must have covering set, be proven composite by full algebra factors, or be proven composite by partial algebra factors, thus, this k is excluded from this Sierpinski/Riesel base.

[sweety439]

Originally Posted by sweety439

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.

In fact, this d equals gcd(k+1,b-1) for Sierpinski problems, and equals gcd(k-1,b-1) for Riesel problems.

Thus, you can think that CRUS only includes the k's such that gcd(k+-1,b-1) = 1 (+ for Sierpinski, - for Riesel), since if gcd(k+-1,b-1) = 1 (+ for Sierpinski, - for Riesel), then this form is completely the same as the form for this k and this Sierpinski/Riesel base in CRUS.

[sweety439]

Originally Posted by sweety439

Conjecture 1 (the strong Sierpinski conjecture): For b>=2, k>=1, if there is an n 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^(9*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^(9*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), 3*2^r (r>=0) or 9*2^r (r>=0) for every prime factor p of (k*b^n+1)/gcd(k+1,b-1)).
(3) this k is not excluded from this Sierpinski base b by the post #265. (the first 6 Sierpinski bases with k's excluded by the post #265 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 is an n 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^(9*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^(9*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), 3*2^r (r>=0) or 9*2^r (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).

We can find the 2nd, 3rd, 4th, ... n such that (k*b^n+-1)/gcd(k+-1,b-1) is prime. (i.e. find the 2nd, 3rd, 4th, ... prime of the form (k*b^n+-1)/gcd(k+-1,b-1) for fixed k and fixed Sierpinski/Riesel base b)

For the n's such that (k*b^n+-1)/gcd(k+-1,b-1) is prime: (excluding MOB, since if k is multiple of the base (b), then k and k / b are from the same family, if k = k' * b^r, then the exponent n for this k is (the first number > r for k') - r, and the correspond prime for this k is the correspond prime for n = (the first number > r for k') for k')

S2:

k = 1: 1, 2, 4, 8, 16, ... (sequence is not in OEIS)
k = 3: A002253
k = 5: A002254
k = 7: A032353
k = 9: A002256
k = 11: A002261
k = 13: A032356
k = 15: A002258
k = 17: A002259
k = 19: A032359
k = 21: A032360
k = 23: A032361
k = 25: A032362
k = 27: A032363
k = 29: A032364
k = 31: A032365
k = 33: A032366
k = 35: A032367
k = 37: A032368
k = 39: A002269
k = 41: A032370
k = 43: A032371
k = 45: A032372
k = 47: A032373
k = 49: A032374
k = 51: A032375
k = 53: A032376
k = 55: A032377
k = 57: A002274
k = 59: A032379
k = 61: A032380
k = 63: A032381

S3:

k = 1: A171381
k = 2: A003306
k = 4: A005537
k = 5: 2, 6, 12, 18, 26, 48, 198, 456, ... (sequence is not in OEIS)
k = 7: 1, 9, 33, 65, 337, ... (sequence is not in OEIS)
k = 8: A005538
k = 10: A005539
k = 11: 1, 3, 21, 39, 651, ... (sequence is not in OEIS)
k = 13: 2, 14, 32, 40, 112, ... (sequence is not in OEIS)
k = 14: A216890
k = 16: 3, 4, 5, 12, 24, 36, 77, 195, 296, 297, 533, 545, 644, 884, 932, ... (sequence is not in OEIS)

S4:

k = 1: 1, 2, 4, 8, ... (sequence is not in OEIS)
k = 2: A127936
k = 3: 1, 3, 4, 6, 9, 15, 18, 33, 138, 204, 219, 267, ... (sequence is not in OEIS)
k = 5: 1, 3, 6, 12, 15, 18, 36, 72, 81, 84, 117, 522, 1023, 1083, 1206, ... (sequence is not in OEIS)
k = 6: 2, 20, 94, 100, 104, 176, 1594, ... (sequence is not in OEIS)
k = 7: A002255

S5:

k = 1: 1, 2, 4, ... (sequence is not in OEIS)
k = 2: A058934
k = 3: 2, 6, 8, 62, 120, 186, 414, 764, ... (sequence is not in OEIS)
k = 4: A204322
k = 6: A143279
k = 7: (covering set {2, 3})
k = 8: 1, 1037, ... (sequence is not in OEIS)

S6:

k = 1: 1, 2, 4, ... (sequence is not in OEIS)
k = 2: A120023
k = 3: A186112
k = 4: A248613
k = 5: A247260
k = 7: 1, 6, 17, 38, 50, 80, 207, 236, 264, 309, 555 ... (sequence is not in OEIS)
k = 8: 4, 10, 16, 32, 40, 70, 254, ... (sequence is not in OEIS)

S10:

k = 1: 1, 2, ... (sequence is not in OEIS)
k = 2: A096507
k = 3: A056807
k = 4: A056806
k = 5: A102940
k = 6: A056805
k = 7: A056804
k = 8: A096508
k = 9: A056797
k = 11: A102975
k = 12: 2, 38, 80, 9230, 25598, 39500, ... (sequence is not in OEIS)
k = 13: A289051
k = 14: A099017
k = 15: 1, 4, 7, 8, 18, 19, 73, 143, 192, 408, 533, 792, 3179, 7709, 9554, 52919, 56021, 61604, ... (sequence is not in OEIS)
k = 16: A273002

R2:

k = 1: A000043
k = 3: A002235
k = 5: A001770
k = 7: A001771
k = 9: A002236
k = 11: A001772
k = 13: A001773
k = 15: A002237
k = 17: A001774
k = 19: A001775
k = 21: A002238
k = 23: A050537
k = 25: A050538
k = 27: A050539
k = 29: A050540
k = 31: A050541
k = 33: A002240
k = 35: A050543
k = 37: A050544
k = 39: A050545
k = 41: A050546
k = 43: A050547
k = 45: A002242
k = 47: A050549
k = 49: A050550
k = 51: A050551
k = 53: A050552
k = 55: A050553
k = 57: A050554
k = 59: A050555
k = 61: A050556
k = 63: A050557

R3:

k = 1: A028491
k = 2: A003307
k = 4: A005540
k = 5: 1, 3, 5, 9, 15, 23, 45, 71, 99, 125, 183, 1143, ... (sequence is not in OEIS)
k = 7: 2, 4, 6, 8, 16, 18, 28, 36, 52, 106, 114, 204, 270, 292, 472, 728, 974, ... (sequence is not in OEIS)
k = 8: A005541
k = 10: A005542
k = 11: 22, 30, 46, 162, ... (sequence is not in OEIS)
k = 13: 1, 5, 25, 41, 293, 337, 569, 1085, ... (sequence is not in OEIS)
k = 14: 1, 11, 16, 80, 83, 88, 136, 187, 328, 397, 776, 992, 1195, ... (sequence is not in OEIS)
k = 16: 1, 3, 9, 13, 31, 43, 81, 121, 235, 1135, 1245, ... (sequence is not in OEIS)

R4:

k = 1: (proven composite by full algebra factors)
k = 2: A146768
k = 3: A272057
k = 5: 1, 2, 4, 5, 6, 7, 9, 16, 24, 27, 36, 74, 92, 124, 135, 137, 210, 670, 719, 761, 819, 877, 942, 1007, 1085, ... (sequence is not in OEIS)
k = 6: 1, 3, 5, 21, 27, 51, 71, 195, 413, ... (sequence is not in OEIS)
k = 7: 2, 3, 5, 12, 14, 41, 57, 66, 284, 296, 338, 786, 894, ... (sequence is not in OEIS)

R5:

k = 1: A004061
k = 2: A120375
k = 3: 1, 2, 4, 9, 16, 17, 54, 64, 112, 119, 132, 245, 557, 774, 814, 1020, 1110, ... (sequence is not in OEIS)
k = 4: A046865
k = 6: A257790
k = 7: 1, 5, 11, 13, 15, 41, 61, 77, 103, 123, 199, 243, 279, 1033, 1145, ... (sequence is not in OEIS)
k = 8: 2, 4, 8, 10, 28, 262, 356, 704, ... (sequence is not in OEIS)

R6:

k = 1: A004062
k = 2: A057472
k = 3: A186106
k = 4: 1, 3, 25, 31, 43, 97, 171, 213, 273, 449, 575, 701, 893, ... (sequence is not in OEIS)
k = 5: A079906
k = 7: 1, 2, 3, 13, 21, 28, 30, 32, 36, 48, 52, 76, 734, ... (sequence is not in OEIS)
k = 8: 1, 5, 35, 65, 79, 215, 397, 845, ... (sequence is not in OEIS)

R10:

k = 1: A004023
k = 2: A002957
k = 3: A056703
k = 4: A056698
k = 5: A056712
k = 6: A056716
k = 7: A056701
k = 8: A056721
k = 9: A056725
k = 11: A111391
k = 12: 5, 3191, 3785, 5513, 14717, ... (sequence is not in OEIS)
k = 13: A056707
k = 14: 1, 2, 3, 4, 5, 16, 21, 23, 62, 175, 195, 206, 261, 347, 448, 494, 689, 987, 1361, 8299, 13225, 21513, 23275, ... (sequence is not in OEIS)
k = 15: 1, 2, 15, 22, 27, 33, 38, 473, 519, 591, 699, 2273, 2476, 2985, 6281, 6947, 11990, 16828, 17096, 26236, 33459, 34963, ... (sequence is not in OEIS)
k = 16: A056714

[sweety439]

Pseudoprimes only exist when gcd(k+-1,b-1) (+ for Sierpinski, - for Riesel) is not 1, since the N-1 method (for Sierpinski when gcd(k+1,b-1) = 1) and N+1 method (for Riesel when gcd(k-1,b-1) = 1) can prove the primality, all of the primes with gcd(k+-1,b-1) (+ for Sierpinski, - for Riesel) = 1 are proven primes.

[sweety439]

The records for the n are: (excluding the k's with covering set and the k's proven composite by full/partial algebra factors)

S2:

1 (1)
4 (2)
12 (3)
16 (4)
19 (6)
31 (8)
47 (583)
383 (6393)
2897 (9715)
3061 (33288)
4847 (3321063)
5359 (5054502)
10223 (31172165)
21181 (>31600000)

S3:

1 (1)
5 (2)
16 (3)
17 (6)
21 (8)
41 (4892)
621 (>10000)

S4:

1 (1)
6 (2)
19 (3)
30 (4)
51 (46)
86 (108)
89 (167)
94 (291)
186 (10458)

(next record k is > 794)

S5:

1 (1)
3 (2)
18 (3)
19 (4)
34 (8)
40 (1036)
61 (6208)
181 (>12000)

S6:

1 (1)
8 (4)
20 (5)
53 (7)
67 (8)
97 (9)
117 (23)
136 (24)
160 (3143)
1296 (>=268435454)

S7:

1 (4)
9 (6)
21 (124)
101 (216)
121 (252)
141 (1044)

(next record k is > 209)

S8:

2 (1)
3 (2)
13 (4)
31 (20)
68 (115)
94 (194)
118 (820)
173 (7771)
256 (>=2863311528)

S9:

1 (1)
6 (2)
17 (3)
21 (4)
26 (6)
40 (9)
41 (2446)
311? (>2000)

S10:

1 (1)
8 (2)
9 (3)
22 (6)
34 (26)
100 (>=33554430)

S11:

1 (2)
10 (10)
20 (35)
45 (40)
47 (545)
194 (>2000)

S12:

1 (1)
2 (3)
12 (>=33554431)

R2:

1 (2)
13 (3)
14 (4)
43 (7)
44 (24)
74 (2552)
659 (800516)
2293 (>8300000)

R3:

1 (3)
11 (22)
71 (46)
97 (3131)
119 (8972)
313 (24761)
1613 (>50000)

R4:

2 (1)
7 (2)
39 (12)
74 (1276)
106 (4553)
659 (400258)

(next record k is > 1114)

R5:

1 (3)
2 (4)
31 (5)
32 (8)
34 (163)
86 (2058)
428 (9704)
638? (>2000)

R6:

1 (2)
37 (4)
54 (6)
69 (10)
92 (49)
251 (3008)
1597 (>5000000)

R7:

1 (5)
31 (18)
59 (32)
73 (127)
79 (424)
139 (468)
159 (4896)
197 (181761)

(next record k is > 457)

R8:

2 (2)
5 (4)
11 (18)
37 (851)
74 (2632)
236 (5258)
239? (>2000)

R9:

2 (1)
11 (11)
53 (536)
119 (4486)
386 (>5000)

R10:

1 (2)
12 (5)
32 (28)
89 (33)
98 (90)
109 (136)
121 (483)
406 (772)
450 (11958)
505 (18470)
1231 (37398)
1803 (45882)
1935 (51836)
2452 (>554789)

R11:

1 (17)
32 (18)
39 (22)
62 (26202)
201? (>2000)

R12:

1 (2)
23 (3)
24 (4)
46 (194)
157 (285)
298 (1676)
1037 (>5000)

[sweety439]

Reserve S80 k=947 (the only remain k not in CRUS for S80).

[sweety439]

Originally Posted by sweety439

The records for the n are: (excluding the k's with covering set and the k's proven composite by full/partial algebra factors)

S2:

1 (1)
4 (2)
12 (3)
16 (4)
19 (6)
31 (8)
47 (583)
383 (6393)
2897 (9715)
3061 (33288)
4847 (3321063)
5359 (5054502)
10223 (31172165)
21181 (>31600000)

S3:

1 (1)
5 (2)
16 (3)
17 (6)
21 (8)
41 (4892)
621 (>10000)

S4:

1 (1)
6 (2)
19 (3)
30 (4)
51 (46)
86 (108)
89 (167)
94 (291)
186 (10458)

(next record k is > 794)

S5:

1 (1)
3 (2)
18 (3)
19 (4)
34 (8)
40 (1036)
61 (6208)
181 (>12000)

S6:

1 (1)
8 (4)
20 (5)
53 (7)
67 (8)
97 (9)
117 (23)
136 (24)
160 (3143)
1296 (>=268435454)

S7:

1 (4)
9 (6)
21 (124)
101 (216)
121 (252)
141 (1044)

(next record k is > 209)

S8:

2 (1)
3 (2)
13 (4)
31 (20)
68 (115)
94 (194)
118 (820)
173 (7771)
256 (>=2863311528)

S9:

1 (1)
6 (2)
17 (3)
21 (4)
26 (6)
40 (9)
41 (2446)
311? (>2000)

S10:

1 (1)
8 (2)
9 (3)
22 (6)
34 (26)
100 (>=33554430)

S11:

1 (2)
10 (10)
20 (35)
45 (40)
47 (545)
194 (>2000)

S12:

1 (1)
2 (3)
12 (>=33554431)

R2:

1 (2)
13 (3)
14 (4)
43 (7)
44 (24)
74 (2552)
659 (800516)
2293 (>8300000)

R3:

1 (3)
11 (22)
71 (46)
97 (3131)
119 (8972)
313 (24761)
1613 (>50000)

R4:

2 (1)
7 (2)
39 (12)
74 (1276)
106 (4553)
659 (400258)

(next record k is > 1114)

R5:

1 (3)
2 (4)
31 (5)
32 (8)
34 (163)
86 (2058)
428 (9704)
638? (>2000)

R6:

1 (2)
37 (4)
54 (6)
69 (10)
92 (49)
251 (3008)
1597 (>5000000)

R7:

1 (5)
31 (18)
59 (32)
73 (127)
79 (424)
139 (468)
159 (4896)
197 (181761)

(next record k is > 457)

R8:

2 (2)
5 (4)
11 (18)
37 (851)
74 (2632)
236 (5258)
239? (>2000)

R9:

2 (1)
11 (11)
53 (536)
119 (4486)
386 (>5000)

R10:

1 (2)
12 (5)
32 (28)
89 (33)
98 (90)
109 (136)
121 (483)
406 (772)
450 (11958)
505 (18470)
1231 (37398)
1803 (45882)
1935 (51836)
2452 (>554789)

R11:

1 (17)
32 (18)
39 (22)
62 (26202)
201? (>2000)

R12:

1 (2)
23 (3)
24 (4)
46 (194)
157 (285)
298 (1676)
1037 (>5000)

The k's either with covering set or proven composite by full/partial algebra factors are:

S4:

k = 419: covering set {3, 5, 7, 13}
k = 659: covering set {3, 5, 13, 17, 241}
k = 794: covering set {3, 5, 7, 13}

S5:

All k = 7, 11 (mod 24): covering set {2, 3}

S7:

k = 209: covering set {2, 3, 5, 13, 43}.

S8:

All k = 47, 79, 83, 181 (mod 195): covering set {3, 5, 13}
All k = m^3: for all n; factors to: (m*2^n + 1) * (m^2*4^n - m*2^n + 1)

S9:

All k = 31, 39 (mod 80): covering set {2, 5}

S10:

k = 989: covering set {3, 7, 11, 13}
k = 1121: covering set {7, 11, 13, 37}
k = 3653: covering set {3, 7, 11, 37}

S11:

All k = 5, 7 (mod 12): covering set {2, 3}

S12:

k = 521: covering set {5, 13, 29}
k = 597: covering set {5, 13, 29}
k = 1143: covering set {5, 13, 29}

R4:

k = 361: covering set {3, 5, 7, 13}
k = 919: covering set {3, 5, 7, 13}
k = 1114: covering set {3, 5, 7, 13}
All k = m^2 for all n; factors to: (m*2^n - 1) * (m*2^n + 1)

R5:

All k = 13, 17 (mod 24): covering set {2, 3}

R7:

k = 457: covering set {2, 3, 5, 13, 19}

R8:

All k = 14, 112, 116, 148 (mod 195): covering set {3, 5, 13}
k = 658: covering set {3, 5, 19, 37, 73}
All k = m^3: for all n; factors to: (m*2^n - 1) * (m^2*4^n + m*2^n + 1)

R9:

All k = 41, 49 (mod 80): covering set {2, 5}
k = 74: covering set {5, 7, 13, 73}
All k = m^2: for all n; factors to: (m*3^n - 1) * (m*3^n + 1)

R10:

k = 334: covering set {3, 7, 13, 37}
k = 343: n = 1 (mod 3): factor of 3; n = 2 (mod 3): factor of 37; n = 0 mod 3: let n=3q; factors to: (7*10^q - 1) * [49*10^(2q) + 7*10^q + 1]
k = 1585: covering set {3, 7, 11, 13}
k = 1882: covering set {3, 7, 11, 13}

R11:

All k = 5, 7 (mod 12): covering set {2, 3}

R12:

k = 376: covering set {5, 13, 29}
k = 742: covering set {5, 13, 29}
k = 1288: covering set {5, 13, 29}
All k where k = m^2 and m = 5, 8 (mod 13): for even n let k = m^2 and let n = 2*q; factors to: (m*12^q - 1) * (m*12^q + 1); odd n: factor of 13
All k where k = 3*m^2 and m = 3, 10 (mod 13): even n: factor of 13; for odd n let k = 3*m^2 and let n = 2*q - 1; factors to: [m*2^n*3^q - 1] * [m*2^n*3^q + 1]

[sweety439]

Originally Posted by sweety439

The 1st, 2nd and 3rd conjecture of Sierpinski/Riesel bases 4, 5, 7, 8, 9, 10, 11, 12, 13, 14 and 16 are:

base    1st    2nd    3rd
S4      419    659    794
S5        7     11     31
S7      209   1463   3305
S8       47     79     83
S9       31     39    111
S10     989   1121   3653
S11       5      7     17
S12     521    597   1143
S13      15     27     47
S14       4     11     19
S16      38    194    524
R4      361    919   1114
R5       13     17     37
R7      457   1291   3199
R8       14    112    116
R9       41     49     74
R10     334   1585   1882
R11       5      7     17
R12     376    742   1288
R13      29     41     69
R14       4     11     19
R16     100    172    211

Now, I only decide to reserve the 1st, 2nd and 3rd conjecture of Sierpinski/Riesel bases 5, 8, 9, 11, 13, 14 and 16, since the 2nd and 3rd conjecture for other bases b<=16 are larger.

Update the text files for the 1st, 2nd and 3rd conjecture of SR12.

R12 has only 2 k's remain: 1037 and 1132, but S12 has many k's remain. (thus, I did not search S12 very far)

[sweety439]

Originally Posted by sweety439

The records for the n are: (excluding the k's with covering set and the k's proven composite by full/partial algebra factors)

S2:

1 (1)
4 (2)
12 (3)
16 (4)
19 (6)
31 (8)
47 (583)
383 (6393)
2897 (9715)
3061 (33288)
4847 (3321063)
5359 (5054502)
10223 (31172165)
21181 (>31600000)

S3:

1 (1)
5 (2)
16 (3)
17 (6)
21 (8)
41 (4892)
621 (>10000)

S4:

1 (1)
6 (2)
19 (3)
30 (4)
51 (46)
86 (108)
89 (167)
94 (291)
186 (10458)

(next record k is > 794)

S5:

1 (1)
3 (2)
18 (3)
19 (4)
34 (8)
40 (1036)
61 (6208)
181 (>12000)

S6:

1 (1)
8 (4)
20 (5)
53 (7)
67 (8)
97 (9)
117 (23)
136 (24)
160 (3143)
1296 (>=268435454)

S7:

1 (4)
9 (6)
21 (124)
101 (216)
121 (252)
141 (1044)

(next record k is > 209)

S8:

2 (1)
3 (2)
13 (4)
31 (20)
68 (115)
94 (194)
118 (820)
173 (7771)
256 (>=2863311528)

S9:

1 (1)
6 (2)
17 (3)
21 (4)
26 (6)
40 (9)
41 (2446)
311? (>2000)

S10:

1 (1)
8 (2)
9 (3)
22 (6)
34 (26)
100 (>=33554430)

S11:

1 (2)
10 (10)
20 (35)
45 (40)
47 (545)
194 (>2000)

S12:

1 (1)
2 (3)
12 (>=33554431)

R2:

1 (2)
13 (3)
14 (4)
43 (7)
44 (24)
74 (2552)
659 (800516)
2293 (>8300000)

R3:

1 (3)
11 (22)
71 (46)
97 (3131)
119 (8972)
313 (24761)
1613 (>50000)

R4:

2 (1)
7 (2)
39 (12)
74 (1276)
106 (4553)
659 (400258)

(next record k is > 1114)

R5:

1 (3)
2 (4)
31 (5)
32 (8)
34 (163)
86 (2058)
428 (9704)
638? (>2000)

R6:

1 (2)
37 (4)
54 (6)
69 (10)
92 (49)
251 (3008)
1597 (>5000000)

R7:

1 (5)
31 (18)
59 (32)
73 (127)
79 (424)
139 (468)
159 (4896)
197 (181761)

(next record k is > 457)

R8:

2 (2)
5 (4)
11 (18)
37 (851)
74 (2632)
236 (5258)
239? (>2000)

R9:

2 (1)
11 (11)
53 (536)
119 (4486)
386 (>5000)

R10:

1 (2)
12 (5)
32 (28)
89 (33)
98 (90)
109 (136)
121 (483)
406 (772)
450 (11958)
505 (18470)
1231 (37398)
1803 (45882)
1935 (51836)
2452 (>554789)

R11:

1 (17)
32 (18)
39 (22)
62 (26202)
201? (>2000)

R12:

1 (2)
23 (3)
24 (4)
46 (194)
157 (285)
298 (1676)
1037 (>5000)

(194*11^3155+1)/5 is (probable) prime!!!

Next possible record k for S11 is 195, tested to n=10000 with no (probable) prime found.

Beside, I also found that 638*5^6974-1 is prime. (next possible record k for R5 is 662)

Also, R8 k=239 and R11 k=201 are tested to n=10000 with no (probable) prime found.

Note: 302*9^2849-1 is prime, but 302 is not a record k for R9, since the first (probable) prime for R9 k=119 is (119*9^4486-1)/2, this (probable) prime is converted by R3 k=119.

[sweety439]

Conjecture: If k is not Sierpinski/Riesel in base b, then k*b is also not Sierpinski/Riesel in base b, i.e. if (k*b+-1)/gcd(k+-1,b-1) is prime, then there is an n>=1 such that ((k*b)*b^n+-1)/gcd(k*b+-1,b-1) is prime. (if this k is not excluded from the Sierpinski/Riesel problem base b)

Some classic quasi-counterexamples:

S2 k=6977 has a prime at n=3, but for k = 55816 = 6977*2^3, this prime k=6977, n=3 would be k=55816, n=0 but n must be > 0 hence it is not allowed so k=55816 must continue to be searched, and k=55816 has a prime at n=14536.

S3 k=23 has a prime at n=3, but for k = 621 = 23*3^3, this prime k=23, n=3 would be k=621, n=0 but n must be > 0 hence it is not allowed so k=621 must continue to be searched, and k=621 has no (probable) prime for n<=10000.

S4 k=11 has a prime at n=2, but for k = 176 = 11*4^2, this prime k=11, n=2 would be k=176, n=0 but n must be > 0 hence it is not allowed so k=176 must continue to be searched, and k=176 has a prime at n=228.

S5 k=8 has a prime at n=1, but for k = 40 = 8*5^1, this prime k=8, n=1 would be k=40, n=0 but n must be > 0 hence it is not allowed so k=40 must continue to be searched, and k=40 has a prime at n=1036.

S7 k=3 has a prime at n=1, but for k = 21 = 3*7^1, this prime k=3, n=1 would be k=21, n=0 but n must be > 0 hence it is not allowed so k=21 must continue to be searched, and k=21 has a prime at n=124.

S22 k=1 has a prime at n=1, but for k = 22 = 1*22^2, this prime k=1, n=1 would be k=22, n=0 but n must be > 0 hence it is not allowed so k=22 must continue to be searched, and k=22 has no prime for n<2^24-1.

R2 k=37 has a prime at n=1, but for k = 74 = 37*2^1, this prime k=37, n=1 would be k=74, n=0 but n must be > 0 hence it is not allowed so k=74 must continue to be searched, and k=74 has a prime at n=2552.

R2 k=337 has a prime at n=1, but for k = 674 = 337*2^1, this prime k=337, n=1 would be k=674, n=0 but n must be > 0 hence it is not allowed so k=674 must continue to be searched, and k=674 has a prime at n=11676.

R6 k=491 has a prime at n=3, but for k = 106056 = 491*6^3, this prime k=491, n=3 would be k=106056, n=0 but n must be > 0 hence it is not allowed so k=106056 must continue to be searched, and k=106056 has a (probable) prime at n=3038.

R10 k=45 has a prime at n=1, but for k = 450 = 45*10^1, this prime k=45, n=1 would be k=450, n=0 but n must be > 0 hence it is not allowed so k=450 must continue to be searched, and k=450 has a prime at n=11958.

R15 k=196 has a prime at n=1, but for k = 2940 = 196*15^1, this prime k=196, n=1 would be k=2940, n=0 but n must be > 0 hence it is not allowed so k=2940 must continue to be searched, and k=2940 has a prime at n=13254.

R18 k=1 has a prime at n=2, but for k = 324 = 1*18^2, this prime k=1, n=2 would be k=324, n=0 but n must be > 0 hence it is not allowed so k=324 must continue to be searched, and k=324 has a (probable) prime at n=25665.

R23 k=10 has a prime at n=1, but for k = 230 = 10*23^1, this prime k=10, n=1 would be k=230, n=0 but n must be > 0 hence it is not allowed so k=230 must continue to be searched, and k=230 has a prime at n=6228.

R27 k=22 has a prime at n=1, but for k = 594 = 22*27^1, this prime k=22, n=1 would be k=594, n=0 but n must be > 0 hence it is not allowed so k=594 must continue to be searched, and k=594 has a prime at n=36624.

R31 k=4 has a prime at n=1, but for k = 124 = 4*31^1, this prime k=4, n=1 would be k=124, n=0 but n must be > 0 hence it is not allowed so k=124 must continue to be searched, and k=124 has a prime at n=1116.

R48 k=8 has a prime at n=1, but for k = 384 = 8*48^1, this prime k=8, n=1 would be k=384, n=0 but n must be > 0 hence it is not allowed so k=384 must continue to be searched, and k=384 has no prime for n<=200000.