A Sierpinski/Riesel-like problem

[sweety439]

Originally Posted by sweety439

Extended to base 539

Note: I only searched the k <= 5000000, if there are <16 Sierpinski/Riesel k's <= 5000000, then this text file only show the Sierpinski/Riesel k's <= 5000000 for this Sierpinski/Riesel base (if there are no Sierpinski/Riesel k's <= 5000000, then this text file do not show any Sierpinski/Riesel k's <= 5000000 for this Sierpinski/Riesel base), also, I only searched the exponent n <= 2000 (for (k*b^n+-1)/gcd(k+-1,b-1), + for Sierpinski, - for Riesel) and only searched the primes <= 100000 (for the prime factor of (k*b^n+-1)/gcd(k+-1,b-1), + for Sierpinski, - for Riesel), thus this text file wrongly shows 1 as Sierpinski number base 125, although (1*125^n+1)/gcd(1+1,125-1) has no covering set, but since (1*125^n+1)/gcd(1+1,125-1) has a prime factor <= 100000 for all n <= 2000

These are the conjectures in the thread https://mersenneforum.org/showthread.php?t=11061 (conjectured smallest prime Sierpinski/Riesel numbers), for the extended Sierpinski/Riesel conjectures (k*b^n+-1)/gcd(k+-1,b-1) (+ for Sierpinski, - for Riesel) for bases 2<=b<=128 and 256, 512, 1024, 2048, 4096, 8192, 16384, 32768, 65536

[sweety439]

searched to base 256 (also base 512, 1024, 2048, 4096, 8192, 16384, 32768, 65536)

[sweety439]

Originally Posted by sweety439

Done to base 2500

All Sierpinski/Riesel bases listed "NA" have CK > 5M (i.e. 5M is the lower bound for these Sierpinski/Riesel bases)

upper bounds for these Sierpinski/Riesel bases <= 600:

S66: 21314443 (if not exactly this number, then must be == 4 mod 5 or == 12 mod 13)
S120: 374876369 (if not exactly this number, then must be == 6 mod 7 or == 16 mod 17)
S156: 18406311208 (if not exactly this number, then must be == 4 mod 5 or == 30 mod 31)
S210: 147840103 (if not exactly this number, then must be == 10 mod 11 or == 18 mod 19)
S280: 82035074042274 (if not exactly this number, then must be == 2 mod 3 or == 30 mod 31)
S330: 16636723 (if not exactly this number, then must be == 6 mod 7 or == 46 mod 47)
S358: 27478218 (if not exactly this number, then must be == 2 mod 3 or == 6 mod 7 or == 16 mod 17)
S456: 14836963 (if not exactly this number, then must be == 4 mod 5 or == 6 mod 7 or == 12 mod 13)
S462: 6880642 (if not exactly this number, then must be == 460 mod 461)
S546: 45119296 (if not exactly this number, then must be == 4 mod 5 or == 108 mod 109)

R66: 101954772 (if not exactly this number, then must be == 1 mod 5 or == 1 mod 13)
R120: 166616308 (if not exactly this number, then must be == 1 mod 7 or == 1 mod 17)
R156: 2113322677 (if not exactly this number, then must be == 1 mod 5 or == 1 mod 31)
R180: 7674582 (if not exactly this number, then must be == 1 mod 179)
R210: 80176412 (if not exactly this number, then must be == 1 mod 11 or == 1 mod 19)
R280: 513613045571841 (if not exactly this number, then must be == 1 mod 3 or == 1 mod 31)
R330: 16527822 (if not exactly this number, then must be == 1 mod 7 or == 1 mod 47)
R358: 27606383 (if not exactly this number, then must be == 1 mod 3 or == 1 mod 7 or == 1 mod 17)
R420: 6548233 (if not exactly this number, then must be == 1 mod 419)
R456: 76303920 (if not exactly this number, then must be == 1 mod 5 or == 1 mod 7 or == 1 mod 13)
R546: 11732602 (if not exactly this number, then must be == 1 mod 5 or == 1 mod 109)
R570: 12511182 (if not exactly this number, then must be == 1 mod 569)

[sweety439]

These are the conjectured first 4 Sierpinski/Riesel numbers, for the power-of-2 bases searched up to b=2^16

[sweety439]

Using the Riesel side as an example:

1. n must be >= 1 for all k

2. If (k*b^n-1)/gcd(kb-1,b-1) where n=1 is prime than k*b (i.e. MOB) will need a different prime because this prime would be (kb*b^0-1)/gcd(kb-1,b-1)

3. If (k*b^n-1)/gcd(kb-1,b-1) where n>1 is prime than k*b will have the same prime (in a slightly different form), i.e. (kb*b^(n-1)-1)/gcd(kb-1,b-1)

4. Assume that (k*b^1-1)/gcd(kb-1,b-1) is prime. (k*b^1-1)/gcd(kb-1,b-1) = (kb-1)/gcd(kb-1,b-1)

5. Conclusion: Per #2 and #4 the only time k*b needs a different prime than k is when (kb-1)/gcd(kb-1,b-1) is prime ((kb+1)/gcd(kb+1,b-1) for Sierp)

[sweety439]

Status for the first 4 Sierpinski/Riesel conjectures (added R100 and R512, R1024 is still running .... now running for k=91)

[sweety439]

Update files to include SR100, SR512, SR1024

the (probable) prime (469*100^4451-1)/gcd(469-1,100-1) is given by https://stdkmd.net/nrr/prime/primedifficulty.txt (the form 521w)

Also see the GitHub page https://github.com/xayahrainie4793/f...el-conjectures for the status (this website also be update for S26, some primes are given by CRUS S676)

[sweety439]

Originally Posted by sweety439

These are the conjectured first 4 Sierpinski/Riesel numbers, for the power-of-2 bases searched up to b=2^16

k's with algebra factors for Sierpinski/Riesel base b=2^n with 9<=n<=16:

S512: all k = m^3
S1024: all k = m^5
S2048: all k = m^11
S4096: all k = m^3 and all k = 4*m^4
S8192: all k = m^13
S16384: all k = m^7 and all k = 2^r with r = 6, 10, 12 mod 14
S32768: all k = m^3 and all k = m^5 and all k = 2^r with r = 7, 11, 13, 14 mod 15
S65536: all k = 4*m^4

R512: all k = m^3
R1024: all k = m^2 and all k = m^5
R2048: all k = m^11
R4096: all k = m^2 and all k = m^3
R8192: all k = m^13
R16384: all k = m^2 and all k = m^7
R32768: all k = m^3 and all k = m^5
R65536: all k = m^2

[sweety439]

Originally Posted by sweety439

Extended Sierpinski problem base b:

Finding and proving the smallest k>=1 such that (k*b^n+1)/gcd(k+1,b-1) is not prime for all integers n>=1. (k-values that make a full covering set with all or partial algebraic factors are excluded from the conjectures)

Extended Riesel problem base b:

Finding and proving the smallest k>=1 such that (k*b^n-1)/gcd(k-1,b-1) is not prime for all integers n>=1. (k-values that make a full covering set with all or partial algebraic factors are excluded from the conjectures)

This b must be >=2, and the b=2 case is the original Sierpinski/Riesel problems, this project extend these Sierpinski/Riesel problems to bases b>2

[sweety439]

Status for 2<=b<=128 and 1<=k<=128:

Sierpinski (k*b^n+1)/gcd(k+1,b-1)

Riesel (k*b^n-1)/gcd(k-1,b-1)

[sweety439]

The records of the n are: (GFNs and half GFNs are excluded)

S2:

3 (1)
7 (2)
12 (3)
19 (6)
31 (8)
47 (583)
383 (6393)
2897 (9715)
3061 (33288)
4847 (3321063)
5359 (5054502)
10223 (31172165)
21181 (>32500000)

S3:

2 (1)
5 (2)
16 (3)
17 (6)
21 (8)
41 (4892)
621 (20820)
1187? (>16300)

S4:

2 (1)
6 (2)
19 (3)
30 (4)
51 (46)
86 (108)
89 (167)
94 (291)
186 (10458)
1238 (>20000)

S5:

2 (1)
3 (2)
18 (3)
19 (4)
34 (8)
40 (1036)
61 (6208)
181 (>20000)

S6:

2 (1)
8 (4)
20 (5)
53 (7)
67 (8)
97 (9)
117 (23)
136 (24)
160 (3143)
1814 (>175600)

S7:

2 (1)
5 (2)
9 (6)
21 (124)
101 (216)
121 (252)
141 (1044)
389 (>3000)

S8:

3 (2)
13 (4)
31 (20)
68 (115)
94 (194)
118 (820)
173 (7771)
259 (27626)
395 (61857)
467 (>833333)

S9:

2 (1)
6 (2)
17 (3)
21 (4)
26 (6)
40 (9)
41 (2446)
311 (15668)
1039? (>5000)

S10:

2 (1)
8 (2)
9 (3)
22 (6)
34 (26)
269 (>100000)

S11:

2 (1)
4 (2)
10 (10)
20 (35)
45 (40)
47 (545)
194 (3155)
195 (>5000)

S12:

2 (3)
17 (78)
30 (144)
37 (199)
261 (644)
378 (2388)
404 (714558)
885? (>25000)

R2:

1 (2)
13 (3)
14 (4)
43 (7)
44 (24)
74 (2552)
659 (800516)
2293 (>10200000)

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)
1810? (>20000)

R5:

1 (3)
2 (4)
31 (5)
32 (8)
34 (163)
86 (2058)
428 (9704)
662 (14628)
1279 (>15000)

R6:

1 (2)
37 (4)
54 (6)
69 (10)
92 (49)
251 (3008)
1597 (>5300000)

R7:

1 (5)
31 (18)
59 (32)
73 (127)
79 (424)
139 (468)
159 (4896)
197 (181761)
679? (>3000)

R8:

2 (2)
5 (4)
11 (18)
37 (851)
74 (2632)
236 (5258)
239 (>20000)

R9:

2 (1)
11 (11)
53 (536)
119 (4486)
386 (>25000)

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? (>5000)

R12:

1 (2)
23 (3)
24 (4)
46 (194)
157 (285)
298 (1676)
1037 (6281)
1132 (>21760)