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2020-08-05, 03:41   #925
sweety439

Nov 2016

7×11×29 Posts

Quote:
 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
Attached Files
 prime Sierpinski.txt (1.1 KB, 7 views) prime Riesel.txt (1.1 KB, 7 views)

Last fiddled with by sweety439 on 2020-08-05 at 03:41

2020-08-06, 03:25   #926
sweety439

Nov 2016

42718 Posts

searched to base 256 (also base 512, 1024, 2048, 4096, 8192, 16384, 32768, 65536)
Attached Files
 prime Sierpinski.txt (2.1 KB, 4 views) prime Riesel.txt (2.2 KB, 4 views)

2020-08-06, 03:44   #927
sweety439

Nov 2016

7×11×29 Posts

Quote:
 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)

2020-08-06, 04:03   #928
sweety439

Nov 2016

1000101110012 Posts

These are the conjectured first 4 Sierpinski/Riesel numbers, for the power-of-2 bases searched up to b=2^16
Attached Files
 conjectured first 4 Sierpinski numbers.txt (6.3 KB, 4 views) conjectured first 4 Riesel numbers.txt (6.3 KB, 5 views)

 2020-08-06, 06:43 #929 sweety439     Nov 2016 7×11×29 Posts 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)
2020-08-06, 08:25   #930
sweety439

Nov 2016

7×11×29 Posts

Status for the first 4 Sierpinski/Riesel conjectures (added R100 and R512, R1024 is still running .... now running for k=91)
Attached Files
 first 4 conjectures.zip (126.1 KB, 4 views)

2020-08-06, 12:34   #931
sweety439

Nov 2016

8B916 Posts

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)
Attached Files
 first 4 SR conjectures.zip (128.1 KB, 3 views)

Last fiddled with by sweety439 on 2020-08-06 at 12:34

2020-08-07, 16:01   #932
sweety439

Nov 2016

7×11×29 Posts

Quote:
 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

2020-08-08, 06:30   #933
sweety439

Nov 2016

8B916 Posts

Quote:
 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

 2020-08-08, 06:36 #934 sweety439     Nov 2016 223310 Posts 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) Last fiddled with by sweety439 on 2020-08-08 at 06:39
 2020-08-08, 07:00 #935 sweety439     Nov 2016 223310 Posts 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) Last fiddled with by sweety439 on 2020-08-14 at 14:16

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