mersenneforum.org A Sierpinski/Riesel-like problem
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2020-08-05, 03:41   #925
sweety439

Nov 2016

11·227 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, 33 views) prime Riesel.txt (1.1 KB, 31 views)

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

2020-08-06, 03:25   #926
sweety439

Nov 2016

11·227 Posts

searched to base 256 (also base 512, 1024, 2048, 4096, 8192, 16384, 32768, 65536)
Attached Files
 prime Sierpinski.txt (2.1 KB, 23 views) prime Riesel.txt (2.2 KB, 32 views)

2020-08-06, 03:44   #927
sweety439

Nov 2016

11×227 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

11×227 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, 27 views) conjectured first 4 Riesel numbers.txt (6.3 KB, 28 views)

 2020-08-06, 06:43 #929 sweety439     Nov 2016 1001110000012 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

249710 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, 31 views)

2020-08-06, 12:34   #931
sweety439

Nov 2016

9C116 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, 25 views)

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

2020-08-07, 16:01   #932
sweety439

Nov 2016

11·227 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

249710 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 11×227 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 11×227 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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