A Sierpinski/Riesel-like problem - Page 26

sweety439 · 2017-06-01, 15:36

Update word files to include the status (compare with CRUS) for SR22, SR46, SR58 and SR63.

sweety439 · 2017-06-01, 15:44

Found these (probable) primes:

(12709*42^1815+1)/41
(563*48^1236+1)/47
(9852*60^1441+1)/59
(1764*42^1317-1)/41
(1599*48^1857-1)/47

Still remain:

S42: k=13283
S48: none
S60: k=4896
R42: k=1600, 6971, 14884
R48: none
R60: k=7671, 16167, 18055

sweety439 · 2017-06-01, 16:56

(7671*60^2239-1)/59 is (probable) prime!!!

Now there are 2 k's remain for R60 with k = 1 mod 59.

sweety439 · 2017-06-01, 18:02

These are text files for SR8 with k<=1024.

These k's are excluded in the text files:

S8: All cube k's (with algebra factors) and all k's = 47, 79, 83, 181 (mod 195) (with covering set {3, 5, 13}).

R8: All cube k's (with algebra factors) and all k's = 14, 112, 116, 148 (mod 195) (with covering set {3, 5, 13}).

Some large n's are given using link: https://www.rieselprime.de/.

Remain k's without known (probable) prime:

S8: 256, 370, 467, 937.
R8: 239, 247, 757.

An interesting one: R8 k=658 with covering set {3, 5, 19, 37, 73} (this is listed in the text file).

sweety439 · 2017-06-02, 16:14

Quote:

Originally Posted by sweety439

These are text files for SR8 with k<=1024.

These k's are excluded in the text files:

S8: All cube k's (with algebra factors) and all k's = 47, 79, 83, 181 (mod 195) (with covering set {3, 5, 13}).

R8: All cube k's (with algebra factors) and all k's = 14, 112, 116, 148 (mod 195) (with covering set {3, 5, 13}).

Some large n's are given using link: https://www.rieselprime.de/.

Remain k's without known (probable) prime:

S8: 256, 370, 467, 937.
R8: 239, 247, 757.

An interesting one: R8 k=658 with covering set {3, 5, 19, 37, 73} (this is listed in the text file).

Found the (probable) prime (937*8^1332+1)/7.

No other (probable) prime for SR8 for these k's was found. (all of these k's are likely tested to n=10000)

sweety439 · 2017-06-02, 17:10

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

Code:

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.

sweety439 · 2017-06-02, 17:53

Quote:

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:

Code:

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.

These text files are the status for these conjectures: (only for bases 5, 8, 9, 11, 13, 14 and 16) (the prime (41*9^2446+1)/2 (S9, k=41) is converted by S3, k=41) (the prime 186*16^5229+1 (S16, k=186) is converted by S4, k=186)

These bases and k's are remain:

Code:

base    k
S13     29
S16     89, 215, 459, 515

Interestingly, there are no k's remain in the Riesel conjectures.

Thus, for all these Sierpinski/Riesel bases except S13 and S16, all of the 1st, 2nd and 3rd conjectures are proven. (the 1st and 2nd conjectures for S13, and the 1st conjecture for S16 are also proven, but the 3rd conjecture for S13, and the 2nd and 3rd conjecture for S16 are not proven)

sweety439 · 2017-06-02, 18:08

Quote:

Originally Posted by sweety439

These text files are the status for these conjectures: (only for bases 5, 8, 9, 11, 13, 14 and 16)

These bases and k's are remain:

Code:

base    k
S13     29
S16     89, 215, 459, 515

Interestingly, there are no k's remain in the Riesel conjectures.

Thus, for all these Sierpinski/Riesel bases except S13 and S16, all of the 1st, 2nd and 3rd conjectures are proven. (the 1st and 2nd conjectures for S13, and the 1st conjecture for S16 are also proven, but the 3rd conjecture for S13, and the 2nd and 3rd conjecture for S16 are not proven)

Note: these k's have algebra factors and should be excluded from the conjectures:

S8: all k = m^3
S16: all k = 4*m^4
R8: all k = m^3
R9: all k = m^2
R14: all k = m^2 and m = 2 or 3 mod 5, and all k = 14*m^2 and m = 2 or 3 mod 5
R16: all k = m^2

For more information for algebra factors, see post #104 (for bases <= 32) and #158 (for 33 <= bases <= 64).

sweety439 · 2017-06-02, 19:49

Quote:

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:

Code:

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.

In fact,

All k = 7 or 11 mod 24 are Sierpinski in base 5. (with covering set {2, 3})
All k = 13 or 17 mod 24 are Riesel in base 5. (with covering set {2, 3})
All k = 47, 79, 83 or 181 mod 195 are Sierpinski in base 8. (with covering set {3, 5, 13})
All k = 14, 112, 116 or 148 mod 195 are Riesel in base 8. (with covering set {3, 5, 13})
All k = 31 or 39 mod 80 are Sierpinski in base 9. (with covering set {2, 5})
All k = 41 or 49 mod 80 are Riesel in base 9. (with covering set {2, 5})
All k = 5 or 7 mod 12 are both Sierpinski and Riesel in base 11. (with covering set {2, 3})
All k = 15 or 27 mod 56 are Sierpinski in base 13. (with covering set {2, 7})
All k = 29 or 41 mod 56 are Riesel in base 13. (with covering set {2, 7})
All k = 4 or 11 mod 15 are both Sierpinski and Riesel in base 14. (with covering set {3, 5})
All k = 31 or 47 mod 96 are Sierpinski in base 17. (with covering set {2, 3})
All k = 49 or 65 mod 96 are Riesel in base 17. (with covering set {2, 3})
All k = 9 or 11 mod 20 are both Sierpinski and Riesel in base 19. (with covering set {2, 5})
All k = 8 or 13 mod 21 are both Sierpinski and Riesel in base 20. (with covering set {3, 7})
All k = 23 or 43 mod 88 are Sierpinski in base 21. (with covering set {2, 11})
All k = 45 or 65 mod 88 are Riesel in base 21. (with covering set {2, 11})
All k = 5 or 7 mod 12 are both Sierpinski and Riesel in base 23. (with covering set {2, 3})
All k = 79 or 103 mod 208 are Sierpinski in base 25. (with covering set {2, 13})
All k = 105 or 129 mod 208 are Riesel in base 25. (with covering set {2, 13})
All k = 13 or 15 mod 28 are both Sierpinski and Riesel in base 27. (with covering set {2, 7})
All k = 4 or 11 mod 15 (with covering set {3, 5}) and all k = 7 or 11 mod 24 (with covering set {2, 3}) and all k = 19 or 31 mod 40 (with covering set {2, 5}) are Sierpinski in base 29.
All k = 4 or 11 mod 15 (with covering set {3, 5}) and all k = 13 or 17 mod 24 (with covering set {2, 3}) and all k = 9 or 21 mod 40 (with covering set {2, 5}) are Riesel in base 29.
All k = 10 or 23 mod 33 are both Sierpinski and Riesel in base 32. (with covering set {3, 11})

sweety439 · 2017-06-02, 20:10

Quote:

Originally Posted by sweety439

In fact,

All k = 7 or 11 mod 24 are Sierpinski in base 5. (with covering set {2, 3})
All k = 13 or 17 mod 24 are Riesel in base 5. (with covering set {2, 3})
All k = 47, 79, 83 or 181 mod 195 are Sierpinski in base 8. (with covering set {3, 5, 13})
All k = 14, 112, 116 or 148 mod 195 are Riesel in base 8. (with covering set {3, 5, 13})
All k = 31 or 39 mod 80 are Sierpinski in base 9. (with covering set {2, 5})
All k = 41 or 49 mod 80 are Riesel in base 9. (with covering set {2, 5})
All k = 5 or 7 mod 12 are both Sierpinski and Riesel in base 11. (with covering set {2, 3})
All k = 15 or 27 mod 56 are Sierpinski in base 13. (with covering set {2, 7})
All k = 29 or 41 mod 56 are Riesel in base 13. (with covering set {2, 7})
All k = 4 or 11 mod 15 are both Sierpinski and Riesel in base 14. (with covering set {3, 5})
All k = 31 or 47 mod 96 are Sierpinski in base 17. (with covering set {2, 3})
All k = 49 or 65 mod 96 are Riesel in base 17. (with covering set {2, 3})
All k = 9 or 11 mod 20 are both Sierpinski and Riesel in base 19. (with covering set {2, 5})
All k = 8 or 13 mod 21 are both Sierpinski and Riesel in base 20. (with covering set {3, 7})
All k = 23 or 43 mod 88 are Sierpinski in base 21. (with covering set {2, 11})
All k = 45 or 65 mod 88 are Riesel in base 21. (with covering set {2, 11})
All k = 5 or 7 mod 12 are both Sierpinski and Riesel in base 23. (with covering set {2, 3})
All k = 79 or 103 mod 208 are Sierpinski in base 25. (with covering set {2, 13})
All k = 105 or 129 mod 208 are Riesel in base 25. (with covering set {2, 13})
All k = 13 or 15 mod 28 are both Sierpinski and Riesel in base 27. (with covering set {2, 7})
All k = 4 or 11 mod 15 (with covering set {3, 5}) and all k = 7 or 11 mod 24 (with covering set {2, 3}) and all k = 19 or 31 mod 40 (with covering set {2, 5}) are Sierpinski in base 29.
All k = 4 or 11 mod 15 (with covering set {3, 5}) and all k = 13 or 17 mod 24 (with covering set {2, 3}) and all k = 9 or 21 mod 40 (with covering set {2, 5}) are Riesel in base 29.
All k = 10 or 23 mod 33 are both Sierpinski and Riesel in base 32. (with covering set {3, 11})

Also,

4 is both Sierpinski and Riesel in all bases b = 14 mod 15. (with covering set {3, 5})
5 is both Sierpinski and Riesel in all bases b = 11 mod 12. (with covering set {2, 3})
6 is both Sierpinski and Riesel in all bases b = 34 mod 35. (with covering set {5, 7})
7 is Sierpinski in all bases b = 5, 11 or 23 mod 24. (with covering set {2, 3})
7 is Riesel in all bases b = 11 mod 12. (with covering set {2, 3})
8 is Sierpinski in all bases b = 20 mod 21 (with covering set {3, 7}) and all bases b = 47 or 83 mod 195. (with covering set {3, 5, 13})
8 is Riesel in all bases b = 20 mod 21 (with covering set {3, 7}) and all bases b = 83 or 307 mod 455. (with covering set {5, 7, 13})
9 is Sierpinski in all bases b = 19 mod 20. (with covering set {2, 5})
9 is Riesel in all bases b = 19, 29 or 39 mod 40. (with covering set {2, 5})
10 is both Sierpinski and Riesel in all bases b = 32 mod 33. (with covering set {3, 11})
11 is Sierpinski in all bases b = 14 mod 15 (with covering set {3, 5}) and all bases b = 19 mod 20 (with covering set {2, 5}) and all bases b = 5 mod 24. (with covering set {2, 3})
11 is Riesel in all bases b = 14 mod 15 (with covering set {3, 5}) and all bases b = 19 mod 20. (with covering set {2, 5})
12 is both Sierpinski and Riesel in all bases b = 142 mod 143. (with covering set {11, 13})

(1, 2 and 3 are neither Sierpinski not Riesel in all small bases)

sweety439 · 2017-06-03, 15:41

Quote:

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:

Code:

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.

The 1st, 2nd and 3rd conjecture of Sierpinski/Riesel bases 17, 18, 19, 20, 21, 23, 25, 26, 27, 29, 31 and 32 are:

Code:

base    1st    2nd    3rd
S17      31     47    127
S18     398    512    571
S19       9     11     29
S20       8     13     29
S21      23     43     47
S23       5      7     17
S25      79    103    185
S26     221    284   1627
S27      13     15     41
S29       4      7     11
S31     239    293    521
S32      10     23     43
R17      49     59     65
R18     246    664    723
R19       9     11     29
R20       8     13     29
R21      45     65    133
R23       5      7     17
R25     105    129    211
R26     149    334   1892
R27      13     15     41
R29       4      9     11
R31     145    265    443
R32      10     23     43

2017-06-01, 16:56	#278
sweety439 "99(4^34019)99 palind" Nov 2016 (P^81993)SZ base 36 101101111010₂ Posts	(767160^2239-1)/59 is (probable) prime!!! Now there are 2 k's remain for R60 with k = 1 mod 59. Last fiddled with by sweety439 on 2017-06-01 at 16:56*

2017-06-02, 17:10	#281
sweety439 "99(4^34019)99 palind" Nov 2016 (P^81993)SZ base 36 2·13·113 Posts	The 1st, 2nd and 3rd conjecture of Sierpinski/Riesel bases 4, 5, 7, 8, 9, 10, 11, 12, 13, 14 and 16 are: Code: 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. Last fiddled with by sweety439 on 2017-06-02 at 19:32

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2017-06-01, 15:44	#277
sweety439 "99(4^34019)99 palind" Nov 2016 (P^81993)SZ base 36 2×13×113 Posts	Found these (probable) primes: (1270942^1815+1)/41 (56348^1236+1)/47 (985260^1441+1)/59 (176442^1317-1)/41 (1599*48^1857-1)/47 Still remain: S42: k=13283 S48: none S60: k=4896 R42: k=1600, 6971, 14884 R48: none R60: k=7671, 16167, 18055