mersenneforum.org - A Sierpinski/Riesel-like problem

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- sweety439 (https://www.mersenneforum.org/forumdisplay.php?f=137)

- - A Sierpinski/Riesel-like problem (https://www.mersenneforum.org/showthread.php?t=21839)

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

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

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

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

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: [URL]https://www.rieselprime.de/[/URL].

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).

[QUOTE=sweety439;460245]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: [URL]https://www.rieselprime.de/[/URL].

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).[/QUOTE]

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)

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
[/CODE]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.

[QUOTE=sweety439;460352]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
[/CODE]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)[/QUOTE]

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 [URL="http://mersenneforum.org/showpost.php?p=451367&postcount=104"]#104[/URL] (for bases <= 32) and [URL="http://mersenneforum.org/showpost.php?p=455591&postcount=158"]#158[/URL] (for 33 <= bases <= 64).

[QUOTE=sweety439;460350]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
[/CODE]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.[/QUOTE]

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})

[QUOTE=sweety439;460364]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})[/QUOTE]

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)