A Sierpinski/Riesel-like problem

[sweety439]

Originally Posted by sweety439

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)

Reserve S8 k=370, R8 k=239 and R8 k=757.

[sweety439]

Originally Posted by sweety439

These files are the status of S5, S9 and S11 for all k<=1024. (tested up to n=2000) (0 if there are no (probable) primes for n<=2000)

These k's with covering set, thus excluded in the files:

S5: k = 7, 11 (mod 24), covering set {2, 3}
S9: k = 31, 39 (mod 80), covering set {2, 5}
S11: k = 5, 7 (mod 12), covering set {2, 3}

Reserve S5 k=181.

[sweety439]

Originally Posted by sweety439

Reserve S8 k=370, R8 k=239 and R8 k=757.

(370*8^8300+1)/7 is (probable) prime!!!

[sweety439]

Originally Posted by sweety439

These files are the status of S5, S9 and S11 for all k<=1024. (tested up to n=2000) (0 if there are no (probable) primes for n<=2000)

These k's with covering set, thus excluded in the files:

S5: k = 7, 11 (mod 24), covering set {2, 3}
S9: k = 31, 39 (mod 80), covering set {2, 5}
S11: k = 5, 7 (mod 12), covering set {2, 3}

For S5, the remain k's are 181, 512, 625, 646 and 905. However, k=512 and 646 are already in CRUS, k=625 is a half GFN, and since 905 = 5 * 181, k=905 will have the same (probable) prime as k=181, thus, we only need to find (probable) prime for k=181.

For S9, the remain k's are 41, 311, 369, 621 and 821. However, the primes for k=41 and 821 are already found by S3 k=41 and S3 k=821: (41*3^4892+1)/2 (= (41*9^2446+1)/2)) and (821*3^5512+1)/2 (= (821*9^2756+1)/2), k=311 is already tested by S81 to n=2K (i.e. n=1K for S81) with no (probable) prime found, and since 369 = 9 * 41, k=369 will have the same (probable) prime as k=41, k=621 is already tested by S3 to n=5K (i.e. n=10K for S3) with no (probable) prime found.

S11 has very more k's remain than S5 and S9.

[sweety439]

Originally Posted by sweety439

These files are the status of R5, R9 and R11 for all k<=1024. (tested up to n=2000) (0 if there are no (probable) primes for n<=2000)

These k's with covering set, thus excluded in the files:

R5: k = 13, 17 (mod 24), covering set {2, 3}
R9: k = 41, 49 (mod 80), covering set {2, 5}, also square k's with full algebra factors
R11: k = 5, 7 (mod 12), covering set {2, 3}

For R5, the remain k's are 638, 662 and 1006. However, all the three k's are already in CRUS.

For R9 (note that k=74 and 666 are Riesel numbers for base 9), the remain k's are 119, 302, 386, 744 and 939. However, the primes for k=119 and 939 are already found by R3 k=119 and R3 k=313: (119*3^8972-1)/2 (= (119*9^4486-1)/2) and (313*3^24761-1)/2 (= (939*9^12380-1)/2), k=302 has a prime at n=2849, thus, we only need to find primes for k=386 and 744.

R11 has very more k's remain than R5 and R9.

[sweety439]

(311*9^15668+1)/8 is (probable) prime!!!

This eliminated k=311 from S9, also k=311 from S81 (since it also equals (311*81^7834+1)/8), and if (311*9^n+1)/8 is prime, then n must be even, since if n is odd, then (311*9^n+1)/8 is even and not prime).

Also, (621*3^20820+1)/2 is (probable) prime!!! This eliminated k=621 from S3!!! (k=621 was the smallest k remain for S3)

[sweety439]

Originally Posted by sweety439

I also reserved S33 and R61 to n=12K (S61 is already proven) and found that (407*33^10961+1)/8 is (probable) prime!!! S33 now has only 2 k's remain.

(407*33^10961+1)/8 is the largest (probable) prime found by this project!!!

No other (probable) prime found for S33 with n<=12K. Also, no (probable) prime found for R61 with n<=10K.

[sweety439]

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})

There are advanced Sierpinski/Riesel problems:

Finding and proving the smallest k such that (k*5^n+1)/gcd(k+1,5-1) is composite for all integers n >= 1 and k != 7, 11 (mod 24).
Finding and proving the smallest k such that (k*8^n+1)/gcd(k+1,8-1) is composite for all integers n >= 1 and k != 47, 79, 83, 181 (mod 195).
Finding and proving the smallest k such that (k*9^n+1)/gcd(k+1,9-1) is composite for all integers n >= 1 and k != 31, 39 (mod 80).
Finding and proving the smallest k such that (k*11^n+1)/gcd(k+1,11-1) is composite for all integers n >= 1 and k != 5, 7 (mod 12).
Finding and proving the smallest k such that (k*5^n-1)/gcd(k-1,5-1) is composite for all integers n >= 1 and k != 13, 17 (mod 24).
Finding and proving the smallest k such that (k*8^n-1)/gcd(k-1,8-1) is composite for all integers n >= 1 and k != 14, 112, 116, 148 (mod 195).
Finding and proving the smallest k such that (k*9^n-1)/gcd(k-1,9-1) is composite for all integers n >= 1 and k != 41, 49 (mod 80).
Finding and proving the smallest k such that (k*11^n-1)/gcd(k-1,11-1) is composite for all integers n >= 1 and k != 5, 7 (mod 12).

[sweety439]

Sierpinski problem base b: Finding and proving the smallest k such that (k*b^n+1)/gcd(k+1,b-1) is composite for all integers n >= 1.

Riesel problem base b: Finding and proving the smallest k such that (k*b^n-1)/gcd(k-1,b-1) is composite for all integers n >= 1.

[sweety439]

A large probable prime n can be proven to be prime if and only if at least one of n-1 and n+1 can be trivially written into a product.

Thus, if n is large, a probable prime (k*b^n+-1)/gcd(k+-1,b-1) can be proven to be prime if and only if gcd(k+-1,b-1) = 1.

[sweety439]

Originally Posted by sweety439

Update the text files for the 1st, 2nd and 3rd conjecture of SR12.

R12 has only 2 k's remain: 1037 and 1132, but S12 has many k's remain. (thus, I did not search S12 very far)

563*12^4020+1 is prime!!!

Reserve k=1037 and 1132 for R12.