These k's are still remain for these bases, they are likely tested to at least n=15K.

S5, k=181
S16, k=89
R8, k=239 and 757

Reserve k=386 and 744 for R9.

Update the file for the status for the 1st, 2nd and 3rd conjecture for SR10.

All the three conjectures for R10 are proven, but S10 has 5 k's remain for k < 3rd CK: 100, 269, 1343, 2573, 3356 (k = 1000 and 2690 are included in the conjectures but excluded from testing, since these k-values will have the same (probable) prime as k = 100 and 269)

1037*12^6281-1 is prime!!!

No prime found for R9 k=386, R9 k=744 and R12 k=1132, they are likely tested to at least n=15K.

Update the current file for Sierpinski bases 5, 8, 9, 11 for all k's <= 1024.

Update the current file for Riesel bases 5, 8, 9, 11 for all k's <= 1024.

[QUOTE=sweety439;468776]Update the file for the status for the 1st, 2nd and 3rd conjecture for SR10.

All the three conjectures for R10 are proven, but S10 has 5 k's remain for k < 3rd CK: 100, 269, 1343, 2573, 3356 (k = 1000 and 2690 are included in the conjectures but excluded from testing, since these k-values will have the same (probable) prime as k = 100 and 269)[/QUOTE]

Reserve S10 (for k=1343, 2573 and 3356).

[QUOTE=sweety439;462272]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)[/QUOTE]

S12 has 9 k's remain:

12, 563, 846, 885, 911, 976, 1041, 1052, 1057. (k = 144 is included in the conjectures but excluded from testing, since this k-value will have the same prime as k = 12)

Since 563*12^4020+1 is prime, k=563 can be removed, I will run other k's (except k=12) after the reservations for S10 were done (see post [URL="http://mersenneforum.org/showpost.php?p=469196&postcount=469"]#469[/URL]).

[QUOTE=sweety439;469196]Reserve S10 (for k=1343, 2573 and 3356).[/QUOTE]

(3356*10^4584+1)/9 is (probable) prime!!!

If k is a rational power of b, then...

* In Riesel case, if the divisor (i.e. gcd(k-1,b-1)) is d, then these numbers are repunit numbers in positive base d+1. (if and only if d+1 is perfect power, then these numbers have algebra factors)

* In Sierpinski case, if the divisor (i.e. gcd(k+1,b-1)) is 1, then these numbers are generalized Fermat numbers in base m, where m is the largest integer such that both k and b are integer powers of m. (if and only if m is perfect odd power, then these numbers have algebra factors)

* In Sierpinski case, if the divisor (i.e. gcd(k+1,b-1)) is 2, then these numbers are half generalized Fermat numbers in base m, where m is the largest integer such that both k and b are integer powers of m. (if and only if m is perfect odd power, then these numbers have algebra factors)

* In Sierpinski case, if the divisor (i.e. gcd(k+1,b-1)) is d and d>=3, then these numbers are repunit numbers in negative base -(d-1). (if and only if d-1 is either perfect odd power or of the form 4*m^4, then these numbers have algebra factors)

[QUOTE=sweety439;469196]Reserve S10 (for k=1343, 2573 and 3356).[/QUOTE]

No (probable) primes found for k=269, 1343 and 2573, all of the k's are likely tested to at least n=15K.