12*312^21162+1 is prime.
Result text file attached.

2*801^n+1 is currently at n=28661, and 7*1004^n+1 is currently at n=31030, both no primes found.

Many of the bases for 8<=k<=12 are only searched to n=5K, I will reserve all such bases to n=25K.

Update current file.

The exclusions are:

[CODE]
Riesel k=2:
none

Riesel k=3:
b==(1 mod 2); factor of 2

Riesel k=4:
b==(1 mod 3); factor of 3
b==(4 mod 5): odd n, factor of 5; even n, algebraic factors
b=m^2 proven composite by full algebraic factors

Riesel k=5:
b==(1 mod 2); factor of 2

Riesel k=6:
b==(1 mod 5); factor of 5
b==(34 mod 35); covering set [5, 7]
b=6*m^2 with m==(2, 3 mod 5): even n, factor of 5; odd n, algebraic factors
(This includes bases 24, 54, 294, 384, 864, 1014.)

Riesel k=7:
b==(1 mod 2); factor of 2
b==(1 mod 3); factor of 3

Riesel k=8:
b==(1 mod 7) has a factor of 7
b==(20 mod 21) has a covering set of [3, 7]
b==(83, 307 mod 455) has a covering set of [5, 7, 13]
(This includes bases 83, 307, 538, 762, 993.)
b=m^3 proven composite by full algebraic factors

Riesel k=9:
b==(1 mod 2) has a factor of 2
b==(4 mod 5): odd n has a factor of 5; even n has algebraic factors
b=m^2 proven composite by full algebraic factors

Riesel k=10:
b==(1 mod 3) has a factor of 3
b==(32 mod 33) has a covering set of [3, 11]

Riesel k=11:
b==(1 mod 2) has a factor of 2
b==(1 mod 5) has a factor of 5
b==(14 mod 15) has a covering set of [3, 5]

Riesel k=12:
b==(1 mod 11) has a factor of 11
b==(142 mod 143) has a covering set of [11, 13]
base 307 has a covering set of [5, 11, 29]
base 901 has a covering set of [7, 11, 13, 19]

Sierp k=2:
b==(1 mod 3); factor of 3
base 512 is a GFN with no known prime

Sierp k=3:
b==(1 mod 2); factor of 2

Sierp k=4:
b==(1 mod 5); factor of 5
b==(14 mod 15); covering set [3, 5]
base 625 proven composite by full algebraic factors
bases 32, 512, and 1024 are GFN's with no known prime

Sierp k=5:
b==(1 mod 2); factor of 2
b==(1 mod 3); factor of 3

Sierp k=6:
b==(1 mod 7); factor of 7
b==(34 mod 35); covering set [5, 7]

Sierp k=7:
b==(1 mod 2); factor of 2

Sierp k=8:
b==(1 mod 3) has a factor of 3
b==(20 mod 21) has a covering set of [3, 7]
b==(47 or 83 mod 195) has a covering set of [3, 5, 13]
(This includes bases 47, 83, 242, 278, 437, 473, 632, 668, 827, 863, 1022.)
base 467 has a covering set of [3, 5, 7, 19, 37]
base 722 has a covering set of [3, 5, 13, 73, 109]
b=m^3 proven composite by full algebraic factors
base 128 is a GFN with no possible prime

Sierp k=9:
b==(1 mod 2) has a factor of 2
b==(1 mod 5) has a factor of 5

Sierp k=10:
b==(1 mod 11) has a factor of 11
b==(32 mod 33) has a covering set of [3, 11]
base 1000 is a GFN with no known prime

Sierp k=11:
b==(1 mod 2) has a factor of 2
b==(1 mod 3) has a factor of 3
b==(14 mod 15) has a covering set of [3, 5]

Sierp k=12:
b==(1 mod 13) has a factor of 13
b==(142 mod 143) has a covering set of [11, 13]
bases 296 and 901 have a covering set of [7, 11, 13, 19]
bases 562, 828, and 900 have a covering set of [7, 13, 19]
base 563 has a covering set of [5, 7, 13, 19, 29]
base 597 has a covering set of [5, 13, 29]
base 12 is a GFN with no known prime
[/CODE]

I have updated the files in post 62.

Now I reserve all bases for k = 8 to 12 that are at n<25K to n=25K and found that 8*284^5266-1 and 10*1020^6944-1 are primes. (using pfgw)

Current at:

Riesel k=8: n=6144
Riesel k=9: n=7445
Riesel k=10: n=7025
Riesel k=11: n=9679
Riesel k=12: n=8690
Sierp k=8: n=6135
Sierp k=9: n=9541
Sierp k=10: n=5828
Sierp k=11: n=9568
Sierp k=12: n=5631

Only found the above two primes.

[QUOTE=sweety439;489251]Now I reserve all bases for k = 8 to 12 that are at n<25K to n=25K and found that 8*284^5266-1 and 10*1020^6944-1 are primes. (using pfgw)

Current at:

Riesel k=8: n=6144
Riesel k=9: n=7445
Riesel k=10: n=7025
Riesel k=11: n=9679
Riesel k=12: n=8690
Sierp k=8: n=6135
Sierp k=9: n=9541
Sierp k=10: n=5828
Sierp k=11: n=9568
Sierp k=12: n=5631

Only found the above two primes.[/QUOTE]

12*826^5786+1 is prime.

8*854^6500-1 is prime.

Currently at:

Riesel k=8: n=6632
Riesel k=9: n=8346
Riesel k=10: n=7750
Riesel k=11: n=11009
Riesel k=12: n=9840
Sierp k=8: n=6599
Sierp k=9: n=10619
Sierp k=10: n=6196
Sierp k=11: n=11120
Sierp k=12: n=5915

8*194^38360-1 is prime

4*312^51565-1 is prime!