[QUOTE=robert44444uk;95550]Is there a case for excluding GFN's? Probably not, as such numbers might produce primes, but we should accept that such bases are going to give us problems.[/QUOTE]

Personally, I like to exclude them, since they are not prime "trivially" (for some weird definition of trivial :wink: ). Plus there is a neat symmetry, since the corresponding -1 series is also excluded due to triviality.

Anyway, FWIW, couple more tests:
[code]
1*22^65536+1 [86924,-94019,-53914,4292] is composite LLR64=025A0D6038FFD624. (e=0.00496 (0.00615895~7.06466e-16@1.019) t=1009.07s)
1*22^131072+1 [-45196,-45619,-74943,30011] is composite LLR64=F8D5A92E929D694B. (e=0.00694 (0.00913339~6.99073e-16@0.998) t=4347.90s)
[/code]
Currently testing the next one. After that, I'll call it quits (maybe I should've sieved these, hmmm... :redface:)

[QUOTE=Xentar;95554]Just found in the prime database, that
18 * 14^70119+1
is prime.
Can I use this in any way for
18 * 18^n +1
?

Or, what have I search for, to make the work easier?[/QUOTE]

No, it is the base b that is critical. For example, if you found a prime on the database k*2^n+/1, (i.e. b=2, n even) then this could be useful in a search for b=4,8,16,32.... as k*4^(n/2)+/-1 would also be prime, because it is the same number, just stated differently.


The prime you found 18*18^70119+1 can be restated in terms of using a different b, as the same number is (18*2^70119)*9^70119+1, so it gives a solution for k of 18*2^70119 for b=9, 18*3^70119 for b=6, and a whopping k of 18*6^70119 for b=3 - all of which are greater k's than those we are looking for.

Similarly if you find primes which are x*3^n+/1 then those are useful for b=9,27...

It is a question of rearranging the terms, to see if k and b are sensible

[QUOTE=axn1;95555]Personally, I like to exclude them, since they are not prime "trivially" (for some weird definition of trivial. Plus there is a neat symmetry, since the corresponding -1 series is also excluded due to triviality.

[/QUOTE]

It is really tempting to define as such, given that half of the b's we are testing are even, they will bug every solution. I just noted that I had indentified base 8, with k=1 as a big problem (k=8 and 64 had produced trivial results)

Does anyone have a list of b^n+1 primes b<100 and even?

Base 23
 

Base 23 is manageable:

Sierpinski 182 [3,5,53], 5 candidates left at n=2000,

8,68,122,124,154

Riesel 476 [3,5,53] 9 candidates left at n=2000

134,194,230,314,328,394,404,464,472

More base 22 eliminated:

4440*22^5999-1 is prime
3426*22^7586-1 is prime
4302*22^7653-1 is prime
2991*22^10884+1 is prime
1335*22^11155-1 is prime
4070*22^11432-1 is prime
185*22^11433-1 is prime

I'll also reserve base 22 and 23 for the record (except for 22: 22 and 22: 484)

so actual status is:
[CODE]Base 6:

Sierpinski
1 to 243417
Reisel
1 to 213409

Base 7:

Totally horrible. Possible covering set with repeat every 24 n is [19,5,43,1201,13,181,193,73], also 5 other sets perming 73, 193 and 409.

Sierpinski and Riesel numbers are both lower than 162643669672445

Work is needed to find a low k value which is Riesel or Sierpinski.

Base 8:

Sierpinski
1
Riesel (done?)

Base 9:

Sierpinski (done ?)
Riesel
4 jasong
16
36
64
Note 16 and 64 are subsets of 4.

Base 10:

Sierpinski
4069*10^n+1
5028*10^n+1
6172*10^n+1
7404*10^n+1
7666*10^n+1
7809*10^n+1
8194*10^n+1
9175*10^n+1 (status not known)
Riesel
1343*10^n-1
1803*10^n-1
1935*10^n-1
3356*10^n-1
4421*10^n-1
6665*10^n-1
7019*10^n-1
8579*10^n-1
10176*10^n-1 (status not known)

Base 11:

Sierpinski
416 tnerual
958 tnerual
Riesel
62
682
862
904
1528
2410
2690
3110
3544
3788
4208
4564

Base 12:

Sierpinski
1 to 14599
Riesel
1 to 16328.

Base 13:

Sierpinski (done)
Riesel
288

Base 14: done

Base 15:

Horrible. A covering set is [241,113,211,17,1489,13,3877], and Sierpinski and Riesel values are therefore less than 7330957703181619. As bad as the base 3 problem.

Base 16:

Sierpinski number not known,
186 (to be removed see post #49 below by citrix)
2158 (tested up to n=4000 by citrix)
2857 (tested up to n=4000 by citrix)
2908 (tested up to n=4000 by citrix)
3061 (tested up to n=4000 by citrix)
4885 (tested up to n=4000 by citrix)
5886 (tested up to n=4000 by citrix)
6348 (tested up to n=4000 by citrix)
6663 (tested up to n=4000 by citrix)
6712 (tested up to n=4000 by citrix)
7212 (tested up to n=4000 by citrix)
7258 (tested up to n=4000 by citrix)
7615 (tested up to n=4000 by citrix)
7651 (tested up to n=4000 by citrix)
7773 (tested up to n=4000 by citrix)
8025 (tested up to n=4000 by citrix)
10001 to 66740
Riesel
1343*16^n-1
1803*16^n-1
1935*16^n-1
2333*16^n-1
3015*16^n-1
3332*16^n-1
4478*16^n-1
4500*16^n-1
4577*16^n-1
5499*16^n-1
5897*16^n-1
6588*16^n-1
6633*16^n-1
6665*16^n-1
7019*16^n-1
7602*16^n-1
8174*16^n-1
8579*16^n-1
10001 to 33965

Base 17:

Sierpinski
92 (LTD)
160 (LTD)
244 (LTD)
262 (LTD)
Riesel (done)


Base 18:

Sierpinski
18 xentar
324 xentar
122 xentar
381 xentar
Riesel (done)

Base 19:
?

Base 20:
?

Base 21:

Sierpinski
118 (checked to n=3500)
riesel (done)

Base 22:

Sierpinski
22 (cedricvonck)
484 (cedricvonck)
942 (michaf)
1611 (michaf)
1908 (michaf)
4233 (michaf)
5061 (michaf)
5128 (michaf)
5659 (michaf)
6234 (michaf)
6462 (michaf)
Riesel
1013 (michaf)
2853 (michaf)
3104 (michaf)
3656 (michaf)
4001 (michaf)
4118 (michaf)

Base 23:

Sierpinski (all tested up to n=2000)
8 (michaf)
68 (michaf)
122 (michaf)
124 (michaf)
154 (michaf)
Riesel (all tested up to n=2000)
134 (michaf)
194 (michaf)
230 (michaf)
314 (michaf)
328 (michaf)
394 (michaf)
404 (michaf)
464 (michaf)
472 (michaf)

[/CODE]

as always i'm not sure with the end of the base 10 range.

if you have any reservation, update, correction, ... feel free to post

Status update:
18 * 18 ^ n + 1 n>110,000
122 * 18 ^ n + 1 n>22,000
324 * 18 ^ n + 1 n>88,000
381 * 18 ^ n + 1 not tested yet
first, checked for small primes - without success
now, I will concentrate on a k = 18

By the way (I'm no mathematician)
when I find a prime for
18 * 18 ^ n + 1
it would mean that
324 * 18 ^ (n-1) + 1
should be prime too, right? 'cause 324 = 18 * 18

Btw: Robert, thank you for your explanation.

Found:

6172*10^10740+1
7809*10^11793+1
4069*10^12095+1
3356*10^13323-1

All remaining base 10 candidates are done to 20,000. I will split base 10 updates to a new thread and suggest the other bases find their homes in another thread. I also suggest that a sticky thread is created for all bases with their current status with updates in threads for each base. Those threads do not need to be stickies.

[QUOTE=Xentar;95614]
By the way (I'm no mathematician)
when I find a prime for
18 * 18 ^ n + 1
it would mean that
324 * 18 ^ (n-1) + 1
should be prime too, right? 'cause 324 = 18 * 18
[/QUOTE]

Correct! So no need to check 324!!

Proof
 

I have been discussing with David Broadhurst whether all bases produce covering sets.

We have proved this for all bases not of the form 2^n-1 !! Furthermore, we have proved that the covering set repeats no greater than every 12n

See [url]http://tech.groups.yahoo.com/group/primeform/message/8214[/url]