Karsten: Any progress on a proof that 5 * b[sup]b[/sup] + 1 is never prime?

well I have more proof that they should be prime for what we've limited it down to but one thing I have to figure out to solve it lol.

Also: @Karsten(2): Haven't you ever considered collecting Proth numbers? (Where k ≥ 50000.)

[QUOTE=3.14159;231392]Karsten: Any progress on a proof that 5 * b[sup]b[/sup] + 1 is never prime?[/QUOTE]

A proof, 5*b^b+1 is composite for all b>=1 can never be done by brute force, this has to be done by a mathmatical proof only.

What I'm doing is to find a counterexample.

And up to b=34000 no one found yet.

[QUOTE=3.14159;231411]Also: @Karsten(2): Haven't you ever considered collecting Proth numbers? (Where k ≥ 50000.)[/QUOTE]

Yes, but not only for k or n above a limit.
If so, like for Riesel [b]all[/b] of them (if possible).
But this only if I got a Database (SQL, PHP) for inserting and listing them, it's too much work manually only.

as far as i can see any (6*n)^(6*n)

is even so 5* that gives a end 0 so they all end in 1

the things I've found using Pari scripts means a pattern of ending with 5 (not prime except 5 anyways) didn't fit 6n+1 as prime, most of the squares knocked out had roots of prime number so they got knocked out as well.

the last set i haven't a pattern for start with 91 to check if it's in that group I need patterns lol.

If you look [url=http://factordb.com/index.php?query=5*%28n*6%29%5E%28n*6%29%2B1&use=n&n=1&sent=Show&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=20&format=1]here[/url] you see, there's no pattern in factoring 5*b^b+1 with b==0 mod 6 (or b multiples of 6).
So there seems not exit a full covering set for those values, the only ones to consider and testing further.

I'm not looking for a covering set I'm simple looking at all non primes of 6n+1 and checking if they fit into a groups i can define so far no group they fit into I can't define another group on the information i have.

[QUOTE=Karsten]Yes, but not only for k or n above a limit.
If so, like for Riesel all of them (if possible).
But this only if I got a Database (SQL, PHP) for inserting and listing them, it's too much work manually only.[/QUOTE]

I say for k ≥ 50000, because smaller values have already been thoroughly searched by PrimeGrid, et. al.

Submissions: 16587*2^45008+1 (13553 digits)

Verification:
Primality testing 16587*2^45008+1 [N-1, Brillhart-Lehmer-Selfridge]
Running N-1 test using base 5
Special modular reduction using all-complex FFT length 4K on 16587*2^45008+1
Calling Brillhart-Lehmer-Selfridge with factored part 99.97%
16587*2^45008+1 is prime! (3.0277s+0.0171s)

17613*2^45008+1 (13554 digits)

Verification:
Primality testing 17613*2^45008+1 [N-1, Brillhart-Lehmer-Selfridge]
Running N-1 test using base 7
Special modular reduction using all-complex FFT length 4K on 17613*2^45008+1
Calling Brillhart-Lehmer-Selfridge with factored part 99.97%
17613*2^45008+1 is prime! (3.0014s+0.0007s)

10025*2^45009+1 (13554 digits)

Verification:
Primality testing 10025*2^45009+1 [N-1, Brillhart-Lehmer-Selfridge]
Running N-1 test using base 3
Special modular reduction using all-complex FFT length 3072 on 10025*2^45009+1
Calling Brillhart-Lehmer-Selfridge with factored part 99.97%
10025*2^45009+1 is prime! (2.2585s+0.0011s)

11143*2^45010+1 (13554 digits)

Verification:
Primality testing 11143*2^45010+1 [N-1, Brillhart-Lehmer-Selfridge]
Running N-1 test using base 3
Special modular reduction using all-complex FFT length 4K on 11143*2^45010+1
Calling Brillhart-Lehmer-Selfridge with factored part 99.97%
11143*2^45010+1 is prime! (2.9564s+0.0007s)

I wonder why, for 10025*2^45009+1, it used 3072 as opposed to 4096 on all the others, for the FFT length.

[QUOTE=3.14159;231424]I say for k ≥ 50000, because smaller values have already been thoroughly searched by PrimeGrid, et. al.[/QUOTE]

PG is searching k<10000. And the results are not shown anywhere, only pages with primes for k<1200 exist.