Quote:
Originally Posted by KEP
Well I did use NewPGen to test base 251 for k=4, for as many n as NewPGen could by curtan verify (before starting to test 2148....), the limit of n is higher than 900M, for k=4 for base 251, but I decided to stop testing there since p<=5 for the entire range removed all n's. Well I guess it has something to do with the trivial factors

Definition of trivial factors for the conjectures: Each and every nvalue has the same factor.
Hence k's with trivial factors cannot be considered the conjecture nor can they beconsidered remaining if they are lower than the conjecture. This is because they will always be composite.
As you found, NewPGen or Srsieve would quickly sieve all nvalues out.
Here: 4*251^n+1 always has a trivial factor of 5. Here's a demonstration:
4*251^1+1 = 1005 = 3*5*67
4*251^2+1 = 252005 = 5*13*3877
4*251^3+1 = 63253005 = 3*5*17*248051
4*251^4+1 = 15876504005 = 5*41*3061*25301
etc.
If factors and covering sets confuse you, try plugging the above into Alperton's excellent
prime factoring web page to get the prime factors of the first few nvalues of a base before starting on it. I learn a lot by looking at the patterns of factors that occur in the various forms. It is also how I determine the smallest covering set after determining the lowest conjectured k.
One more thing: If you still plan on testing Sierp base 255 to n=2500, be sure and remove the following k's before reporting primes remaining:
k==(1 mod 2) [odd k values]
k==(126 mod 127) [k's that leave a remainder of 126 after dividing by 127, i.e. 126, 253, 380, etc.]
When starting new bases, it's essential to understand how all of the trivial factors work or you wind up with k's remaining that you shouldn't and you end up searching things that are proven composite for all nvalues.
I hope this helps.
Gary