Thanks for the info. guys. It's kind of fun doing those new bases isn't it?

After a short review, I'll post it on the web pages.

KEP, I'm not sure I understand your large covering sets. Most will have small covering sets of 6 or less factors. Examples:

For Sierp base 254, k=4 has a covering set of {3, 5}. That is the factors of 3 and 5 knock out all n-values for k=4.

You are correct, it is proven because:

k n-prime

2 1

3 2

k=1 is a GFN and is not considered, although it does have a prime at n=4.

On Sierp base 251, your later analogy is correct. k=4 has a covering set of {3, 5}. As you stated, it is proven:

k n-prime

2 1

k=1 and 3 have trivial factors of 2.

One more thing, you do not need to test k=1 on any Sierp base. k=1 makes the form a Generalized Fermat number (GFN). GFN's can only have a prime when n is a perfect power of 2.

Based on the above, on your base 252, if you wanted to test k=1, you would only need to test n=1, 2, 4, 8, 16, 32, etc. There would be no need to test any other n-values as they would yield composites. But it's not necessary to test at all to prove the conjecture because it is a GFN. So per your analysis, for base 252, only k=27 is considered remaining.

Gary