Question on k's marked with *
 

I understand why the k's are marked with *. And, no, I don't have any answer as to whether n=0 is acceptable or not (though I would guess *not*).

However, I am not sure whether k's divisible by 5 qualify for Sierpiski/Riesel base 5 numbers. I mean, these k's don't contribute to any new unique series. The equivalent situation in regular Sierpinski/Rieslel (base 2) would be to consider even k's (which they don't consider, btw). So why should we consider the k's divisible by 5 for our search?

Any ideas?

The two definitions for base 5 Sierpinski number that have been suggested so far are:

1. k is a base 5 Sierpinski number if k*5^n+1 is composite for every positive integer n.

2. k is a base 5 Sierpinski number if k*5^n+1 is composite for every non-negative integer n.

The few candidates of the form k=5*r in the reservation list come about because k+1=r*5^1+1 is prime (and is thus r is eliminated under either definition), but no larger prime in this sequence has been found and so k=5*r is not yet eliminated under definition 1.

If we want to allow definition 1 as a possibility then we really need to eliminate these candidates, since it is possible that 5*r+1 is the last prime in that sequence. (At least, I don't know of a proof that this is impossible).

The candidates k=5*r, where r is still an open candidate, are not in the reservation lists because they will be eliminated under either definition when a prime is eventually found for r*5^n+1. (whenever someone finds a prime I add any multiples of 5 to the primes.txt file at the same time).


However, perhaps we need a new definition? We could extend the pattern of definitions above in this way:

2'. k > 0 is a base 5 Sierpinski number if k*5^n+1 is not prime for any integer n >= -1.

2''. k > 0 is a base 5 Sierpinski number if k*5^n+1 is not prime for any integer n >= -2.

and so on. This leads to the obvious definition:

3. k > 0 is a base 5 Sierpinski number if k*5^n+1 is not prime for any integer n.

I think this definition better fits your idea that only candidates that determine unique sequences should be considered.

For the purposes of this project it makes no difference whether we use definition 2 or definition 3.

Also, if we can find primes for all the candidates with stars then it will make no difference to the project whether we use definition 1 or definition 3 either, and then we will be able to refer unambiguously to 'the smallest even base 5 Sierpinski number' regardless of which definition is preferred.


(Everything above applies similarly to the even base 5 Riesel candidates).