"Provable" Sierpinski and Riesel numbers

[David Wilson]

This is a question about Sierpinski and Riesel numbers.

Sierpinski numbers are numbers k such that 2^n k + 1 is composite for all n
>= 1.
Riesel numbers are numbers k such that 2^n k - 1 is composite for all n >=
1.

Let us call the sequence Sk = (2k+1, 4k+1, 8k+1, 16k+1, ...) the Sierpinski
trajectory
of k.  k is proven to be non-Sierpinski by finding a prime element of Sk.
The
standard proof the k is Sierpinski is by finding a periodic sequence Dk with
Each Dk[i] a nontrivial divisor of Sk[i].

For example, with k = 78557, we have Sk = (157115, 314229, 628457, 1256913,
...).
We can show that the periodic sequence Dk with period
(5,3,73,3,5,3,7,3,5,3,13,3,5,3,19,3,5,3,7,3,5,3,13,3,5,3,37,3,5,3,7,3,5,3,13
,3) gives
Dk[i] a nontrivial divisor of Sk[i], so that every element of Sk is
composite
and k is Sierpinski.

Riesel numbers are proved and disproved similarly.

My question is:

Is there a way to show that a given number k does not admit a periodic
sequence Dk with Dk[i] divides Sk[i], except to exhibit a prime element
of Sk?