Consider the sequence of numbers ranging over $n$ of the form $k\cdot a^n + 1$ for a fixed, odd natural number $k$. The number $k$ is considered a Sierpinski number if the sequence determined by that $k$ contains no primes.
One way to show that such a $k$ is a Sierpinski number is to find a covering system of congruences for the sequence proving there are no primes in the sequence. The way to show that such a $k$ is not a Sierpinski number is to find one prime among the numbers in the sequence.
For those sequences where a prime has been found, that is, $k$ is not a Sierpinski number, is it known whether there could be infinitely many primes in that sequence?
If only a finite number of primes are expected to be in the sequence, then a covering system of congruences might work after a certain $n$ to show that the rest of the sequence contains only composite numbers.
Sierpinski showed that there are an infinite number of $k$ whose sequences contain no primes. My conjecture is that there can be at most a finite number of primes for any $k$, but I wonder if this has not already been resolved by someone or if someone has already made the conjecture.