Sierpinski
[Si60] proved that there are infinitely many Sierpinski numbers, using covering systems to construct suitable covering sets for any $m$ satisfying a certain congruence. This establishes that there is a positive density set of such $m$.
The smallest Sierpinski number is believed to be $78557$, which was found by Selfridge.
Erdős and Graham
[ErGr80] asked whether there are Sierpinski numbers for which a covering system is not 'responsible', for which the best interpretation seems to be the above question. This is formulated precisely in problem F13 of Guy's collection
[Gu04]. Erdős and Graham thought the answer is yes (in that there are such Sierpinski numbers), since otherwise this would imply there are infinitely many Fermat primes.
There is now further evidence with a concrete example: an argument of Izotov
[Iz95], given in more detail by Filaseta, Finch, and Kozek
[FFK08], suggests that $m=734110615000775^4$ is a Sierpinski number without a covering set. (Izotov proved that this $m$ is indeed a Sierpinski number.)
Filaseta, Finch, and Kozek
[FFK08] give a revised conjecture, suggesting that every Sierpinski number is either a perfect power or else has a finite covering set of primes. They also prove that for every $l\geq 1$ there is an $m$ such that $2^km^i+1$ is composite for all $1\leq i\leq l$ and $k\geq 0$.
See also
[203], and
[276] for another problem in which the question is whether covering systems are always responsible.
