Covering set

Definition

Let $S$ be a subset of the set of integers. A covering set for $S$ is a set $P$ of primes such that every element of $S$ is divisible by at least one of those primes. Note that any prime number in $S$ must be contained in $P$. Thus:

• If $P$ and $S$ are disjoint, then that implies that every element of $S$ is composite.
• If $P$ and $S$ have a finite intersection, then $S$ has only finitely many primes.

The notion of covering set is useful for proving that a number is a Sierpinski number or a Riesel number.