Summary: Using an idea of Erdős, Sierpiński proved that there exist infinitely many odd positive integers $k$ such that $k\bullet 2^{n}+1$ is composite for all positive integers $n$. In this paper we give a brief discussion of Sierpiński's theorem and some variations that have been examined, including the work of Riesel, Brier, Chen, and most recently, Filaseta, Finch and Kozek. The majority of the paper is devoted to the presentation of some new results concerning our own variations of Sierpiński's original theorem.

11B25, 11B07, 11B99

sierpiński number, arithmetic progression, primitive divisor