Generalized Sierpiński Numbers

Abstract

A Sierpiński number is a positive odd integer $k$ such that $k\cdot 2^n+1$ is composite for all positive integers $n$. Fix an integer $A$ with $2\le A$. We show that there exists a positive odd integer $k$ such that $k\cdot a^n+1$ is composite for all integers $a\in [2,A]$ and all $n\in {\mathbb{Z}}^{+}$.