Generalized Sierpiński numbers
Volume 174 / 2023
Colloquium Mathematicum 174 (2023), 191-201
MSC: Primary 11A07; Secondary 11B25, 11N13.
DOI: 10.4064/cm9156-9-2023
Published online: 17 November 2023
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 $n \in \mathbb {Z}^+$.
