Korselt Rational Bases and Sets

Abstract

For a positive integer $N$ and $\mathbb{A}$ a subset of $\mathbb{Q}$, let $\mathbb{A}$-$\mathcal{KS}(N)$ denote the set of $\alpha=\dfrac{\alpha_{1}}{\alpha_{2}}\in \mathbb{A}\setminus \{0,N\}$ verifying $\alpha_{2}p-\alpha_{1}$ divides $\alpha_{2}N-\alpha_{1}$ for every prime divisor $p$ of $N$. The set $\mathbb{A}$-$\mathcal{KS}(N)$ is called the set of Korselt bases of $N$ in $\mathbb{A}$ or simply the $\mathbb{A}$-Korselt set of $N$. In this paper, we prove that for each squarefree composite number $N\in\mathbb{N}\setminus\{0,1\}$ the $\mathbb{Q}$-Korselt set of $N$ is finite where we provide an upper and lower bounds for each Korselt base of $N$ in $\mathbb{Q}$. Furthermore, we give a necessary and a sufficient condition for the upper bound of a Korselt base to be reached.