Korselt Rational Bases and Sets

Abstract

For a positive integer $N$ and $\mathbb{A}$ a subset of $\mathbb{Q}$, let $\mathbb{A}$-$\mathcal{K}\mathcal{S}(N)$ denote the set of $\alpha =\frac{{\alpha }_1}{{\alpha }_2}\in \mathbb{A}\setminus \{0,N\}$ verifying ${\alpha }_{2}p-{\alpha }_1$ divides ${\alpha }_{2}N-{\alpha }_1$ for every prime divisor $p$ of $N$. The set $\mathbb{A}$-$\mathcal{K}\mathcal{S}(N)$ 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.