The $\mathbb{Q}$-Korselt Set of $\mathrm{pq}$

Abstract

Let $N$ be a positive integer, $\mathbb{A}$ be a nonempty subset of $\mathbb{Q}$ and $\alpha =\frac{{\alpha }_1}{{\alpha }_2}\in \mathbb{A}\setminus \{0,N\}$. $\alpha$ is called an $N$-Korselt base (equivalently $N$ is said an $\alpha$-Korselt number) if ${\alpha }_{2}p-{\alpha }_1$ is a divisor of ${\alpha }_{2}N-{\alpha }_1$ for every prime $p$ dividing $N$. The set of all Korselt bases of $N$ in $\mathbb{A}$ is called the $\mathbb{A}$-Korselt set of $N$ and is simply denoted by $\mathbb{A}$-$\mathcal{K}\mathcal{S}(N)$. Let $p$ and $q$ be two distinct prime numbers. In this paper, we study the $\mathbb{Q}$-Korselt bases of $pq$, where we give in detail how to provide $\mathbb{Q}$-$\mathcal{K}\mathcal{S}(pq)$. Consequently, we finish the incomplete characterization of the Korselt set of $pq$ over $\mathbb{Z}$ given in [4], by supplying the set $\mathbb{Z}$-$\mathcal{K}\mathcal{S}(pq)$ when $q<2p$.