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

Let $N$ be a positive integer, $\mathbb{A}$ be a nonempty subset of $\mathbb{Q}$ and $\alpha=\dfrac{\alpha_{1}}{\alpha_{2}}\in $\mathbb{A}$\setminus {0,N}$\mathrm{.}$\alpha$iscalledan$N$-Korseltbase(equivalently$N$issaidan$\alpha$-Korseltnumber)if$\alpha_{2}p-\alpha_{1}$isadivisorof$\alpha_{2}N-\alpha_{1}$foreveryprime$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 $q<2p$.