Connections on the rational Korselt set of $pq$

Abstract

For a positive integer $N$ and $\mathbb{A}$, a subset of $\mathbb{Q}$, let $\mathbb{A}$-$\mathcal{KS}(N)$ denote the set of $\alpha=\dfrac{\alpha_{1}}{\alpha_{2}}\in \mathbb{A\setminus} \{0,N\}$, where $\alpha_{2}r-\alpha_{1}$ divides $\alpha_{2}N-\alpha_{1}$ for every prime divisor $r$ of $N$. The set $\mathbb{A}$-$\mathcal{KS}(N)$ is called the set of $N$-Korselt bases in $\mathbb{A}$. Let $p, q$ be two distinct prime numbers. In this paper, we prove that each $pq$-Korselt base in $\mathbb{Z\setminus}\{ q+p-1\}$ generates at least one other in $\mathbb{Q}$-$\mathcal{KS}(pq)$. More precisely, we prove that if $(\mathbb{Q\setminus}\mathbb{Z})$-$\mathcal{KS}(pq)=\emptyset$, then $\mathbb{Z}$-$\mathcal{KS}(pq)=\{ q+p-1\}$.