Connections on the Rational Korselt Set of pq

Abstract

For a positive integer $N$ and $\mathbb{A}$ a subset of $\mathbb{Q}$, let $\mathbb{A}$-$\mathcal{K}\mathcal{S}(N)$ denote the set of $\alpha =\frac{{\alpha }_1}{{\alpha }_2}\in \mathbb{A}\setminus \{0,N\}$ verifying ${\alpha }_{2}r-{\alpha }_1$ divides ${\alpha }_{2}N-{\alpha }_1$ for every prime divisor $r$ of $N$. The set $\mathbb{A}$-$\mathcal{K}\mathcal{S}(N)$ 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 other(s) in $\mathbb{Q}$-$\mathcal{K}\mathcal{S}(pq)$. More precisely, we will prove that if $(\mathbb{Q}\setminus \mathbb{Z})$-$\mathcal{K}\mathcal{S}(pq)=\varnothing$ then $\mathbb{Z}$-$\mathcal{K}\mathcal{S}(pq)=\{q+p-1\}$.