Numbers with empty rational Korselt sets

Abstract

Let $N$ be a positive integer, and $\alpha=\dfrac{\alpha_{1}}{\alpha_{2}}\in \mathbb{Q}\setminus \{0,N\}$ with $\gcd(\alpha_{1}, \alpha_{2})=1$. $N$ is called an $\alpha$-Korselt number, equivalently $\alpha$ is said an $N$-Korselt base, if $\alpha_{2}p-\alpha_{1}$ divides $\alpha_{2}N-\alpha_{1}$ for every prime divisor $p$ of $N$. The set of $N$-Korselt bases in $\mathbb{Q}$ is denoted by $\mathbb{Q}$-$\mathcal{KS}(N)$ and called the set of rational Korselt bases of $N$.In this paper rational Korselt bases are deeply studied, where we give in details their belonging sets and their forms in some cases. This allows us to deduce that for each integer $n\geq 3$, there exist infinitely many squarefree composite numbers $N$ with $n$ prime factors and empty rational Korselt sets.