Numbers with empty rational Korselt sets

Authors : Nejib GHANMİ

Pages : 83-94

Doi:10.15672/hujms.761213

View : 17 | Download : 7

Publication Date : 2022-02-14

Article Type : Research Paper

Abstract :Let $N$ be a positive integer, and $\alpha=\dfrac{\alpha_{1}}{\alpha_{2}}\in \mathbb{Q}\setminus \{0,N\}$ with $\gcdinsert ignore into journalissuearticles values(\alpha_{1}, \alpha_{2});=1$. $N$ 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 $\mathbb{Q}$-$\mathcal{KS}insert ignore into journalissuearticles values(N);$ 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.
Keywords : prime number, squarefree composite number, Korselt number, Korselt set, Korselt base, Carmichael number