$\mathbb{Q}$-Korselt numbers

Authors : Nejib GHANMI

Pages : 2752-2762

View : 23 | Download : 3

Publication Date : 0000-00-00

Article Type : Research Paper

Abstract :Let $\alpha=\dfrac{\alpha_{1}}{\alpha_{2}}\in \mathbb{Q}\setminus \{0\}$; a positive integer $N$ is said to be an \emph{$\alpha$-Korselt number} insert ignore into journalissuearticles values(\emph{$K_{\alpha}$-number}, for short); if $N\neq \alpha$ and $\alpha_{2}p-\alpha_{1}$ divides $\alpha_{2}N-\alpha_{1}$ for every prime divisor $p$ of $N$. In this paper we prove that for each squarefree composite number $N$ there exist finitely many rational numbers $\alpha$ such that $N$ is a $K_{\alpha}$-number and if $\alpha\leq1$ then $N$ has at least three prime factors. Moreover, we prove that for each $\alpha\in \mathbb{Q}\setminus \{0\}$ there exist only finitely many squarefree composite numbers $N$ with two prime factors such that $N$ is a $K_{\alpha}$-number.
Keywords : Prime number, Carmichael number, Korselt number, squarefree composite number, Korselt set, Korselt weight