Connections on the rational Korselt set of $pq$

Authors : Nejib GHANMİ

Pages : 135-143

Doi:10.15672/hujms.659265

View : 21 | Download : 10

Publication Date : 2021-02-04

Article Type : Research Paper

Abstract :For a positive integer $N$ and $\mathbb{A}$, a subset of $\mathbb{Q}$, let $\mathbb{A}$-$\mathcal{KS}insert ignore into journalissuearticles values(N);$ 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 $\mathbb{A}$-$\mathcal{KS}insert ignore into journalissuearticles values(N);$ 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 $\mathbb{Q}$-$\mathcal{KS}insert ignore into journalissuearticles values(pq);$. More precisely, we prove that if $insert ignore into journalissuearticles values(\mathbb{Q\setminus}\mathbb{Z});$-$\mathcal{KS}insert ignore into journalissuearticles values(pq);=\emptyset$, then $\mathbb{Z}$-$\mathcal{KS}insert ignore into journalissuearticles values(pq);=\{ q+p-1\}$.
Keywords : prime number, Carmichael number, squarefree composite number, Korselt base, Korselt number, Korselt set