The $\mathbb{Q}$-Korselt Set of $\mathrm{pq}$

TL;DR

This paper investigates the set of rational Korselt bases for the product of two primes, providing a detailed characterization of these bases and completing the understanding of Korselt sets over integers for certain prime pairs.

Contribution

It offers a detailed description of the $Q$-Korselt bases for $pq$ and completes the characterization of the Korselt set over integers when $q<2p$, advancing number theory knowledge.

Findings

Explicit description of $Q$-$KS(pq)$ provided.

Complete characterization of $Z$-$KS(pq)$ for $q<2p$.

Advances understanding of Korselt bases in number theory.

Abstract

Let $N$ be a positive integer, $\mathbb{A}$ be a nonempty subset of $\mathbb{Q}$ and $\alpha=\dfrac{\alpha_{1}}{\alpha_{2}}\in \mathbb{A}\setminus \{0,N\}$. $\alpha$ is called an $N$-Korselt base (equivalently $N$ is said an $\alpha$-Korselt number) if $\alpha_{2}p-\alpha_{1}$ is a divisor of $\alpha_{2}N-\alpha_{1}$ for every prime $p$ dividing $N$. The set of all Korselt bases of $N$ in $\mathbb{A}$ is called the $\mathbb{A}$-Korselt set of $N$ and is simply denoted by $\mathbb{A}$-$\mathcal{KS}(N)$. Let $p$ and $q$ be two distinct prime numbers. In this paper, we study the $\mathbb{Q}$-Korselt bases of $pq$, where we give in detail how to provide $\mathbb{Q}$-$\mathcal{KS}(pq)$. Consequently, we finish the incomplete characterization of the Korselt set of $pq$ over $\mathbb{Z}$ given in [4], by supplying the set $\mathbb{Z}$-$\mathcal{KS}(pq)$ when $q <2p$.