Korselt Rational Bases and Sets

TL;DR

This paper investigates Korselt bases for positive integers, proving finiteness of the set of rational Korselt bases for squarefree composite numbers and establishing bounds and conditions for these bases.

Contribution

It introduces the concept of Korselt bases in rational numbers and proves the finiteness of these sets for squarefree composite numbers, providing bounds and conditions.

Findings

The set of rational Korselt bases for squarefree composite numbers is finite.

Bounds for Korselt bases in rational numbers are established.

Necessary and sufficient conditions for the bounds to be reached are provided.

Abstract

For a positive integer $N$ and $\mathbb{A}$ a subset of $\mathbb{Q}$, let $\mathbb{A}$-$\mathcal{KS}(N)$ denote the set of $\alpha=\dfrac{\alpha_{1}}{\alpha_{2}}\in \mathbb{A}\setminus \{0,N\}$ verifying $\alpha_{2}p-\alpha_{1}$ divides $\alpha_{2}N-\alpha_{1}$ for every prime divisor $p$ of $N$. The set $\mathbb{A}$-$\mathcal{KS}(N)$ is called the set of Korselt bases of $N$ in $\mathbb{A}$ or simply the $\mathbb{A}$-Korselt set of $N$. In this paper, we prove that for each squarefree composite number $N\in\mathbb{N}\setminus\{0,1\}$ the $\mathbb{Q}$-Korselt set of $N$ is finite where we provide an upper and lower bounds for each Korselt base of $N$ in $\mathbb{Q}$. Furthermore, we give a necessary and a sufficient condition for the upper bound of a Korselt base to be reached.