Korselt Rational Bases of Prime Powers

TL;DR

This paper characterizes Korselt bases for prime power numbers, explicitly determines the Korselt set over rationals, and explores the relationship between rational and integer Korselt bases, revealing conditions for their existence.

Contribution

It provides explicit descriptions of Korselt bases for prime powers over rationals and establishes connections with integer Korselt bases, including conditions for their non-existence in certain intervals.

Findings

The set of Korselt bases over rationals for prime powers is explicitly determined.

The Korselt set over the interval [-1,1) is empty for prime squares.

Every nonzero rational is a Korselt base for infinitely many prime power numbers.

Abstract

Let $N$ be a positive integer, $\mathbb{A}$ be a subset of $\mathbb{Q}$ and $\alpha=\dfrac{\alpha_{1}}{\alpha_{2}}\in \mathbb{A}\setminus \{0,N\}$. $N$ 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 $N$. By the Korselt set of $N$ over $\mathbb{A}$, we mean the set $\mathbb{A}$-$\mathcal{KS}(N)$ of all $\alpha\in \mathbb{A}\setminus \{0,N\}$ such that $N$ is an $\alpha$-Korselt number. In this paper we determine explicitly for a given prime number $q$ and an integer $l\in \mathbb{N}\setminus \{0,1\}$, the set $\mathbb{Q}$-$\mathcal{KS}(q^{l})$ and we establish some connections between the $q^{l}$-Korselt bases in $\mathbb{Q}$ and others in $\mathbb{Z}$. The case of $\mathbb{A}=\mathbb{Q}\cap[-1,1[$ is studied where we prove that $(\mathbb{Q}\cap[-1,1[)$-$\mathcal{KS}(q^{l})$ is empty if and only if $l=2$. Moreover, we show that each nonzero rational $\alpha$ is an $N$-Korselt base for infinitely many numbers $N=q^{l}$ where $q$ is a prime number and $l\in\mathbb{N}$.