Numbers with empty rational Korselt sets

Year 2022, Volume: 51 Issue: 1, 83 - 94, 14.02.2022

Nejib Ghanmi

https://doi.org/10.15672/hujms.761213

Abstract

Let $N$ be a positive integer, and $\alpha=\dfrac{\alpha_{1}}{\alpha_{2}}\in \mathbb{Q}\setminus \{0,N\}$ with $\gcd(\alpha_{1}, \alpha_{2})=1$. $N$ 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 $N$. The set of $N$-Korselt bases in $\mathbb{Q}$ is denoted by $\mathbb{Q}$-$\mathcal{KS}(N)$ and called the set of rational Korselt bases of $N$.

In this paper rational Korselt bases are deeply studied, where we give in details their belonging sets and their forms in some cases. This allows us to deduce that for each integer $n\geq 3$, there exist infinitely many squarefree composite numbers $N$ with $n$ prime factors and empty rational Korselt sets.

Keywords

prime number, squarefree composite number, Korselt number, Korselt set, Korselt base, Carmichael number

References

• [1] K. Bouallegue, O. Echi and R. Pinch, Korselt Numbers and Sets, Int. J. Number Theory 6, 257-269, 2010.
• [2] O. Echi and N. Ghanmi, The Korselt Set of pq, Int. J. Number Theory, 8 (2), 299-309, 2012.
• [3] N. Ghanmi, $\mathbb{Q}$-Korselt Numbers, Turkish J. Math. 42, 2752-2762, 2018.
• [4] N. Ghanmi, Korselt Rationel Bases of Prime Powers, Studia Sci. Math. Hungar. 56 (4), 388-403, 2019.
• [5] N. Ghanmi, The $\mathbb{Q}$-Korselt Set of pq, Period. Math. Hungar. 81 (2), 174-193, 2020.
• [6] N. Ghanmi and I. Al-Rassasi, On Williams Numbers With Three Prime Factors, Miss. J. Math. Sc. 25 (2), 134-152, 2013.
• [7] N. Ghanmi, O. Echi and I. Al-Rassasi, The Korselt Set of a Squarefree Composite Number, Math. Rep. Cand. Aca. Sc. 35 (1), 1-15, 2013.
• [8] A. Korselt, Problème chinois, L’intermediaire des Mathématiciens 6, 142–143, 1899.