تقرير
The set of $k$-units modulo $n$
العنوان: | The set of $k$-units modulo $n$ |
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المؤلفون: | Castillo, John H., Mainguez, Jhony Fernando Caranguay |
المصدر: | Involve 15 (2022) 367-378 |
سنة النشر: | 2017 |
المجموعة: | Mathematics |
مصطلحات موضوعية: | Mathematics - Number Theory, 11A05, 11A07, 11A15, 16U60 |
الوصف: | Let $R$ be a ring with identity, $\mathcal{U}(R)$ the group of units of $R$ and $k$ a positive integer. We say that $a\in \mathcal{U}(R)$ is $k$-unit if $a^k=1$. Particularly, if the ring $R$ is $\mathbb{Z}_n$, for a positive integer $n$, we will say that $a$ is a $k$-unit modulo $n$. We denote with $\mathcal{U}_k(n)$ the set of $k$-units modulo $n$. By $\text{du}_k(n)$ we represent the number of $k$-units modulo $n$ and with $\text{rdu}_k(n)=\frac{\phi(n)}{\text{du}_k(n)}$ the ratio of $k$-units modulo $n$, where $\phi$ is the Euler phi function. Recently, S. K. Chebolu proved that the solutions of the equation $\text{rdu}_2(n)=1$ are the divisors of $24$. The main result of this work, is that for a given $k$, we find the positive integers $n$ such that $\text{rdu}_k(n)=1$. Finally, we give some connections of this equation with Carmichael's numbers and two of its generalizations: Kn\"odel numbers and generalized Carmichael numbers. |
نوع الوثيقة: | Working Paper |
DOI: | 10.2140/involve.2022.15.367 |
URL الوصول: | http://arxiv.org/abs/1708.06812 |
رقم الأكسشن: | edsarx.1708.06812 |
قاعدة البيانات: | arXiv |
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