An extension of Wilson's Theorem
Fuente:
arXiv
Saved in:
| Main Author: | |
|---|---|
| Format: | Preprint |
| Published: |
2023
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866916160056852480 |
|---|---|
| author | Gaitanas, Konstantinos |
| author_facet | Gaitanas, Konstantinos |
| contents | Let $\mathcal{N}[k]$ be the multiset containing the $\binom{n-1}{k}$ products of $k$-subsets of $\{1,\ldots, n-1\}$. We show that if $n\geq (2c+3)^2$, then \begin{gather*}\left((-1)^c+\sum_{M\in \mathcal{N}[n-1-c]}M\right)\cdot(c+1)\equiv 0\pmod{n},\end{gather*} if and only if $n=(c+1)p$, where $p$ is prime. This provides a combinatorial extension of Wilson's Theorem, which is the special case where $c=0$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_09644 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | An extension of Wilson's Theorem Gaitanas, Konstantinos General Mathematics 11A07, 11N80 Let $\mathcal{N}[k]$ be the multiset containing the $\binom{n-1}{k}$ products of $k$-subsets of $\{1,\ldots, n-1\}$. We show that if $n\geq (2c+3)^2$, then \begin{gather*}\left((-1)^c+\sum_{M\in \mathcal{N}[n-1-c]}M\right)\cdot(c+1)\equiv 0\pmod{n},\end{gather*} if and only if $n=(c+1)p$, where $p$ is prime. This provides a combinatorial extension of Wilson's Theorem, which is the special case where $c=0$. |
| title | An extension of Wilson's Theorem |
| topic | General Mathematics 11A07, 11N80 |
| url | https://arxiv.org/abs/2403.09644 |