An extension of Wilson's Theorem

Konstantinos Gaitanas

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arXiv · 2403.09644

An extension of Wilson's Theorem

Konstantinos Gaitanas

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Abstract

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 $c=0$.

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Konstantinos Gaitanas. 2023-10-14. An extension of Wilson's Theorem. https://arxiv.org/abs/2403.09644

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