An extension of Wilson's Theorem

TL;DR

This paper extends Wilson's Theorem by establishing a new combinatorial congruence involving products of subsets, valid for large enough n, and characterizes when n is a multiple of a prime based on this extension.

Contribution

It introduces a combinatorial generalization of Wilson's Theorem, linking subset product sums to the primality of n for large n.

Findings

The congruence holds if and only if n=(c+1)p with p prime.

The result generalizes Wilson's Theorem to a broader combinatorial context.

Provides conditions on n for the congruence to be valid, involving prime factors.

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 $c=0$.