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An extension of Wilson's Theorem

Konstantinos Gaitanas

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 $c=0$.

An extension of Wilson's Theorem

Abstract

Let be the multiset containing the products of -subsets of . We show that if , 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 , where is prime. This provides a combinatorial extension of Wilson's Theorem, which is the special case where .
Paper Structure (4 sections, 6 theorems, 45 equations)

This paper contains 4 sections, 6 theorems, 45 equations.

Table of Contents

  1. Introduction
  2. Notation and preliminaries
  3. Some useful Lemmas
  4. Main results

Key Result

Theorem 1.1

A positive integer $n>1$ is prime if and only if

Theorems & Definitions (12)

  • Theorem 1.1
  • proof
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • Theorem 4.1
  • proof
  • Theorem 4.2
  • proof
  • ...and 2 more