An elementary (number theory) proof of Touchard's congruence

TL;DR

This paper presents an elementary number theory proof of Touchard's congruence for Bell numbers, utilizing recursive formulas to establish the congruence without advanced methods.

Contribution

It provides a new, elementary proof of Touchard's congruence based on recursive properties of Bell numbers, simplifying previous approaches.

Findings

Proves Touchard's congruence using elementary number theory.

Introduces a recursive approach to Bell numbers for congruence proof.

Simplifies understanding of Bell number properties and their congruences.

Abstract

Let B_n denote the nth Bell number. We use well-known recursive expressions for B_n to give a generalizing recursion that can be used to prove Touchard's congruence.