Backward Touchard congruence

TL;DR

This paper explores divisibility properties of Bell numbers, extending classical congruences to higher powers of primes and relating them to derangement numbers and the Sun-Zagier congruence.

Contribution

It introduces new divisibility relations for Bell numbers involving higher powers of primes and connects these to derangement numbers and the Sun-Zagier congruence.

Findings

Relation between Bell numbers and derangement numbers established.

Extended divisibility properties involving higher powers of primes.

Results on the periodicity of Bell numbers modulo p.

Abstract

The celebrated Touchard congruence states that $B_{n+p}\=B_n+B_{n+1}$ modulo $p$, where $p$ is a prime number and $B_n$ denotes the Bell number. In this paper we study divisibility properties of $B_{n-p}$ and their generalizations involving higher powers of $p$ as well as the $r$-Bell numbers. In particular, we show a closely relation of the considered problem to the Sun-Zagier congruence, which is additionally improved by deriving \mbox{a new} relation between $r$-Bell and derangement numbers. Finally, we conclude some results on the period of the Bell numbers modulo $p$.