Backward Touchard congruence

Abstract

The celebrated Touchard congruence states that Bn+p ≡ Bn + Bn+1 modulo p, where p is a prime number and Bn denotes the Bell number. In this paper we study divisibility properties of Bn−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 a new relation between r-Bell and derangement numbers. Finally, we conclude some results on the period of the Bell numbers modulo p.