Given a prime number $p$, we deduce from a formula of Barsky and Benzaghou and from a result of Coulter and Henderson on trinomials over finite fields, a simple necessary and sufficient condition $\beta(n) =k\beta(0)$ in $\mathbb{F}_{p^p}$ in order to resolve the congruence $B(n) \equiv k \pmod{p}$, where $B(n)$ is the $n$-th Bell number, and $k$ is any fixed integer. Several applications of the formula and of the condition are included, in particular we give equivalent forms of the conjecture of Kurepa that $B(p-1)$ is $\neq 1$ modulo $p$.
@article{10_37236_3532,
author = {Luis H. Gallardo and Olivier Rahavandrainy},
title = {Bell numbers modulo a prime number, traces and trinomials},
journal = {The electronic journal of combinatorics},
year = {2014},
volume = {21},
number = {4},
doi = {10.37236/3532},
zbl = {1353.11046},
url = {http://geodesic.mathdoc.fr/articles/10.37236/3532/}
}
TY - JOUR
AU - Luis H. Gallardo
AU - Olivier Rahavandrainy
TI - Bell numbers modulo a prime number, traces and trinomials
JO - The electronic journal of combinatorics
PY - 2014
VL - 21
IS - 4
UR - http://geodesic.mathdoc.fr/articles/10.37236/3532/
DO - 10.37236/3532
ID - 10_37236_3532
ER -
%0 Journal Article
%A Luis H. Gallardo
%A Olivier Rahavandrainy
%T Bell numbers modulo a prime number, traces and trinomials
%J The electronic journal of combinatorics
%D 2014
%V 21
%N 4
%U http://geodesic.mathdoc.fr/articles/10.37236/3532/
%R 10.37236/3532
%F 10_37236_3532
Luis H. Gallardo; Olivier Rahavandrainy. Bell numbers modulo a prime number, traces and trinomials. The electronic journal of combinatorics, Tome 21 (2014) no. 4. doi: 10.37236/3532
