Bell Numbers Modulo a Prime Number, Traces and Trinomials

Luis H. Gallardo, Olivier Rahavandrainy

Abstract


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 $p$.


Keywords


Finite Fields; Trinomials; Artin-Schreier extension; Bell numbers; Stirling numbers; Kurepa's Conjecture

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