Bell Numbers Modulo a Prime Number, Traces and Trinomials

  • Luis H. Gallardo
  • Olivier Rahavandrainy
Keywords: Finite Fields, Trinomials, Artin-Schreier extension, Bell numbers, Stirling numbers, Kurepa's Conjecture


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 $p$.

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