Bull. Korean Math. Soc. 2008 Vol. 45, No. 1, 143-155 Published online March 1, 2008

Mireille Car, Luis H. Gallardo, Olivier Rahavandrainy, and Leonid N. Vaserstein University Aix-Marseille III, University of Brest, University of Brest, and The Pennnsylvania State University

Abstract : Let $p$ be a prime number. It is known that the order $o(r)$ of a root $r$ of the irreducible polynomial $x^p-x-1$ over $\mathbb F_p$ divides $g(p)=\frac{p^p-1}{p-1}.$ Samuel Wagstaff recently conjectured that $o(r)=g(p)$ for any prime $p.$ The main object of the paper is to give some subsets $S$ of $\{1,\ldots,g(p)\}$ that do not contain $o(r).$

Keywords : Bell numbers modulo a prime, extension of prime degree p of $\mathbb F_p$