Let p be a prime number. It is known that the order o(r) of a root r of the irreducible polynomial $x^p-x-l$ 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,...,g(p)} that do not contain o(r).