Abstract: The present paper is concerned with the order o(α_p) of a root α_p of the polynomial x^p-x-1 over a prime field F_p. We propose a conjecture that o(α_p)=m_p:=(p^p-1)/(p-1) for any prime p. Then we introduce two methods to verify the conjecture in certain cases.

Key words: finite field, order of a root, polynomial, finite field, order of a root