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> **Possible Duplicate:**  
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> [Multiplicative order of zeros of the Artin-Schreier Polynomial](https://mathoverflow.net/questions/46133/multiplicative-order-of-zeros-of-the-artin-schreier-polynomial) 
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I will be grateful for any reference to some literature on the following question (to the best of my knowledge the answer is not known):


Is multiplicative order of roots of $x^p - x - 1$ equal to $\frac{p^p-1}{p-1}$ over $GF(p)$?


Brief Note:
$P(x) = x^p - x - 1$ is irreducible. (Proof: Observe that composition with $x+1$ does not change $P$, so if polynomial Q(x) is a factor than Q(x+a) is also a factor.) Norm($x$) = 1, so the maximum possible order for $x$ is $\frac{p^p-1}{p-1}$.



Finite fields: Is multiplicative order of $x^p - x - 1$ equal to $\frac{p^p-1}{p-1}$ over GF(p)?
see papers of mine (on Bell numbers) with coauthors and papers of sam wagstaff.


