This question was asked on NMBRTHRY by Kurt Foster:


If $p$ is a prime number and $\mathbb{F}\_p$ the field of $p$ elements, the zeroes of the Artin-Schreier polynomial


$x^p - x - 1 \in \mathbb{F}\_p[x]$


obviously have multiplicative order dividing $1 + p + p^2 + \dots + p^{p-1} = (p^p - 1)/(p-1)$ (express the norm as the product of the compositional powers of the Frobenius map)


Once upon a time, long long ago, I read that it had been conjectured (by Shafarevich IIRC) that this is the *exact* multiplicative order for every prime $p$. Can anyone supply a reference?



Multiplicative order of zeros of the Artin-Schreier Polynomial
I've never seen it ascribed to Shafarevich, but it is an old question. As a question, it equivalent to determining the period mod p of the sequence of Bell numbers discussed, e.g. in:


Levine, Jack; Dalton, R. E.
Minimum periods, modulo p, of first-order Bell exponential integers.
Math. Comp. 16 1962 416–423. 


But they refer to even older papers. Any conjecture is wishful thinking since we can't get past $p=29$ or so with current technology. (Edit: As Kevin points out in the comments, I am seriously underestimating current technology, so this comment applies only to last century.)


Incidentally, I proved that the order is at least $2^{2.54p}$ in JTNB 16 (2004) 233-239.


