TITLE: Multiplicative order of zeros of the Artin-Schreier Polynomial
QUESTION [12 upvotes]: 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?

REPLY [10 votes]: The current computational status is that this is known for all p < 126 and also for p = 137, 149, 157, 163, 167 and 173. See Peter L. Montgomery, Sangil Nahm and Samuel S. Wagstaff Jr., "The period of the Bell numbers modulo a prime", Math. Comp., 79, 271, July 2010, 1793-1800.  The method used requires a factorization of (p^p-1)/(p-1) which is hard for larger p :-)