TITLE: Explicit formulas for primitive roots?
QUESTION [5 upvotes]: For a Fermat prime or an "upper" Sophie Germain prime a primitive root is explicitly known. Are there further results when the factorization of p-1 is known? Is it unlikely that we ever get explicit formulas for larger classes of primes, not only for such quite special cases?

REPLY [2 votes]: Yes, there is an explicit formula, given by Horst Bergmann in "Über eine Formel für primitive
Kongruenzwurzeln" (Elemente der Mathematik, 1983). 
For every odd prime p:
$w = \sum_{r = 2}^{p-1} r P(r) \prod_{s = 1}^{r-1}(1 - P(s))$, with $P(t) = \prod_{m = 1}^{p-2}(t^m - 1)$
It works, and although it isn't very useful for computation, I find it still very strange that this amazing result seems to be nearly forgotten (e.g. isn't mentioned in Wikipedia or Mathworld)