Artin's Conjecture: There are infinitely many primes $p$ for which 2 is a primitive root, i.e., 2 generates the multiplicative group of ${\mathbb Z}/p{\mathbb Z}$.

[The conjecture][1] is actually a bit more general, but we should at least be able to say what happens with 2! The [OEIS lists][2] the first several such primes.


  [1]: http://en.wikipedia.org/wiki/Artin's_conjecture_on_primitive_roots
  [2]: http://oeis.org/A001122