How can I prove, that for any positive integer $n>0$ there is a prime $p$, such that the multiplicative group of the residue ring $Z_p^*$ contains an element $a$ of order $n$? No ideas at all...
closed as too localized by Mark Sapir, Felipe Voloch, Peter Mueller, Charles, Derek Holt Jun 19 '13 at 21:40This question is unlikely to help any future visitors; it is only relevant to a small geographic area, a specific moment in time, or an extraordinarily narrow situation that is not generally applicable to the worldwide audience of the internet. For help making this question more broadly applicable, visit the help center.If this question can be reworded to fit the rules in the help center, please edit the question. 


$Z_p^*$ is of order $p1$ so what you are really asking is for a prime in the arithmetic progression $n+1, 2n+1, 3n+1, \ldots$. This is true by Dirichlet's theorem, see http://en.wikipedia.org/wiki/Dirichlet%27s_theorem_on_arithmetic_progressions 

