# For any n and some prime p there is an elemnet in Zp* of order n [closed]

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...

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## closed as too localized by Mark Sapir, Felipe Voloch, Peter Mueller, Charles, Derek Holt Jun 19 '13 at 21:40

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Is this a homework problem? – Mark Sapir Jun 19 '13 at 18:16
Start by investigating the case that n is prime, and look at factors of a^n - 1. When you handle that, try n a prime power. Gerhard "Eventually, Look Up Zsigmondy's Theorem" Paseman, 2013.06.19 – Gerhard Paseman Jun 19 '13 at 18:18
Voted to close. – Mark Sapir Jun 19 '13 at 18:31
Not research level, but also probably not homework. Please do not delete; there is information in the answer and in the comments which could be useful for mathematicians. – Todd Trimble Jun 24 '13 at 0:49

$Z_p^*$ is of order $p-1$ 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