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Let $a=\text{ord}(x,m) : $ $a$ is the minimum value for which $x^a \equiv 1\pmod{m} $.

By inspection it appears that $$\text{ord}(x,p^b) = p^{b-1} \cdot \text{ord}(x,p)$$ where $x,p,b,$ belong to $\mathbb{Z}^+$, $x>1$, $p$ is odd prime.

  1. Is this assertion actually true?
  2. If so, what is a simple proof of it?

I may have a "proof" of my own, but even if it is correct (which I doubt) it seems much too complicated.

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    Note it is obviously not true if $x=1$, for example. What is true is that $o(x,p^b)=p^k o(x,p)$ for some $0\leq k < b$.2012-06-10

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