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The following facts about primitive roots of an odd prime seem to be well known. For example, they both appear as exercises in Burton's Elementary Number Theory.

Let $p$ be an odd prime. Then:
(1) Any primitive root of $p^2$ is a primitive root of $p^k$ for every positive integer $k$.
(2) Any odd primitive root of $p^k$ is a primitive root of $2p^k$.

I thought these facts might be from Gauss' Disquisitiones Arithmeticae, but I couldn't find them there. Does anyone know the origin of these two facts?

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For what it's worth, Dickson, in his History of the Theory of Numbers, Chapter VII, page $186$, credits Jacobi (Canon Arithmeticus, $1839$) with the result that if $p$ is an odd prime, a primitive root of $p^2$ is a primitive root of $p^k$ for all $k> 2$.

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    Dear André: Your answer looks convincing to me. I think $(2)$ is an almost trivial consequence of $(1)$. +1!2012-01-13
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See page 23 and page 24 of Alan Baker's A Concise Introduction to the Theory of Numbers, or see this answer.

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    Dear @A$n$ononym: Sorry, I didn't read your question carefully enou$g$h. It's a good question: +1. I'll still leave my "answer" because, even i$f$ it doesn't *answer* the question, it adds a tiny bit to it (at least I hope so) by providin$g$ explicit re$f$erences.2012-01-13