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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?

Anononym
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2 Answers2

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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 $k> 2$.

André Nicolas
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  • Dickson says: Jacobi proved that, if $n$ is an odd prime, any primitive root of $n^2$ is a primitive root of any higher power of $n$. That gives us most of (1). I'm going to have to take a closer look at that chapter in Dickson to see if the rest is there. Thanks for this. – Anononym Jan 13 '12 at 07:31
  • @Anononym: Yes, I only saw the prime power part in Dickson. The fact that it is also a primitive root of $p$ is barely worth noting, since it is clear that we can always go down. – André Nicolas Jan 13 '12 at 07:44
  • Dear André: Your answer looks convincing to me. I think $(2)$ is an almost trivial consequence of $(1)$. +1! – Pierre-Yves Gaillard Jan 13 '12 at 08:06
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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.

Community
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Pierre-Yves Gaillard
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  • Thanks for the response, but I'm not looking for a proof of those two facts about primitive roots. I'm looking for their origin. Specifically, I'd like to know who first proved these results. – Anononym Jan 13 '12 at 06:24
  • Dear @Anononym: Sorry, I didn't read your question carefully enough. It's a good question: +1. I'll still leave my "answer" because, even if it doesn't answer the question, it adds a tiny bit to it (at least I hope so) by providing explicit references. – Pierre-Yves Gaillard Jan 13 '12 at 06:47